Categorical structures for
higher-dimensional universal algebra
Habilitation Thesis
Masaryk University
John Bourke
January 28, 2021
Contents
1 Introduction 2
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Two-dimensional universal algebra . . . . . . . . . . . . . . . 2
1.3 Algebraic weak factorisation systems . . . . . . . . . . . . . . 8
1.4 Skew monoidal categories . . . . . . . . . . . . . . . . . . . . . 12
1.5 Higher groupoids, monads, nerves and theories . . . . . . . . . 16
1.6 Homotopy enriched category theory and higher-dimensional
universal algebra . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 Two-dimensional monadicity 29
3 Algebraic weak factorisation systems I: Accessible AWFS 71
4 Skew structures in 2-category theory and homotopy theory 117
5 Monads and theories 166
6 Adjoint functor theorems for homotopically enriched categories
208
1
Chapter 1
Introduction
1.1 Overview
This thesis gives a snapshot of five different areas in which I have worked to
date, including one paper from each area. In this introduction, we will walk
through the different topics chronologically, and I will tell the idiosyncratic
stories of how these papers came about (which rarely make it to the final
version but often form an important part of the research process). At the
same time, I will emphasise the many connections between the different areas
and point out some of the open questions and future directions to be explored.
The second to fifth of these topics involve categorical structures and settings
that arose in my research on higher dimensional structures: in turn,
algebraic weak factorisation systems, skew monoidal categories, nervous monads
and homotopically enriched category theory. We will see how concrete
examples and questions led us to develop the theory of these categorical
structures, often leading in new directions beyond the original goals. To begin
with, we take a look at a topic that leads naturally to each of the later
ones — namely, two-dimensional universal algebra.
1.2 Two-dimensional universal algebra
Classical universal algebra describes algebraic structures borne by sets —
for instance, monoids, groups, rings, vector spaces and (most of) the usual
structures of algebra. To capture this in general, one has a pair (Ω, E)
consisting of a signature Ω of operations and a set E of equations — then
2
one can consider (Ω, E)-algebras and their homomorphisms. These form a
category (Ω, E)-Alg equipped with a forgetful functor U : (Ω, E)-Alg→ Set
to the category of sets.
There are several categorical approaches to the subject, and perhaps the
best known is the approach via monads T on Set. In this context, one can
speak of a T-algebra — a set X together with a function x : TX → X
satisfying two equations similar to those for a module over a ring — and
their homomorphisms f : (X, x) → (Y, y) which satisfy the commutativity
y ◦Tf = f ◦x. These form a category SetT
, the Eilenberg-Moore category of
algebras, and as before, we have a forgetful functor UT
: SetT
→ Set, which
has a left adjoint.
The appeal of the monad-theoretic approach is partially due to its generality:
monads can be considered on any category, not just Set, and so provide
a kind of generalised universal algebra. It is also due to the fact that monads
arise from adjunctions, the ubiquitous central concept in category theory.
Indeed, given an adjunction
A
U
QQ B
F
ss
with F U we obtain a monad T = UF on B and canonical comparison
K : A → BT
to the category of algebras such that the triangle
A
U 11
K GG BT
UT
~~
B
commutes. The fundamental result here is Beck’s monadicity theorem [3],
which famously gives an easily checkable criterion that characterises those
adjunctions for which the canonical functor
K : A → BT
is an equivalence. In particular, this is the case for the adjunctions of classical
universal algebra so that we obtain an equivalence of categories (Ω, E)-Alg
∼= SetT
over Set.1
Therefore, anything one can prove about categories of the
1
In fact, the categories of universal algebra are precisely those of the form SetT
for T
a filtered colimit preserving (a.k.a. finitary) monad on Set.
3
form SetT
holds for any variety of universal algebra — this is the categorical
approach to universal algebra via monads.
What about two-dimensional universal algebra? Here the objects of interest
are categories with algebraic structure, such as monoidal categories.
In this context there are four kinds of morphism that naturally arise: the
strict monoidal functors F : A → B which preserve the tensor product ⊗
and unit i on the nose; the pseudo (or strong) monoidal functors which involve
coherence isomorphisms Fa ⊗ Fb ∼= F(a ⊗ b) and i ∼= Fi; the lax
monoidal functors which involve morphisms Fa⊗Fb → F(a⊗b) and i → Fi
and the colax ones, in which the orientation of these morphisms is reversed.
With monoidal transformations as 2-cells, these assemble into four distinct
2–categories, which are interconnected and sit over Cat as in the diagram
below left.
MonCats MonCatp
MonCatl
MonCatc
Cat
GG
UU
99
Vs
11
Vp

Vl
ÖÖ
Vc
{{
T-Algs T-Algp
T-Algl
T-Algc
Cat
GG
UU
99
Us
11
Up

Ul
ÖÖ
Uc
{{
One of the themes of two-dimensional universal algebra is that the 2categories
of strict morphisms, like MonCats, are as easily understood as
their 1-dimensional counterparts. But the 2-categories of pseudo, lax and
colax morphisms are less well behaved and exhibit much subtler behaviour.
How about the about the abstract approach? The approach favoured
by 2-category theorists is via 2-monads — just enriched monads — on the
2-category Cat [9]. There are the usual strict algebras2
and four kinds of
morphism — strict, pseudo, lax and colax — much as above. All told, we
obtain a diagram of 2-categories as above, labelled by the kind of morphisms
they contain, with algebra transformations between them in each case.
Comparing the diagrams left and right above we see that a monadicity
theorem in this setting ought to match each 2-category on the left with the
2
There are also weaker kinds but these are less central to the subject than one might
at first guess.
4
corresponding one on the right in a compatible way. More precisely, there
should exist a 2-monad T on Cat and, for each w ∈ {s, p, l, c}, an equivalence
of 2-categories Ew : MonCatw → T-Algw over Cat, as left below
MonCatw T-Algw
Cat
Ew GG
Vw
''
Uw
ÓÓ
and these equivalences should be natural in the various inclusions. Establishing
a monadicity theorem of this kind was the content of my paper Twodimensional
monadicity, included in this collection and published in Advances
in Mathematics [11] in 2014.
How should we approach this problem? The starting point is that the
world of strict morphisms is easily understood: by the enriched version of
Beck’s theorem, there exists a 2-monad T and an equivalence MonCats
T-Algs over Cat. Therefore what needs to be done is to extend from strict
morphisms to each of the different flavours. Two clearly related questions
are
(1) What are the characterising properties of pseudo, lax and colax morphisms
in two-dimensional universal algebra?
(2) Are each of the weaker flavours of morphism uniquely determined by
their relationship with the strict ones?
What I will do now is explain the answers to these questions, in particular
the second of them, from which the main monadicity theorem — Theorem
25 of [11] — easily follows.
The first question was one which had been in the back of my head shortly
after completing my Phd. A lightbulb flashed on reading Boardman and
Vogt’s desiderata of the properties of homotopy invariant morphisms in the
introduction to their classic book [10], namely:
• if f : X → Y is a structured map of structured spaces and g homotopic
to f then g can be structured;
5
• if f : X → Y is a structured map of structured spaces and a homotopy
equivalence, then any homotopy inverse to f can be structured.3
Certainly 2-categorical analogues of these homotopy invariance properties
hold for pseudomorphisms of algebras, and can be expressed as lifting properties
of the forgetful 2-functor Up : T-Algp → Cat. For lax morphisms, the
first property holds much as before whilst I realised that the appropriate version
of the second property above should be related to the following fragment
of Kelly’s doctrinal adjunction [40]4
:
• if f : (X, x) → (Y, x) is a strict morphism of algebras whose underlying
functor f has a right adjoint, then the right adjoint can be given the
structure of a lax morphism (in a unique way for which the adjunction
lifts to T-Algl).
The problem with this property is that one needs to quantify over the strict
morphisms — so that it is not a property of the forgetful 2-functor Ul :
T-Algl → Cat.
The solution was to pass from 2-categories and 2-functors to the recently
introduced F-categories and F-functors of Lack and Shulman [43]. These
used enriched category theory to formalise the idea, already appreciated in
[9], that to study weak morphisms it is useful to consider them in tandem
with strict ones. Now an F-enriched category is a 2-category with two kinds
of 1-cell, one a subclass of the other so that, for instance, there is an Fcategory
T-Algl of algebras, strict and lax morphisms. As F-categories also
include 2-categories in a trivial way, there is moreover a forgetful F-functor
Ul : T-Algl → Cat
and we could now express doctrinal adjunction as a lifting property of Ul.
Ultimately, rather than being about lifting general adjunctions, the necessary
fragment of doctrinal adjunction turned out to concern only those
adjunctions with identity counit. Such adjunctions, or their left adjoints f,
are often called lalis.5
The reason that these are sufficient is that each lax
3
Boardman and Vogt had a third condition regarding homotopy invariance for objects
not relevant here.
4
Doctrinal adjunction, in its most general form, actually states a relationship between
colax and lax morphisms.
5
The paper actually uses the terminology l-reflection rather than lali.
6
morphism f : (X, x) (Y, y) could be represented as a span
(Cf , cf )(X, x) (Y, y)
pf
oo
qf
GG (1.2.1)
of strict morphisms, where the left leg pf : Cf → X is the left adjoint of a lali
in Cat, from which the lax morphism f could be recovered using doctrinal
adjunction.
This led to the notion of an l-doctrinal F-functor: one, such as Ul :
T-Algl → Cat, which had the doctrinal adjunction property for lalis as well
as a couple of other related properties. Considering the evident factorisation
T-Algs
Us
RR
incl GG T-Algl
Ul
GG Cat
of the forgetful 2-functor Us : T-Algs → Cat, we can now express the key
point of the paper: namely, the factorisation is an orthogonal decomposition
(in the sense of factorisation systems) where the right class consisted of the
l-doctrinal F-functors. To see that the left leg belongs to the left class used
the covering span of (1.2.1).
A generalisation of this result, valid not just for 2-monads, appears as
Corollary 22 of the paper and the main result on monadicity, Theorem 25,
is a straightforward consequence. The monadicity theorem has been used
to good effect by Gurski, Johnson and Osorno [34] to understand the lax
morphisms of n-fold monoidal categories in their work on spectra.
The following table is intended to express the idea, hinted at for lax
morphisms above, that each flavour of weak morphism is parametrised by a
particular class of adjunction or equivalence.
Two-dimensional universal algebra
lax morphisms lalis
pseudomorphisms surjective equivalences
colax morphisms ralis
Table 1.1: Parametrising classes of maps
We will return to this table in the following section on algebraic weak
factorisation systems.
7
1.3 Algebraic weak factorisation systems
Going back to the early days of category theory [48] there have been various
notions of factorisation system around. A factorisation system involves two
classes of maps E and M which are, first of all, orthogonal in the sense that
given a solid square as on the outside of
X
e∈E

GG A
m∈M

Y GG
∃!
VV
B
there exists a unique diagonal filler as depicted, and such that each morphism
factors as an E followed by an M. This captures the relationship
between, for instance, surjective and injective homomorphisms in algebraic
categories. Weak orthogonality, in which one requires only the existence but
not uniqueness of the diagonal filler, and the associated weak factorisation
systems are much more important in homotopy theory — capturing, as they
do, the relationship between the cofibrations and trivial fibrations in a model
category [54].
The uniqueness condition ensures that factorisations are functorial. However,
in the case of weak orthogonality functoriality is not guaranteed. Despite
this, Quillen’s small object argument [54] ensures that most weak factorisation
sytems can be equipped with functorial factorisations. Motivated
by this gap between theory and practice, Grandis and Tholen introduced natural
weak factorisation systems in 2006 [31]. Their definition was tweaked
and perfected by Garner [30] and these structures are now commonly known
as algebraic weak factorisation systems (hence awfs’). As is clear with a
decade of hindsight, awfs’ are the correct categorical structure combining
weak orthogonality and functoriality of factorisations.
The definition of an awfs is, by all accounts, rather complicated — it
involves a functorial factorisation, with its left and right components respectively
extended to comonads and monads L and R on the arrow category,
interacting via a distributive law. Given this, one can then consider the
coalgebras for the comonad L and algebras for the monad R, which are best
thought of as morphisms of C equipped with extra structure witnessing that
they belong to the “left and right classes” of the awfs. This point of view
was clarified by Garner [30], who modified Quillen’s small object argument
to show that given a set J of morphisms in a suitable category C, one can
8
cofibrantly generate an awfs (L, R) whose R-algebras are maps f : A → B
equipped with a lifting function ϕ
X
j∈J

r GG A
f

Y s
GG
ϕl,r,s
VV
B
providing lifts for each lifting problem as above. Of course, once we equip
morphisms with a choice of liftings, the new possibility emerges of encoding
further compatibilities between the liftings — in particular, given a commuting
square between morphisms of J, one can ask for the horizontal compati-
bility
X
u GG
j

X
j

r GG A
f

Y v
GG Y s
GG
ϕj,r,s
VV
B
=
X
j

ru GG A
f

Y sv
GG
ϕj ,ru,sv
VV
B
(1.3.1)
expressing the equality of the two diagonal fillers Y → B. This stronger
notion was captured by Garner’s notion of cofibrant generation by a category
of morphisms.
Building on these results, Emily Riehl introduced the notion of an algebraic
model category [55] — in which the two weak factorisations are made
(compatibly) algebraic — and showed that any cofibrantly generated model
category could be made algebraic.
For our purposes, a guiding example is the model structure for twodimensional
universal algebra introduced by Steve Lack [42]. Lack showed
that for T a suitable 2-monad, the 2-category T-Algs of algebras and strict
morphisms had a model structure whose trivial fibrations are those morphisms
sent by U : T-Algs → Cat to surjective equivalences, henceforth Usurjective
equivalences. Lack furthermore showed that the model structure
was closely connected to the classical construction of pseudomorphism classifiers
[9]: the pseudomorphism classifier QpA of an algebra A is defined by
the property that pseudomorphisms A B correspond to strict morphisms
QpA → B. In particular, the resulting adjunction induces a comonad Qp
on T-Algs and Lack showed that the counit pA : QpA → A was a cofibrant
replacement of A in this model structure.
The correspondence between pseudomorphisms and homotopy theory was
further sharpened by Garner [30] by passing from the model structure to
9
the algebraic model structure. To see this, note that each awfs C has a
canonical cofibrant replacement comonad Q and so induces a Kleisli category
CQ whose morphisms are thought of as weak morphisms. In the case of
the awfs for U-surjective equivalences on T-Algs, Garner showed that the
cofibrant replacement comonad coincides with the pseudomorphism classifier
so that, in particular, (T-Algs)Q = T-Algp. In other words, the algebraic
model structure encodes precisely the correct notion of pseudomorphism in
two-dimensional universal algebra.
Bearing in mind Table 1.1 of the previous section, a natural question
to ask was whether there is a corresponding story to be told for the lax
morphisms of T-Algl, on switching from surjective equivalences to lalis. To
begin with, there is an awfs on Cat whose algebras consist of the lalis —
this awfs is, indeed, the dual of one of the early examples from Grandis and
Tholen’s paper [31]. Thus, the natural goal became to lift this to an awfs on
T-Algs whose algebras are the U-lalis, and to show that T-Algl was the Kleisli
2-category for the associated cofibrant replacement comonad on T-Algs.
The problem here is that the usual approach would be to apply the left
adjoint F : Cat → T-Algs to the generating set or category of cofibrations J
and cofibrantly generate the appropriate awfs on T-Algs. However the awfs
for lalis did not appear to be cofibrantly generated in any sense considered
to date. Indeed, whilst all of the awfs arising from classical homotopy theory
considered by Riehl are cofibrantly generated by a set, with Garner we realised
that this awfs, arising from lax aspects of 2-category theory — namely,
adjunctions — was not cofibrantly generated even by a small category.
This led us in our 2016 publication [14], included in this collection and
published in the JPAA, to extend the notion of cofibrant generation beyond
the horizontal compatibilities of Diagram (1.3.1) to encode the vertical compability
that equates the two ways of lifting against a composable pair in J,
as depicted below.
X
j

u GG A
f

Y
k

VV
ϕ(j,u,vk)
Z v
GG
dd
ϕ(k,ϕ(j,u,vk),v)
B
=
X
j

u GG A
f

Y
k

Z v
GG
dd
ϕ(kj,u,v)
B
This stronger notion can be understood as cofibrant generation by a double
10
category and, in particular, captures the awfs for lalis. It is also needed to
capture other important types of functor involving universal properties, such
as Grothendieck fibrations. One of the main results — Theorem 25 of [14]
— is that each small double category of morphisms on a locally presentable
category generates an awfs and each reasonable (accessible) awfs arises in
this way — thus, this notion of cofibrant generation is, in a sense, the most
general possible.
The second main result — Theorem 6 of [14] — was a recognition theorem
for awfs comparable to Beck’s theorem for monads. The recognition
theorem characterises awfs in terms of their double categories of algebras.
Just as Beck’s theorem enables us to recognise and work with monads via
their categories of algebras, so the recognition theorem enables us to recognise
awfs far more easily in the wild. Because it shifts attention away from
the complex syntactic definition of an awfs to its semantics, it enabled us to
give simpler proofs of many old results about awfs — such as the existence
of cofibrantly generated awfs — as well as a number of important new ones,
such as the existence of projective and injective liftings of awfs along right
and left adjoints. These results have been applied to good effect by Hess et
al [36] to construct accessible model structures on categories of a coalgebraic
flavour.
They were also applied to construct the accessible awfs on T-Algs for
U-lalis in the followup [15], establishing T-Algl as its 2-category of weak
maps for the associated cofibrant replacement comonad, as was the original
motivation. The same paper also described how weak maps for an awfs (L, R)
could be viewed as certain equivalence classes of spans, thus capturing the
description of lax morphisms as spans of strict ones in (1.2.1). Similar results
about weak maps were given for dg-categories.
The results of these papers have also been influential in the homotopy
type theory community, for instance in the work of Gambino and coauthors
[29] and most recently in the notion of effective Kan fibration of van-denBerg
and Faber [5], which uses the stronger notion of cofibrant generation
by a small double category.
There are several open questions in this area, particularly about algebraic
model categories. There are various reasons to expect that one might want an
algebraic notion of weak equivalence to complement the algebraic fibrations
and algebraic trivial fibrations of an algebraic model category. I explored this
question of algebraic weak equivalences in Equipping weak equivalences with
algebraic structure [19], published in Mathematische Zeitschrift. There are
11
surprising results here — for instance, there is a monad such that detects
weak homotopy equivalences of topological spaces. Algebraic structure on
weak equivalences also naturally arise in homotopy type theory, where each
property should be witnessed by structure, and has led to a stronger notion of
algebraic model category [61], though it seems to me that stronger definitions
may yet emerge. An open question is to understand the analogue of Smith’s
theorem for accessible model categories, which I anticipate being related to
the above questions about algebraic model categories.
1.4 Skew monoidal categories
One of the cornerstones of modern category theory is the framework of enriched
category theory. Here the hom objects
C(A, B)
of the categories under consideration are not merely sets, but objects of
some monoidal category V — for instance, they may be abelian groups, chain
complexes, categories or other sorts of structured objects. In his foundational
book [41] on the topic, Kelly made the assumption that V is a symmetric
monoidal closed category — this generality includes many examples, such as
the above ones, and furthermore has the desirable property that the theory
of V-enriched categories works much as in the classical case of V = Set.
That said, this assumption on V is rather strong. Indeed Foltz, Kelly and
Lair [27] showed that algebraic categories often admit few if any monoidal
biclosed (and, in particular, symmetric monoidal closed) structures. For instance,
Cat admits exactly two — the cartesian and “funny” tensor product,
whilst the category of monoids, rings and groups admit none. In a similar
direction, with Nick Gurski [12] we showed that the category of Graycategories,
the well known model of semi-strict 3-categories [33], admits no
monoidal biclosed structure that models weak higher dimensional transformations
or which is homotopically well behaved — this rules out the possibility
of using monoidal biclosed or symmetric monoidal closed categories as
a base of enrichment for a good notion of semistrict 4-category. A long term
interest of mine has been to understand what sort of monoidal structure the
category of Gray-categories does support.
In 2012, motivated by structures such as bialgebroids, Szlachanyi introduced
the notion of a skew monoidal category [60]. These involve a tensor
12
product ⊗ and I together with coherence constraints of the form
(A ⊗ B) ⊗ C
α GG A ⊗ (B ⊗ C) I ⊗ A
l GG A A
r GG A ⊗ I
satisfying five axioms. When the three constraints are invertible, one obtains
the usual monoidal categories. The orientations of the arrows are certainly
obscure, but interesting consequences ensue: for instance, tensoring on one
side is a monad, on the other a comonad, and there is a distributive law
between the two. (In fact, this hints at a curious analogy with algebraic
weak factorisation systems.)
The concept came to my attention with the publication of Street’s 2013
paper [59], which describes the closed correlate of skew monoidal categories:
here the basic data are a hom-functor [−, −] : Vop
× V → V and unit object
I, together with the relatively strange looking coherence constraints
[A, B] L GG [[C, A], [C, B]] [I, A] i GG A I
j
GG [A, A]
satisfying 5 axioms. A closed category in the sense of Eilenberg and Kelly is
one of these for which
i : [I, A] → A
and the induced function
V(A, B) → V(I, [A, B]) : f → [A, f] ◦ jA (1.4.1)
is invertible. In the original treatise [26] of Eilenberg and Kelly on enriched
category theory, closed categories rather than monoidal ones were taken as
the base of enrichment — perhaps because, in practise, the structure of
the internal hom is often more easily understood the corresponding tensor
product. Against this, the computations, involving iterated contravariance,
are harder to parse.
In the presence of a unit I, tensor product ⊗ and hom [−, −] related by
a natural isomorphism
V(A ⊗ B, C) ∼= V(A, [B, C])
Street showed that there is a perfect correspondence between skew monoidal
structures on (V, ⊗, I) and skew closed ones on (V, [−, −], I) — the resulting
structures are called skew monoidal closed categories.
13
Now, whilst the invertibility of the two unit morphisms l and r in a
monoidal category correspond to the invertibility of i : [I, A] → A and (1.4.1)
in a closed category, invertibility of the associator (A⊗B)⊗C → A⊗(B⊗C)
does not appear to correspond to the invertibility of any morphism on the
closed side (at least one that is expressible purely in terms of the closed
structure). Therefore, in some sense, the correspondence in the skew setting
repairs an asymmetry in the classical setting.
With the above in mind, my own interest in the skew story arose on
considering the value gained by allowing the function
V(A, B) → V(I, [A, B])
of (1.4.1) to be non-invertible, or equivalently the morphism l : I ⊗ A → A.
This function allows us to view morphisms f : A → B as “elements” of the
hom [A, B]. In the classical case of a closed category, its invertibility says
that elements of [A, B] are precisely the morphisms A → B but, in the case it
is not invertible, it allows the possibility that [A, B] could contain more than
simply the morphisms from A to B — for instance, it might contain weak
maps from A to B, such as the pseudo or lax maps considered in previous
sections.
This perspective is developed in my paper Skew structures in 2-category
theory and homotopy theory [13] published in the Journal of Homotopy and
Related Structures, and included in this series. In this paper, I showed that in
2-category theory there are many natural examples of skew monoidal closed
categories that arise in this way, and used them to solve an old problem
about the construction of monoidal bicategories, which I will illustrate with
an example.
A permutative category is a symmetric strict monoidal category — here
the monoidal structure is strict but the braiding A ⊗ B ∼= B ⊗ A is not. Permutative
categories capture all symmetric monoidal categories up to equivalence.
Since these structures form a part of 2-dimensional universal algebra,
there are 2-categories Perms, Permp and so on, of permutative categories
and strict or pseudomorphisms. The idea is that Permp should be a (symmetric)
monoidal bicategory, categorifying the symmetric monoidal structure
on commutative monoids, but the details are complicated — see, for instance
[58].
The evident structure here is the internal hom Permp(A, B) between
permutative categories, whose morphisms are pseudomorphisms, preserving
14
the tensor product and unit up to isomorphism. Given two such F, G : A
B, their pointwise tensor product (F · G)a = Fa · Ga is again pseudo, via
the isomorphisms
(F·G)(a·b) = F(a·b)G(a·b) ∼= Fa·Fb·Ga·Gb ∼= Fa·Ga·Fb·Gb = (F·G)a(F·G)b.
This is the key in making the category Permp(A, B) into a permutative
category. Considering the second isomorphism in the above, observe that
even if F and G are strict, their pointwise tensor product F.G needn’t be —
therefore, it is not possible to make Perms(A, B) into a permutative category.
However, since in the skew setting we have seen that elements of the
internal hom can be more general than mere morphisms A → B, it is perfectly
possible that Permp(A, B) forms the internal hom object of a skew monoidal
closed structure on the 2-category of strict morphisms Perms. Indeed this is
the case. Moreover, the skew monoidal closed structure on Perms interacts
well with the Quillen model structure on Perms and so descends to a skew
monoidal structure on the homotopy category, which is in fact genuinely
monoidal — I called skew monoidal categories on Quillen model categories
with this property homotopy monoidal. In particular on restricting to the
cofibrant objects, the constraints αA,B,C : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C),
lA : I ⊗ A → A and rA : A → A ⊗ I are all equivalences, and the resulting
monoidal bicategory structure is, up to equivalence, the monoidal bicategory
structure on Permp that we wanted to construct at the start.
This is a good example of the approach championed by Max Kelly, of
studying the non-strict world as strictly as possible: this line of thinking,
interpreted for monoidal bicategories such as Permp, leads naturally into
the world of skew monoidal categories.
More generally, the above analysis extends to any pseudo-commutative
2-monad T on Cat, thus giving the first detailed proof of a result of HylandPower
[37] on the construction of monoidal bicategories in this setting. This
is Theorem 6.7 of [13]. Other examples of skew structures described in the
paper include one for bicategories.
This paper has been influential in a number of ways. Firstly, it established
that skew monoidal categories naturally arise in, and have the potential to
solve problems in, higher category theory and homotopy theory, thus providing
a completely new class of examples to the bialgebroid examples of
Szlachanyi [60], which had been the guiding examples until that time. The
examples of [13] played an influential role in Campbell’s development of skew
15
enriched categories [25]. In Sydney 2016-2017, Steve Lack and I built on ideas
in my paper [13] to develop the notion of skew multicategory [18], which has
aroused the interest of workers in theoretical computer science [62]. We also
introduced the notion of a braiding on a skew monoidal category [21], which
curiously involves an isomorphism (A ⊗ B) ⊗ C → (A ⊗ C) ⊗ B.
At the point of writing this thesis, skew monoidal categories have naturally
appeared in a broad range of settings, and I anticipate that we will see
many more examples and applications of them in years to come. A major
project that I am involved in at the moment is to use skew monoidal categories
as a base of enrichment for semistrict higher categories, moving us
beyond the case of Gray-categories in dimension 3.
1.5 Higher groupoids, monads, nerves and
theories
In category theory, there are two approaches to classical universal algebra.
One of these, as discussed in Section 1.2, is through finitary monads on the
category of sets. The other is via Lawvere theories [45]. which are identity
on objects finite coproduct preserving functors F → T where F → Set is
the full subcategory consisting of the finite cardinals {0, 1, 2, . . . ..}. That the
two approaches are equivalent was established by Linton [46], who showed
that the categories of finitary monads on Set and of Lawvere theories are
equivalent and that the equivalence respects semantics — that is, the passage
to categories of models.
This story has been generalised in many ways — until recently, the most
general of these was via the monads with arities of Weber [64] and the theories
with arities of [7]. The basic context here is a small dense full subcategory
A → E of a category E, generalising the case of F → Set appropriate to
classical universal algebra. A guiding example was the full subcategory Θ0 →
[Gop
, Set] of globular sets consisting of the globular cardinals, of which two
are drawn below.
• •GG •GG • • •GG
00
dd
GG


Just as the finite cardinals parametrise the shapes of operations of classical
universal algebra, so the globular cardinals parametrise the shapes of opera-
16
tions of (globular) higher dimensional category theory. A guiding result here
was the nerve theorem of Berger [6]. This takes as input a globular operad T
(a certain monad on globular sets capturing Batanin’s ω-categories [2]) and
forms the associated globular theory JT : Θ0 → ΘT obtained by taking the
(identity on objects,fully faithful)-factorisation
KT ◦ JT : Θ0 → ΘT → T-Alg
of the composite FT
KT : A → E → T-Alg of the inclusion with the free
algebra functor to the Eilenberg-Moore category of algebras. Berger’s nerve
theorem asserts that for such T the nerve functor
T-Alg(KT −, 1) : T-Alg → [(ΘT )op
, Set]
is fully faithful and moreover characterises its essential image as those presheaves
satisfying a generalised Segal condition. (This could be viewed as a higher dimensional
generalisation of the classical nerve theorem which identifies small
categories as simplicial sets satisfying the Segal condition.)
Weber in [64] observed that Berger’s construction make sense for any
monad T on E equipped with a dense full subcategory K : A → E —
thus one can ask whether a given monad satisfies the nerve theorem. He
introduced a class of monads called monads with arities A, which include the
globular operads, and showed that these do satisfy the nerve theorem — this
is Weber’s nerve theorem. Later, the theories corresponding to monads with
arities were identified by Berger, Mellies and Weber [7] and called theories
with arities A. In particular [7] established an equivalence between monads
and theories with arities, generalising Linton’s equivalence between finitary
monads and Lawvere theories. Pleasantly, Weber’s nerve theorem becomes
the fact that the equivalence respects semantics — that is, algebras for the
monad with arities T amount to models of the associated theory A → AT .
My own role in this story began in late 2015 when I read Maltsiniotis’ paper
[53]. This brought to attention a definition of globular weak ω-groupoid
essentially defined by Grothendieck at the beginning of Pursuing Stacks [32].
These are based on essentially the same globular theories Θ0 → T as those of
Berger, which can be defined as identity on objects functors preserving globular
sums — certain wide pushouts presenting globular cardinals as colimits
of representables. The theories for weak ω-groupoids were singled out as the
cellular contractible ones. It struck me that this definition was remarkably
simple compared to existing notions of globular weak ω-groupoid. With the
17
aim of justifying this claim, I wrote a very short paper [16] giving a reinterpretation
of Garner and van-den-Berg’s construction [63] of the globular
∞-groupoid associated to the identity type in homotopy type theory using
Grothendieck ∞-groupoids instead of Batanin ∞-groupoids, shortening the
argument considerably.
If the reader permits me a final digression on this topic, let me mention
that at this time I became somewhat obsessed by these Grothendieck
ω-groupoids. Indeed, a significant and long standing open problem is to
bridge the gap between globular and simplicial models of ∞-groupoids and
∞-categories. The ∞-groupoid part of this is Grothendieck’s homotopy hypothesis
(still unproven, in this form.) It struck me that with the relative
simple definition of Grothendieck ω-groupoid, it might be possible. To get
started, what one would like is to put a Quillen model structure6
on the
category Mod(T) of Grothendieck ω-groupoids. There are obvious notions
of fibration and weak equivalence and the problem quickly boils down to
proving that the pushout of the generating trivial cofibrations
F(ρn) : F(G(−, n)) → F(G(−, n + 1))
are weak equivalences. Here ρn : n → n + 1 is the inclusion in the globe
category G and F : [Gop
, Set] → Mod(T) the free algebra functor. After
some promising ideas, I spent a week at the State Library in Berlin trying to
prove this result, but did not get much further. However, my approach via
viewing Grothendieck ω-groupoids as iterated algebraic injectives did enable
to me to prove the faithfulness conjecture of Maltsiniotis, resulting ultimately
in the publication [20]. Despite further work on these questions by Henry
and Lanari amongst others, the problem of constructing the model structure
remains very much open.
Returning to the main story, in Sydney in 2016 I continued to think about
these higher dimensional groupoids. A key observation was that the globular
theories for higher groupoids do not have arities Θ0 in the sense of Berger
et al [7]. Therefore, in order to have a notion of theory capturing both the
classical notion of Lawvere theory and of globular theory, it was necessary
to generalise the notion of a theory with arities.
This is achieved by the simple notion of an A-theory, which appears
in my joint paper Monads and theories [17] with Richard Garner, included
in this series and published in Advances in Mathematics in 2019. Having
6
Or perhaps a semi-model structure.
18
identified the correct class of A-theories, we quickly realised that the monads
which correspond to them are those for which the conclusion of Weber’s
nerve theorem [64] holds, which we hence dubbed the A-nervous monads.
In particular we established a semantics respecting equivalence between Anervous
monads and A-theories, showing it to be best possible in a precise
sense. All of this works in the enriched setting, and subsumes most if not
all monad-theory correspondences considered to date. Apart from this, the
main achievement of the paper is that it establishes the excellent properties
of A-nervous monads, A-theories and of their enriched categories of models.
One of the deeper results in this regard is Theorem 38, which shows that
the A-nervous monads are precisely precisely those which can be presented
using colimits of free monads on signatures. In particular, if one is interested
in presentations of general kinds of algebraic structures, this means that the
setting of A-nervous monads and A-theories is ideally suited to the purpose.
1.6 Homotopy enriched category theory and
higher-dimensional universal algebra
How do we know that algebraic categories, like the categories Grp and
Rng of groups and rings, are cocomplete? And how do we know that forgetful
functors between algebraic categories, such as the forgetful functor
U : Rng → Grp from rings to groups, have left adjoints?
In this section I will describe a project that develops techniques to solve
the higher dimensional analogues of these questions in which we, for instance,
replace Rng by the 2-category MonCatp of monoidal categories and
pseudomaps, or by the ∞-cosmos QCatprod of quasicategories with finite
products [38, 47, 57].
There are several approaches to answering these questions for classical
algebraic categories, one of which is via the theory of accessible and locally
presentable categories [50, 1]. Because filtered colimits are well behaved in
such categories, it is not too hard to see that these categories are (finitely)
accessible — meaning, suitably generated through filtered colimits by a set
of finitely presentable objects — and easy to see that they are complete.
Therefore we can obtain cocompleteness via the following well known and
important result — see, for instance, [1, Theorem 4.11].
(1) An accessible category is complete if and only if it is cocomplete.
19
For similar reasons, we can easily apply the following theorem, described
in Theorem 1.66 of [1], to obtain the desired left adjoints.
(2) A functor between complete accessible categories has a left adjoint if
and only if preserves limits and (sufficiently highly) filtered colimits.
In seeking to extend these results to higher dimensions in a manner applicable
to higher dimensional structures and their pseudomaps, one might
expect it to be necessary to consider weaker versions of the limits and filtered
colimits under consideration, having their universal properties only up
to equivalence.
Surprisingly, in many settings, this turns out not to be the case. To
begin with, from the work of the Australian school it has long been known
that 2-categories such as MonCatp admit certain genuine weighted limits —
namely, flexible (or cofibrantly weighted) limits [8]. Building on work of
Makkai [51, 52], in my recent paper [22] I established that such 2-categories
are, in fact, genuinely accessible. Accessibility in the 2-categorical setting
turns out to be closely related to weakness (or cofibrancy) of the categorical
structure involved — for instance, whilst the 2-category MonCatp of monoidal
categories is accessible, its full subcategory SMonCatp of strict monoidal
categories is not.7
The problem is that the notion of strict monoidal category
is too strict.
The infinite dimensional analogue of this question considers ∞-cosmoi
— certain simplicially enriched categories — of (∞, 1)-categories with structure,
and the structures considered in practise in the ∞-setting are invariably
weak. Therefore, by analogy with the two-dimensional case described above,
it is natural to conjecture that ∞-cosmoi such as QCatprod are genuinely
accessible. Indeed this is the case, as is shown for a number of such examples
in our 2020 preprint Adjoint functor theorems for homotopically enriched
categories [23] with Steve Lack and Luk´aˇs Vokˇr´ınek, included in this collection,
and to be established in much greater generality in a forthcoming paper
Accessible infinity-cosmoi with Lack [24].
What I would like to describe here is the framework for cocompleteness
and adjoint functor theorems developed in [23] with such examples in mind,
since it holds in a wide range of contexts and has a broad range of applica-
tions.
7
Indeed — the idempotent completion of SMonCatp is MonCatp.
20
The basic framework is inspired by the paper Enriched Weakness [44]
of Lack and Rosick´y, which introduced the notions of E-weak left adjoint
and E-weak colimit relative to a class of morphisms E in a suitable base for
enrichment V: namely, the enriched functor U : A → B is said to have an
E-weak left adjoint if for each X ∈ B there exists an object FX ∈ A and
morphism ηX : X → UFX for which the induced morphism
A(FX, Y ) U GG B(UFX, UY )
B(ηX ,UY )
GG B(X, UY )
of V belongs to E. The notion of E-weak weighted colimit is similarly
defined. In particular, in the case of V = Set with E the class of isomorphisms
one obtains the usual notions of adjoint functor and colimit. For V = Set
with E the class of surjections, one obtains the notions of weak left adjoint
and weak colimit, which have the existence but not uniqueness part of the
appropriate universal property.
Our paper [23] takes the base V to be a monoidal model category. In this
setting, the limits of relevance to our E-weak adjoint functor theorem and
E-cocompleteness theorem are powers and cofibrantly weighted limits. The
class of morphisms E is the class of so-called shrinking morphisms which are,
assuming all objects are cofibrant, certain weak equivalences. The main result
of the paper, Theorem 7.8, is our adjoint functor theorem for homotopically
enriched categories. From this, the other results involving accessibility and
cocompleteness, as discussed above in the classical setting, naturally follow.
Rather than describing the most general results in detail, I will only mention
a few applications, which are discussed in greater depth in Section 9 of the
paper.
• When V is equipped with the trivial model structure, in which the weak
equivalences are the isomorphisms, we obtain the classical theory. In
particular, we obtain the classical adjoint functor theorem [49] and,
when specialised to the accessible setting, the classical results (1) and
(2) described at the beginning of this section.
• When V is equipped with what we call the split model structure, in
which all maps are weak equivalences and fibrations are the split epis,
we obtain weak enriched category theory. In particular, when V = Set,
we obtain results such as Kainen’s weak adjoint functor theorem [39]
and that an accessible category has products if and only if it has weak
colimits [1, Theorem 4.11].
21
• When V = Cat equipped with the natural model structure, whose
weak equivalences are the equivalences of categories, we obtain a new
2-categorical adjoint functor theorem. Since, as described earlier, 2categories
such as MonCatp are accessible [22] the adjoint functor theorem
in this setting applies naturally to forgetful 2-functors such as
U : MonCatp → Cat, and more broadly to 2-categories that lie outside
the scope of two-dimensional monad theory such as the 2-categories of
regular categories, pretopoi and so forth. Our results on cocompleteness
also give simple proofs that such 2-categories have (a rather strong
form of) bicolimits.
• Our major motivation is where V = SSet is the category of simplicial
sets equipped with the Joyal model structure. In this setting, we
obtain a new homotopical adjoint functor theorem applicable to the
infinity-cosmoi of Riehl and Verity [56, 57]. This covers examples such
as the forgetful cosmological functor QCatprod → QCat from quasicategories
with finite products to quasicategories. Moreover, we obtain
the new result that an accessible infinity cosmos, such as QCatprod,
automatically has flexibly weighted homotopy colimits.
The results of this paper are very recent and will be built upon in the
forthcoming paper Accessible Infinity Cosmoi [24] (where, once again, the
lalis of Sections’ 1.2 and 1.3 play a decisive role!). Beyond this, the goal of this
ongoing project is to continue using the interplay between enriched category
theory and homotopy theory to develop simple approaches to handling weak
higher dimensional structures.
22
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28
Chapter 2
Two-dimensional monadicity
This chapter contains the article Two-dimensional monadicity by John Bourke,
published in Advances in Mathematics 252 (2014) 707–747.
29
TWO-DIMENSIONAL MONADICITY
JOHN BOURKE
Abstract. The behaviour of limits of weak morphisms in 2-dimensional universal
algebra is not 2-categorical in that, to fully express the behaviour that
occurs, one needs to be able to quantify over strict morphisms amongst the
weaker kinds. F-categories were introduced to express this interplay between
strict and weak morphisms. We express doctrinal adjunction as an F-categorical
lifting property and use this to give monadicity theorems, expressed using the
language of F-categories, that cover each weaker kind of morphism.
1. Introduction
The category of monoids sits over the category of sets via a forgetful functor
U : Mon → Set. This functor is monadic in the sense that it has a left adjoint F
and the canonical comparison E : Mon → SetT
to the category of algebras for the
induced monad T = UF is an equivalence of categories. So if you are interested in
monoids you can set about proving some theorem about algebras for an abstract
monad T and be sure it holds for monoids, or any variety of universal algebra for
that matter: this is the categorical approach to universal algebra via monads.
Before going down this path one thing must be established – namely, the monadicity
of U. To this end the fundamental theorem is Beck’s monadicity theorem [2]
which asserts that a functor U : A → B is monadic just when it admits a left
adjoint, is conservative and creates U-absolute coequalisers. What makes the theorem
so useful in practice is that the conditions, up to the existence of a left
adjoint, are cast entirely in terms of the typically simple U – these conditions are
clearly met for monoids or indeed any variety (see Section 6.8 of [18]).
Now our interest is not in universal algebra, but in two dimensional universal
algebra and 2-monads, and monadicity as appropriate to this setting. What do
we mean by the varieties of 2-dimensional universal algebra? Monoidal structure
borne by categories provides a basic example: one observes that associated to this
notion are several kinds of structure. On the objects front we have at least strict
monoidal categories and monoidal categories of interest. Between these are strict,
strong, lax and colax monoidal functors all commonly arising, between which we
have just one kind of monoidal transformation. Restricting ourselves to just one
kind of object, let us take the monoidal categories, we still find that we are presented
with four related 2-categories MonCatw, where w ∈ {s, p, l, c}, living over
2000 Mathematics Subject Classification. Primary: 18D05.
Supported by the Grant agency of the Czech Republic under the grant P201/12/G028.
1
2 JOHN BOURKE
the 2-category of categories Cat as on the left below.
(1) MonCats MonCatp
MonCatl
MonCatc
Cat
GG
UUoooooo
99yyyyyy
Vs
11cccccccccccccccc
Vp

Vl
ÖÖ
Vc
{{wwwwwwwwww
T-Algs T-Algp
T-Algl
T-Algc
Cat
GG
UUooooo
99yyyyy
Us
11cccccccccccccccc
Up

Ul
ÖÖ
Uc
{{wwwwwwwwww
The objects in each of these are monoidal categories; the morphisms are respectively
strict, strong (or pseudo), lax and colax monoidal functors with monoidal
transformations between them in each case. The inclusions witness that strict morphisms
can be viewed as pseudomorphisms (s ≤ p), which can in turn be viewed
as lax or colax (p ≤ l) and (p ≤ c).
The situation corresponds with that of a 2-monad T based on Cat, associated with
which are several kinds of algebra, including strict and pseudo-algebras. We will
only ever consider the strict algebras: note that even non-strict monoidal categories
are the strict algebras for a 2-monad, as further discussed below. Between
strict algebras are strict, pseudo, lax and colax morphisms, with again a single
notion of algebra transformation. As before we obtain a diagram of 2-categories1
and 2-functors over Cat, as on the right above.
Comparing the diagrams left and right above we see that a monadicity theorem
in this setting ought to match each 2-category on the left with the corresponding
one on the right in a compatible way. More precisely, there should exist a
2-monad T on Cat and, for each w ∈ {s, p, l, c}, an equivalence of 2-categories
Ew : MonCatw → T-Algw over Cat, as left below
(2) MonCatw T-Algw
Cat
Ew
GG
Vw
''UUUUUUUUUUU
Uw
ÓÓ§§§§§§§§§§§
MonCatw1
T-Algw1
MonCatw2
T-Algw2
Ew1
GG

Ew2
GG
and these equivalences should be natural in the inclusions w1 ≤ w2 for w1, w2 ∈
{s, p, l, c}, as on the right.
Now it is well known that such a 2-monad does indeed exist, with, moreover,
each comparison Ew : MonCatw → T-Algw an isomorphism of 2-categories. Likewise
many of the other varieties of 2-dimensional universal algebra are monadic
in this sense. With such varieties as the primary object of study the subject of
2-dimensional monad theory was developed, notably in [3]. General results were
obtained such as those enabling one to deduce that each inclusion MonCats →
MonCatw has a left 2-adjoint, or establish the bicategorical completeness and cocompleteness
of MonCatp.
1The objects in each of these 2-categories are the strict algebras.
TWO-DIMENSIONAL MONADICITY 3
Of course to apply such abstract results one must first establish monadicity. How
is this known? Here the subject diverges substantially from the 1-dimensional
approach of Beck’s theorem. The standard approach is that of colimit presentations
[15][11]. Here one explicitly constructs the intended 2-monad T as an
iterated colimit of free ones, and then performs lengthy calculations with the universal
property of the colimit T to establish monadicity in the sense described for
monoidal categories above.
Now although the natural analogue of Beck’s theorem has been obtained for pseudomonads
[17] and pseudoalgebra pseudomorphisms this does not specialise to
capture monadicity in the precise sense described above, even when w = p.
Our objective in the present paper is to establish 2-dimensional monadicity theorems,
in which monadicity is recognised not by using an explicit description of a
2-monad, but by analysing the manner in which the varieties of 2-dimensional universal
algebra sit over the base 2-category – as in Diagram 1. Apart from characterising
such monadic situations our main results, Theorem 23 through Theorem 25,
enable one to establish monadicity when a workable description of a 2-monad is
not easily forthcoming.
In seeking to understand monadicity in the above sense the first important observation
is that the world of strict morphisms is easily understood: one can check
that Vs : MonCats → Cat has a left 2-adjoint – say, by an adjoint functor theorem
– and then apply the enriched version of Beck’s monadicity theorem [4] to establish
strict monadicity. Having observed this to be the case the natural question to ask
is: Which properties of the commutative triangle
MonCats MonCatw
Cat
GG
Vs
''UUUUUUUUUUU
Vw
ÓÓ§§§§§§§§§§§
ensure that the canonical isomorphism E : MonCats → T-Algs extends to an
isomorphism Ew : MonCatw → T-Algw over the base? We answer this question in
Theorem 25 but not using the language of 2-categories. For it turns out that these
determining properties are not 2-categorical in nature: they cannot be expressed
as properties of the 2-categories or 2-functors in the above diagram considered
individually. Rather, to express these properties we must be able to single out
strict morphisms amongst each weaker kind. Consequently we treat the inclusion
MonCats → MonCatw as a single entity, an F-category MonCatw, and the above
triangle as a single F-functor V : MonCatw → Cat.
Let us now give an overview of the paper and of our line of argument. F-categories
were introduced in [16] in order to explain certain relationships between strict and
weak morphisms in 2-dimensional universal algebra. We recall some basic facts
about F-categories in Section 2, in particular discussing F-categories T-Algw of
algebras for a 2-monad and F-categories MonCatw of monoidal categories – we
will use monoidal categories as our running example throughout the paper.
Given a strict monoidal functor F and an adjunction ( , F G, η) the right adjoint
G obtains a unique lax monoidal structure such that the adjunction becomes a
monoidal adjunction. This is an instance of doctrinal adjunction, the main topic
4 JOHN BOURKE
of Section 3, and the first key F-categorical property that we meet. We express
three variants of doctrinal adjunction – w-doctrinal adjunction for w ∈ {l, p, c} –
as lifting properties of an F-functor, so that the case just described asserts that
the forgetful F-functor V : MonCatl → Cat satisfies l-doctrinal adjunction. We
define the closely related class w-Doct of w-doctrinal F-functors and analyse the
relationships between the different notions for w ∈ {l, p, c}; each such class of Ffunctor
is shown to be an orthogonality class in the category of F-categories.
In the fourth section we turn to the reason F-categories were introduced in [16]
– namely, because of the interplay between strict and weak morphisms, tight and
loose, that occurs when considering limits of weak morphisms in 2-dimensional
universal algebra. The crucial limits are ¯w-limits of loose morphisms for w ∈
{l, p, c} – after defining these we describe the illuminating case of the colax limit
of a lax monoidal functor. We then examine how such limits allow one to represent
loose morphisms by tight spans – the nature of this representation is analysed in
detail.
This analysis allows us to prove the key result of the paper, Theorem 21 of Section
5, an orthogonality result which has nothing to do with 2-monads at all. Its
immediate consequence, Corollary 22, ensures that the decomposition in F-CAT
MonCats MonCatw Cat
j
GG V GG = MonCats
MonCats
MonCatw
Cat
1 RRiiiii
j
CC†††††
j

Vs
BB……………………
Vw
QQhhhhhhh
of the forgetful 2-functor Vs : MonCats → Cat is an orthogonal (⊥w-Doct, w-Doct)decomposition.
Likewise for a 2-monad T on Cat the F-category T-Algw is obtained
as a (⊥w-Doct, w-Doct)-factorisation of Us : T-Algs → Cat.
Our monadicity results, given in Section 6, use the uniqueness of (⊥w-Doct, w-Doct)decompositions
to extend our understanding of monadicity in the strict setting
to cover each weaker kind of morphism. For instance, the isomorphism E :
MonCats → T-Algs over Cat induces a commuting diagram as on the outside
of
MonCats MonCatw Cat
j
GG V GG
T-Algs T-Algw Cat
E

Ew

j
GG
U
GG
1

with each horizontal path an orthogonal (⊥w-Doct, w-Doct)-decomposition. The
two outer vertical isomorphisms then induce a unique invertible filler Ew : MonCatw →
T-Algw, so establishing monadicity in each weaker context. This is the idea behind
the main monadicity result, Theorem 25. Naturality in different weaknesses (as in
Diagram 2 above) is treated in Theorem 24.
In the seventh and final section we describe examples and applications of our results.
We begin by completing the example of monoidal categories before moving
on to more complex cases. In Theorem 27 we give an example of the kind of
monadicity result that cannot be established using techniques, like presentations,
that require explicit knowledge of a 2-monad.
TWO-DIMENSIONAL MONADICITY 5
Acknowledgments. The author thanks Richard Garner, Stephen Lack, Ignacio
L´opez Franco, Michael Shulman and Luk´aˇs Vokˇr´ınek for useful feedback during the
preparation of this work, and Michael Shulman for carefully reading a preliminary
draft. Thanks are due to the referee whose insightful observations enabled us
to remove unnecessary hypotheses from the main results. Thanks also to the
organisers of the PSSL93 and the Workshop on category theory, in honour of
George Janelidze for providing the opportunity to present it, and the members of
the Brno Category Theory Seminar for listening to a number of talks about it.
2. F-categories in 2-dimensional universal algebra
In this section we recall the notion of F-category, introduced in [16], and a few
basic facts about them.
2.1. F-categories. An F-category A is a 2-category with two kinds of 1-cell: those
of the 2-category itself which are called loose and a subcategory of tight morphisms
containing all of the identities. A second perspective is that an F-category A is
specified by a pair of 2-categories Aτ and Aλ connected by a 2-functor
j : Aτ → Aλ
which is the identity on objects, faithful and locally fully faithful. Here Aλ contains
all of the morphisms, which is to say the loose ones, and all 2-cells between them,
whilst Aτ contains the tight morphisms together with all 2-cells between them
in Aλ. Loose morphisms in A are drawn with wavy arrows A B and tight
morphisms with straight arrows A → B, so that a typical diagram in A would be
A B
C
f
GGGoGoGoGoGo
g

h
111•1•1•1•1•1•1•1•
α{Ó 
For each pair of objects A, B ∈ A the inclusion of hom categories
jA,B : Aτ (A, B) → Aλ(A, B)
constitutes an injective on objects fully faithful functor. In fact F-categories are
precisely categories enriched in F, the full subcategory of the arrow category Cat2
whose objects are those functors which are both injective on objects and fully
faithful. F is a complete and cocomplete cartesian closed category so that the full
theory of enriched categories [8] can be applied to the study of F-categories.
To begin with we have F-CAT, the 2-category of F-categories, F-functors and Fnatural
transformations. An F-functor F : A → B consists of a pair of 2-functors
Fτ : Aτ → Bτ and Fλ : Aλ → Bλ rendering commutative the square
Aτ Aλ
Bτ Bλ
jA
GG
Fλ

Fτ

jB
GG
6 JOHN BOURKE
This equally amounts to a 2-functor Fλ : Aλ → Bλ which preserves tightness. An
F-natural transformation η : F → G is a 2-natural transformation η : Fλ → Gλ
with tight components.
2.2. 2-categories as F-categories. Each 2-category A may be viewed as an Fcategory
in two extremal ways: as an F-category in which only the identities are
tight, or as an F-category in which all morphisms are tight, whereupon the induced
F-category has the form
1 : A → A
When we view a 2-category A as an F-category it will always be in this second
sense and we again denote it by A.
With this convention established we can treat 2-categories as special kinds of Fcategories
and unambiguously speak of F-functors F : A → B from 2-categories
to F-categories, or from F-categories to 2-categories as in G : B → C. These
F-functors appear as triangles
A
Bτ Bλ
Fλ
&&RRRRRRR
Fτ
ÔÔ
jB
GG
Bτ Bλ
C
jA
GG
Gλ
ÔÔ
Gτ
&&RRRRRRR
with an F-functor between 2-categories just a 2-functor. Observe that each Fcategory
A induces an F-functor from its 2-category of tight morphisms
j : Aτ → A
which is the identity on tight morphisms and j : Aτ → Aλ on loose ones – we
abuse notation by using j in either situation.
2.3. F-categories of monoidal categories. In two-dimensional universal algebra
one encounters morphisms of four different flavours and so F-categories naturally
arise. Here we recall the various F-categories associated to the notion of
monoidal structure – we recall the defining equations for monoidal functors as we
will use these later on.
The data for a monoidal category consists of a tuple A = (A, ⊗, iA, λA, ρA
l , ρA
r )
where we use juxtaposition for the tensor product. A lax monoidal functor (F, f, f0) :
A B consists of a functor F : A → B, coherence constraints fa,b : Fa ⊗ Fb →
F(a ⊗ b) natural in a and b and a comparison f0 : iB → FiA, all satisfying the
three conditions below
(Fa ⊗ Fb) ⊗ Fc F(a ⊗ b) ⊗ Fc F((a ⊗ b) ⊗ c)
Fa ⊗ (Fb ⊗ Fc) Fa ⊗ F(b ⊗ c) F(a ⊗ (b ⊗ c))
fa,b⊗1
GG
fa⊗b,c
GG
1⊗fb,c
GG
fa,b⊗c
GG
λB
F a,F b,F c

FλA
a,b,c

iB ⊗ Fa FiA ⊗ Fa F(iA ⊗ a)
Fa Fa
f0⊗1
GG
fi,a
GG
ρB
l

FρA
l

(Fa) ⊗ iB
Fa ⊗ FiA F(a ⊗ iA)
Fa Fa
1⊗f0
GG
fa,i
GG
ρB
r

FρA
r

TWO-DIMENSIONAL MONADICITY 7
which we call the associativity, left unit and right unit conditions. We call (F, f, f0)
strong or strict monoidal just when the constraints fa,b and f0 are invertible or
identities respectively; reversing these constraints we obtain the notion of a colax
monoidal functor.
Fa ⊗ Fb Ga ⊗ Gb
F(a ⊗ b) G(a ⊗ b)
ηa⊗ηb
GG
ηa⊗b
GG
fa,b

ga,b

iB FiA
GiA
f0
GG
g0 77vvvvvvvvv
ηiA

Between lax monoidal functors are monoidal transformations η : (F, f, f0) →
(G, g, g0): these are natural transformations η : F → G satisfying the two conditions
above, which we call the tensor and unit conditions for a monoidal transfor-
mation.
For w ∈ {s, l, p, c} we thus have w-monoidal functors (with p-monoidal meaning
strong monoidal). Together with monoidal categories and monoidal transformations
these form a 2-category MonCatw which sits over Cat via a forgetful 2-functor
Vw : MonCatw → Cat. Now strict monoidal functors are strong (s ≤ p) and strong
monoidal functors can be viewed as lax (p ≤ l) or colax (p ≤ c). Whenever w1 ≤ w2
it follows that we have an F-category MonCatw1,w2 of monoidal categories with
tight and loose morphisms the w1 and w2-monoidal functors respectively, as specified
by the inclusion j : MonCatw1 → MonCatw2 . Each such F-category comes
equipped with a forgetful F-functor V : MonCatw1,w2 → Cat where Vτ = Vw1 and
Vλ = Vw2 – see the commuting triangles in Diagram 1 of the Introduction. Of
particular importance will be those F-categories MonCats,w for s ≤ w, which we
denote by MonCatw.
2.4. F-categories of algebras. Of prime importance are those F-categories associated
to a 2-monad T on a 2-category C. Each 2-monad has various associated
flavours of algebra and morphism. We will only be interested in strict algebras and
will call them algebras. Between algebras we have strict, pseudo, lax and colax
morphisms – as with monoidal functors we specify these using s, p, l and c.
Drawn in turn from left to right below is the data (f, f) for a strict, pseudo, lax
or colax morphism of algebras (f, f) : (A, a) (B, b)
TA TB
A B
Tf
GG
b

a

f
GG
TA TB
A B
Tf
GG
b

a

f
GG
∼=
f
TA TB
A B
Tf
GG
b

a

f
GG
f
TA TB
A B
Tf
GG
b

a

f
GG
f
uƒ
Thus f is an identity 2-cell in the first case, invertible in the second, and points
into or out of f in the lax or colax cases; in all cases these 2-cells are required to
satisfy two coherence conditions [12].
There is a single notion of 2-cell between any kind of algebra morphisms; for
instance given a pair of lax morphisms (f, f), (g, g) : (A, a) (B, b) an algebra
8 JOHN BOURKE
2-cell α : (f, f) ⇒ (g, g) is a 2-cell α : f ⇒ g satisfying
TA TB
A B
b

f
77
g
WW
a

Tf
GG
α 
f 
=
TA TB
A B
b

g
GG
a

Tf
88
Tg
VV
g
Tα 
whilst the equation in the colax case looks like the lax case with the directions
reversed.
Algebras, w-algebra morphisms and transformations live in 2-categories T-Algw
for w ∈ {s, p, l, c}, each of which comes equipped with an evident forgetful 2-functor
to the base, which we always denote by Uw : T-Algw → C. Each strict morphism
is a pseudo morphism (s ≤ p), and each pseudomorphism can be viewed either
as lax (p ≤ l) or colax (p ≤ c). It follows that for each pair w1, w2 ∈ {s, p, l, c}
satisfying w1 ≤ w2 we have an F-category T-Algw1,w2
with tight morphisms the
w1-morphisms, and loose morphisms the w2-morphisms. Each comes equipped
with a forgetful F-functor U : T-Algw1,w2
→ C (so Uτ = Uw1 and Uλ = Uw2 ) – see
the commuting triangles of Diagram 1 of the Introduction.
Of particular importance are those F-categories T-Algs,w whose tight morphisms
are the strict ones, and following [16] we abbreviate these by T-Algw. As well as
using j : T-Algs → T-Algw for the defining inclusion and U : T-Algw → C for
the forgetful F-functor, we occasionally use jw or Uw if we are in the presence of
multiple j’s or U’s.
2.5. Duality between lax and colax morphisms. Colax algebra morphisms
are lax algebra morphisms with 2-cells reversed. This statement can be made
precise using the covariant duality 2-functor (−)co : 2-CAT → 2-CAT which takes
a 2-category C to the 2-category Cco with the same underlying category but with 2cells
reversed. Likewise it takes a 2-monad T : C → C to a 2-monad Tco : Cco → Cco
and one then has, as noted in [6], an equality Tco
-Algl = T-Algco
c which restricts
to Tco
-Algs = T-Algco
s . The (−)co duality naturally extends to a 2-functor
(−)co
: F-CAT → F-CAT
under whose action we have that T-Algco
c = Tco
-Algl and moreover that
Uco
: T-Algco
c → Cco
equals U : Tco
-Algl → Cco
.
A consequence of this duality is that each theorem about lax morphisms has a
dual version concerning colax morphisms. Indeed all of our definitions and results
in the colax case will be dual to those in the lax setting – though we will state
these results for colax morphisms it will always suffice to prove results only in the
lax setting.
2.6. Equivalence of F-categories. Our monadicity theorems in Section 6 will
assert that certain F-categories are equivalent to F-categories of algebras for a
TWO-DIMENSIONAL MONADICITY 9
2-monad. By an equivalence of F-categories we mean an equivalence in the 2category
of F-categories F-CAT, which is to say an equivalence of V-categories for
V = F.
Recall from [8] that a V-functor F : A → B is an equivalence just when it is
essentially surjective on objects and fully faithful in the enriched sense. When
V = F the first condition amounts to Fτ : Aτ → Bτ being essentially surjective on
objects, in the usual sense. This also implies the weaker statement that Fλ : Aλ →
Bλ is essentially surjective on objects. Enriched fully faithfulness of F amounts
to Fτ A,B : Aτ (A, B) → Bτ (FA, FB) and FλA,B : Aλ(A, B) → Bλ(FA, FB) being
isomorphisms of categories for each pair A, B ∈ A.
We conclude that F is an equivalence of F-categories just when both 2-functors
Fτ and Fλ are essentially surjective on objects and 2-fully faithful, which is to say
that both Fτ and Fλ are 2-equivalences, or equivalences of 2-categories.
3. Doctrinal adjunction and F-categorical lifting properties
If a strict monoidal functor has a right adjoint that right adjoint admits a unique
lax monoidal structure such that the adjunction lifts to a monoidal adjunction.
This is an instance of doctrinal adjunction – the topic of the present section. We
begin by recalling Kelly’s treatment of doctrinal adjunction in the setting of 2monads,
recasting the notion in F-categorical terms, so that the above special case
becomes the assertion that the forgetful F-functor V : MonCatl → Cat satisfies
l-doctrinal adjunction – we treat the cases w ∈ {l, p, c}. In Section 3.2 we define
the closely related notion of a w-doctrinal F-functor before showing that the wdoctrinal
F-functors form an orthogonality class in F-CAT.
3.1. Doctrinal adjunction F-categorically. Doctrinal adjunction was first studied
in Kelly’s paper [6] of the same name. Motivated by the example of adjunctions
between monoidal categories amongst others, all known to be describable using 2monads
via clubs [7] or other techniques, he gave his treatment in the setting of
2-dimensional monad theory. Let us now recall the relevant aspects of this. Given
T-algebras (A, a) and (B, b) and a morphism f : A → B together with an adjunction
( , f g, η) in the base, his Theorem 1.2 asserts that there is a bijection
between colax algebra morphisms of the form (f, f) : (A, a) (B, b) and lax morphisms
of the form (g, g) : (B, b) (A, a). The structure 2-cells f : f.a ⇒ b.Tf
and g : a.Tg ⇒ g.b are expressed in terms of one another as mates as below
TA TB
A B
TA
B
b

f
GG
a

Tf
GG
1

Tg
wwoooooooooooooo
g
wwoooooooooooooooo
1

g
'5
cccc
Tη
CQ
CQ
TB TA
B A
TB
A
a

g
GG
b

Tg
GG
1

Tf
wwoooooooooooooo
f
wwoooooooooooooooo
1

f‘™cccc
Tks
η
ks
Since lax and colax morphisms cannot be composed this relationship cannot be expressed
2-categorically or indeed F-categorically – it can be captured using double
10 JOHN BOURKE
categories as in Example 5.4 of [22]. However if we start with (f, f) a pseudomorphism
then it does live in the same 2-category as the resultant lax morphism
(g, g) : (B, b) (A, a). Moreover it is shown in Proposition 1.3 of [6] that the
unit and counit η and then become algebra 2-cells η : 1 ⇒ (g, g) ◦ (f, f) and
: (f, f) ◦ (g, g) ⇒ 1 in T-Algl and so yield an adjunction ( , (f, f) (g, g), η) in
T-Algl.
Dually, if (g, g) were a pseudomorphism, then upon equipping f with the corresponding
colax structure (f, f) the adjunction ( , f g, η) lifts to an adjunction
( , (f, f) (g, g), η) in the 2-category T-Algc.
The invertibility of f does not imply the invertibility of its mate g, or vice-versa.
However, if both η and are invertible, then f is invertible just when g is, which
is to say that if ( , f g, η) is an adjoint equivalence and either f or g admits the
structure of a pseudomorphism the adjoint equivalence lifts to an adjoint equivalence
in T-Algp.
Let us now abstract these lifting properties of adjunctions and adjoint equivalences
into properties of the forgetful F-functors U : T-Algw1,w2
→ C. By an
adjunction or adjoint equivalence in an F-category A we will mean an adjunction
or adjoint equivalence in its 2-category Aλ of loose morphisms. Given an F-functor
H : A → B and an adjunction ( , f g, η) in B a lifting of this adjunction along
H is an adjunction ( , f g , η ) in A such that Hf = f, Hg = g, H = and
Hη = η. We will likewise speak of liftings of adjoint equivalences along H. An
F-functor H : A → B is said to satisfy
• weak l-doctrinal adjunction if for each tight arrow f : A → B ∈ A each
adjunction ( , Hf g, η) in B lifts along H to an adjunction in A with left
adjoint f.
• weak p-doctrinal adjunction if for each tight arrow f : A → B ∈ A each
adjoint equivalence ( , Hf g, η) in B lifts along H to an adjoint equivalence
in A with left adjoint f.2
• weak c-doctrinal adjunction if for each tight arrow f : A → B ∈ A each
adjunction ( , g Hf, η) in B lifts along H to an adjunction in A with right
adjoint f.
The lifting properties for algebras described above can be rephrased as asserting
exactly that the forgetful F-functor U : T-Algp,w → C satisfies weak w-doctrinal
adjunction for w ∈ {l, p, c}. Each of these statements asserts that if we are given a
pseudomorphism of algebras whose underlying arrow has some kind of adjoint, then
that adjunction lifts in a certain way; of course if the starting pseudomorphism
were in fact strict then the same lifting will exist so that, in particular, the forgetful
F-functor U : T-Algw → C satisfies weak w-doctrinal adjunction for w ∈ {l, p, c}.
In fact such forgetful F-functors lift these adjunctions uniquely – as will follow
upon considering the following simple lifting properties. Recall that a 2-functor
H : A → B reflects identity 2-cells or is locally conservative when it reflects the
property of a 2-cell being an identity or an isomorphism, and is locally faithful if
it reflects the equality of parallel 2-cells. Let us say that an F-functor H : A → B
has any of these three local properties when its loose part Hλ : Aλ → Bλ has
2This lifting property appears biased but of course is not, since the left adjoint of an adjoint
equivalence is equally its right adjoint.
TWO-DIMENSIONAL MONADICITY 11
them: this means that H has these properties with respect to all 2-cells and not
just those between tight morphisms. The following is evident from the definition
of an algebra 2-cell.
Proposition 1. Given w1, w2 ∈ {s, l, p, c} satisfying w1 ≤ w2 the forgetful Ffunctor
U : T-Algw1,w2
→ C reflects identity 2-cells, is locally conservative and
locally faithful. In particular these properties are true of each U : T-Algw → C.
Definition 2. Let w ∈ {l, p, c}. An F-functor H : A → B is said to satisfy wdoctrinal
adjunction if it satisfies the unique form of weak w-doctrinal adjunction.
For instance H : A → B satisfies l-doctrinal adjunction if for each tight arrow
f : A → B ∈ A each adjunction ( , Hf g, η) in B lifts uniquely along H to an
adjunction in A with left adjoint f. Let us note that, since the (−)co duality interchanges
left and right adjoints in an F-category, H satisfies c-doctrinal adjunction
just when Hco satisfies l-doctrinal adjunction. Since adjoint equivalences are fixed
H satisfies p-doctrinal adjunction just when Hco does.
Proposition 3. Let w ∈ {l, p, c} and consider H : A → B.
(1) If H satisfies weak w-doctrinal adjunction and reflects identity 2-cells it satisfies
w-doctrinal adjunction.
(2) If H is locally conservative and satisfies l-doctrinal adjunction or c-doctrinal
adjunction then H satisfies p-doctrinal adjunction.
Proof. (1) To prove the cases w = l and w = p it will suffice to show that any two
adjunctions ( , f g, η) and ( , f g , η ) in A with common left adjoint and
common image under H necessarily coincide in A. Since adjoints are unique
up to isomorphism we have m : g ∼= g given by the composite
B
A A
B
1 GG
f
y
y
y
y
g
YYY{Y{Y{Y{Y{Y{Y{
1
GG
g
YYY{Y{Y{Y{Y{Y{Y{
η 
Using the triangle equations for f g and f g we see that (m.f) ◦ η = η
and that ◦ (f.m) = . Now the image of the 2-cell m under H is an identity
by one of the triangle equations for Hf Hg = Hg . Therefore m is an
identity and g = g . Since m.f and f.m are identities it follows that η = η
and = too. The case w = c is dual.
(2) Suppose that H satisfies l-doctrinal adjunction and is locally conservative.
Then given a tight arrow f ∈ A each adjoint equivalence ( , Hf g, η) in
B lifts uniquely to an adjunction ( , f g , η ) in A. Since H is locally
conservative both and η are invertible because their images are. Therefore
the lifted adjunction is an adjoint equivalence so that H satisfies p-doctrinal
adjunction. The c-case is dual.
Corollary 4. Let w ∈ {l, p, c}. Then U : T-Algw → C satisfies w-doctrinal
adjunction. Furthermore both U : T-Algl → C and U : T-Algc → C satisfy pdoctrinal
adjunction.
12 JOHN BOURKE
Proof. Since U : T-Algw → C satisfies weak w-doctrinal adjunction and reflects
identity 2-cells it follows from Proposition 3.1 that U satisfies w-doctrinal adjunction.
Since it is locally conservative the second part of the claim follows from
Proposition 3.2.
Example 5. In the concrete setting of monoidal categories doctrinal adjunction
is well known. Here we describe only those aspects relevant to our needs: namely,
that the forgetful F-functors V : MonCatw → Cat satisfy w-doctrinal adjunction
for w ∈ {l, p, c}. Consider then a strict monoidal functor F : A → B and an
adjunction of categories ( , F G, η). The right adjoint G obtains the structure of
a lax monoidal functor G = (G, g, g0) : B A with constraints gx,y and g0 given
by the composites
Gx ⊗ Gy GF(Gx ⊗ Gy) G(FGx ⊗ FGy) G(x ⊗ y)
ηGx⊗Gy
GG G( x⊗ y)
GG
and ηiA : iA → GFiA = GiB. It is straightforward to show that, with respect to
these constraints, the natural transformations and η become monoidal transformations.
Therefore we obtain the lifted adjunction ( , F (G, g, g0), η) in MonCatl
whose uniqueness follows, using Proposition 3.1, from the fact that V reflects identity
2-cells; thus V : MonCatl → Cat satisfies l-doctrinal adjunction. The cases
w = p and w = c are entirely analogous.
Note that unless w = l the forgetful F-functor V : MonCatw → Cat does not
satisfy l-doctrinal adjunction. That V : MonCatl → Cat itself does so is due
to the fact that the constraints fa,b : Fa ⊗ Fb → F(a ⊗ b) and f0 : iB → FiA
point in the correct direction – into F – and are not invertible. Whilst l-doctrinal
adjunction captures, to some extent, this laxness it does not determine in any way
the coherence axioms for a lax monoidal functor. This is illustrated by the fact that
there is an F-category of monoidal categories, strict and incoherent lax monoidal
functors (these have components f and f0 oriented as above but satisfying no
equations) and this too sits over Cat via a forgetful F-functor satisfying l-doctrinal
adjunction.
3.2. Reflections and doctrinal F-functors. Although each forgetful F-functor
U : T-Algw → C satisfies w-doctrinal adjunction it turns out that, insofar as this
property characterises such F-functors, the only relevant adjunctions take a more
specialised form. We will begin by treating the cases w = l and w = p, before
treating the case w = c by duality.
Let us call an adjunction (1, f g, η) with tight left adjoint and identity counit
an l-reflection. If, in addition, the unit η is invertible then we call the adjunction
a p-reflection. A p-reflection is of course an adjoint equivalence.
We remark that any adjunction ( , f g, η) is determined by three of its four
parts: in particular the unit η is the unique 2-cell 1 ⇒ g.f satisfying the triangle
equation ( .f) ◦ (f.η) = 1. In a w-reflection (1, f g, η) the unit η is therefore
uniquely determined by the adjoints f g and the knowledge that the counit
is the identity – we can thus faithfully abbreviate a w-reflection (1, f g, η) by
f g if convenient. Whilst we only need to consider liftings of w-reflections we
also need to consider liftings of morphisms of w-reflections. Consider w-reflections
TWO-DIMENSIONAL MONADICITY 13
(1, f1 g1, η1) and (1, f2 g2, η2) and a tight commutative square (r, s) : f1 → f2
as left below.
A C
B D
r GG
f2
f1

s
GG
A C
B D
r GG
g2
yyy
yg1
yyy
y
s
GG
We call (r, s) : f1 → f2 a morphism of w-reflections if the above square of right
adjoints also commutes and furthermore the compatibility with units r.η1 = η2.r
is met. Compatibility with the identity counits is automatic. The following lemma
is sometimes useful for recognising morphisms of w-reflections.
Lemma 6. Let w ∈ {l, p}. Consider w-reflections (1, f1 g1, η1) and (1, f2
g2, η2) and a tight commuting square (r, s) : f1 → f2. Then (r, s) is a morphism
of w-reflections just when its mate m(r, s) : g2.s ⇒ r.g1 is an identity 2-cell.
Proof. It suffices to consider the case w = l. Given a morphism of l-reflections
(r, s) : f1 → f2
A B
C D
r GG
f2

f1

s
GG C
BA B
C D
r GG
s
GG
f1

f2

g1
YYY{Y{Y{Y{Y{Y{Y{
1
GG
1 GG
g2
YYY{Y{Y{Y{Y{Y{Y{
η2 
C
A A
C
B
D
1 GG
f1

g1
YYY{Y{Y{Y{Y{Y{Y{
r GG
1
GG
g1
YYY{Y{Y{Y{Y{Y{Y{
η1 
s
GG
g2
YYY{Y{Y{Y{Y{Y{Y{
consider the mate of the left square: the central composite above. As (r, s) is
a morphism of l-reflections we have η2.r = r.η1 so that the mate reduces to the
rightmost composite: this is an identity by the triangle equation for f1 g1.
Conversely suppose that the mate of (r, s) : f1 → f2 is an identity. Then (r, s) certainly
commutes with the right adjoints so that it remains to verify compatibility
with the units. This is straightforward.
Note that each F-functor preserves morphisms of w-reflections. The following
lifting properties are crucial. For w ∈ {l, p} we say that an F-functor H : A → B
satisfies:
• w-Refl if given a tight arrow f : A → B ∈ A each w-reflection (1, Hf g, η)
lifts uniquely along H to a w-reflection (1, f g , η ) in A.
• w-Morph if given w-reflections (1, f1 g1, η1) and (1, f2 g2, η2) a tight
commuting square (r, s) : f1 → f2 is a morphism of w-reflections just when
its image (Hr, Hs) : Hf1 → Hf2 is one.
We say that H : A → B satisfies c-Refl or c-Morph if Hco : Aco → Bco satisfies lRefl
or l-Morph respectively. We remark that these c-variants concern c-reflections
which are adjunctions with tight right adjoint and identity unit.
Definition 7. Let w ∈ {l, p, c}. An F-functor H : A → B is said to be w-doctrinal
if it satisfies w-Refl, w-Morph and is locally faithful. We denote the class of wdoctrinal
F-functors by w-Doct.
The condition that W be locally faithful may seem somewhat unnatural – see
the discussion after Theorem 21 for our reasons for including it.
Evidently we have that H is c-doctrinal just when Hco is l-doctrinal. Furthermore
14 JOHN BOURKE
H is p-doctrinal just when Hco is p-doctrinal. Let us now compare these lifting
properties with those of Section 3.1.
Lemma 8. Let w ∈ {l, p, c} and consider H : A → B.
(1) If H satisfies w-doctrinal adjunction and reflects identity 2-cells then it satisfies
w-Refl and w-Morph.
(2) If H satisfies w-doctrinal adjunction, reflects identity 2-cells and is locally
faithful then it is w-doctrinal.
(3) If H is locally conservative, reflects identity 2-cells, is locally faithful and
satisfies either l or c-doctrinal adjunction then it is p-doctrinal.
Proof. We will only consider the l-case of (1) and (2), all being essentially identical,
with the l and c cases dual.
(1) Consider a tight morphism f : A → B ∈ A and l-reflection (1, Hf g, η).
Because H satisfies l-doctrinal adjunction this lifts uniquely to an adjunction
( , f g , η ) in A. Since H = 1 and H reflects identity 2-cells is an
identity. This verifies l-Refl. For l-Morph consider l-reflections (1, f1 g1, η1)
and (1, f2 g2, η2) in A and a tight morphism (r, s) : f1 → f2 such that
(Hr, Hs) : Hf1 → Hf2 is a morphism of l-reflections; we must show (r, s) :
f1 → f2 is one too. By Lemma 6 this is equally to say that the mate mr,s :
r.g1 ⇒ g2.s of this square is an identity 2-cell, so giving a commutative square
(s, r) : g1 → g2. So it will suffice to check Hmr,s is an identity. But Hmr,s =
mHr,Hs which is an identity since (Hr, Hs) is a morphism of l-reflections.
(2) Since w-doctrinal just means w-Refl and w-Morph together with local faithfulness
this follows immediately from Part 1.
(3) Since by Proposition 3.2 such an F-functor satisfies p-doctrinal adjunction
this follows from Part 2.
Corollary 9. Let T be a 2-monad on C. For each w ∈ {l, p, c} the forgetful Ffunctor
U : T-Algw → C is w-doctrinal. Furthermore both U : T-Algl → C and
U : T-Algc → C are p-doctrinal.
Proof. Since U : T-Algw → C satisfies w-doctrinal adjunction, reflects identity 2cells
and is locally conservative the first claim follows from Lemma 8.2; since each
such U is also locally conservative the second claim follows from Lemma 8.3.
Let us conclude by mentioning a further evident example of w-doctrinal F-
functors.
Proposition 10. Let w ∈ {l, p, c}. If H : A → B is such that Hλ : Aλ → Bλ is
2-fully faithful then H is w-doctrinal for each w. In particular each equivalence of
F-categories is w-doctrinal for each w.
3.3. A small orthogonality class. Though not strictly necessary in what follows
let us remark that, with the exception of weak w-doctrinal adjunction, all of the
lifting/reflection properties so far considered are expressible as orthogonal lifting
properties in F-CAT. Certainly it is not hard to see that this is true of the property
of reflecting identity 2-cells, or of being locally faithful or locally conservative.
Less obvious is that this is true of the notion of w-doctrinal adjunction or of the
conditions w-Refl and w-Morph, in particular of the condition w-Morph concerning
TWO-DIMENSIONAL MONADICITY 15
liftings of morphisms of adjunctions. We describe the conditions l-Refl and l-Morph
here – the p and c cases being similar. To this end consider the following F-category
Adjl depicted in its entirety on the left below
0
0
1
1
1

f
GG
g
zz zX zX zX zX zX zX zX
f
GG
1

η
CQ
2 Adjl
A B
j
GG
 
H
GGzz
F-CAT(Adjl, A) F-CAT(Adjl, B)
F-CAT(2, A) F-CAT(2, B)
H∗
GG
j∗

j∗

H∗
GG
where g.f is loose, f.g = 1, f.η = 1 and η.g = 1. This is the free adjunction with
identity counit and tight left adjoint f. It has a single non-identity tight arrow f
so that Adjlτ equals the free tight arrow 2; therefore the inclusion Adjlτ → Adjl
is the F-functor j : 2 → Adjl selecting f. Now to give a commutative square as in
the middle above is to give a tight arrow f ∈ A and adjunction (1, Hf g, η) in B.
To give a filler is to lift the adjunction along H to an adjunction (1, f g , η ) in
A; thus H is orthogonal to j when condition l-Refl is met. The conditions l-Refl
and l-Morph together assert exactly that j and H are orthogonal in F-CAT as a 2category
– this means that the right square above is a pullback in CAT. In fact by
a standard argument the conditions l-Refl and l-Morph jointly amount to ordinary
(1-categorical) orthogonality against the single F-functor j × 1 : 2 × 2 → Adjl × 2,
of which j is a retract.
Therefore the class of w-doctrinal F-functors, w-Doct, forms an orthogonality class
in the category of F-categories and F-functors. The following section is geared
towards establishing sufficient conditions on an F-category A under which the
inclusion j : Aτ → A belongs to ⊥w-Doct – is orthogonal to each w-doctrinal
F-functor.
4. Loose morphisms as spans
We now consider completeness properties of F-categories appropriate to those
of the form T-Algw. In an F-category A with such completeness properties we
can represent loose morphisms in A by certain tight spans. In the lax setting, for
instance, each loose morphism f : A B is represented as a tight span Cf : A → B
CfA B
pf rf
oo
qf
GG
by taking its colax limit. The tight left leg pf : Cf → A is part of an l-reflection
(1, pf rf , ηf ) from which f can be recovered as the composite qf .rf : A Cf →
B. In the present section we analyse this representation of loose morphisms by
tight spans in detail, beginning with a discussion of the relevant limits.
4.1. Limits of loose morphisms. The main reason that F-categories were introduced
in [16] was to capture the behaviour of limits in 2-categories of the form
T-Algw for w ∈ {l, p, c}. In such 2-categories many 2-categorical limits have an
F-categorical aspect – namely, that the projections from the limit are strict and
jointly detect strictness. Interpreted in the F-category T-Algw this asserts that
the limit projections are tight and jointly detect tightness. This latter property is
exactly that which distinguishes F-categorical limits (in the sense of F-enriched
category theory) from 2-categorical ones (see Proposition 3.6 of [16]). We will
16 JOHN BOURKE
only need a few basic F-categorical limits and our treatment, in what follows, is
elementary.
There are three limits to consider here – colax, pseudo and lax limits of loose morphisms
– these correspond, in turn, to lax, pseudo and colax morphisms. As we
always lead with the case of lax morphisms we focus here primarily on colax limits.
Given a loose morphism f : A B ∈ A its (colax/pseudo/lax)-limit consists of
an object Cf /Pf /Lf and a (colax/pseudo/lax)-cone (pf , λf , qf ) as below
Cf
A B
pf
 qf
11cccccccc
f
GGGoGoGoGoGoGoGoGoGo
λf 
Pf
A B
pf
 qf
11cccccccc
f
GGGoGoGoGoGoGoGoGoGo
∼=λf
Lf
A B
pf
 qf
11cccccccc
f
GGGoGoGoGoGoGoGoGoGo
λf
uƒ
with both projections pf and qf tight.
The colax limit Cf
3
is required to be the usual 2-categorical colax limit of an arrow
[13] in Aλ: this means that it has the following two universal properties.
(1) Given any colax cone (r, α, s) as below
X
A B
r
 c c c c c
s
111•1•1•1•1•
f
GGGoGoGoGoGoGoGoGoGo
α 
there exists a unique t : X Cf satisfying
pf .t = r, qf .t = s and λf .t = α
(2) Given a pair of colax cones (r, α, s) and (r , α , s ) with common base X together
with 2-cells θr : r ⇒ r ∈ Aλ(X, A) and θs : s ⇒ s ∈ Aλ(X, B)
satisfying
s s
fr fr
θs
CQ
αα

fθr
CQ
there exists a unique 2-cell φ : t ⇒ t ∈ Aλ(X, Cf ) between the induced
factorisations such that
pf .φ = θr and qf .φ = θs .
For Cf to be the colax limit of f in the F-categorical sense we must also have
(3) A morphism t : X Cf is tight just when pf .t and qf .t are tight – the
projections jointly detect tightness.
In the case of the pseudolimit Pf the 2-cell λf is required to be invertible. If we
call those colax cones with an invertible 2-cell pseudo-cones then the universal
properties of (1) and (2) above are only changed by replacing colax cones by
pseudo-cones – thus (pf , λf , qf ) is the universal pseudo-cone. The F-categorical
3Colax limits of arrows are usually called oplax limits of arrows. We prefer colax limits here
since they are lax limits in Aco
rather than Aop
and, similarly, sit better with our usage of lax
and colax morphisms.
TWO-DIMENSIONAL MONADICITY 17
aspect of (3) remains the same. The lax limit of f is simply the colax limit in Aco.
For w ∈ {l, p, c} let us write w-limit of a loose morphism as an abbreviation, so
that c-limit stands for colax limit, for instance. Now we mentioned above that lax
morphisms correspond to colax limits and so on. To capture this let us set l = c,
¯p = p and ¯c = l as in [16]: the correspondence is captured by the following result.
Proposition 11. Let w ∈ {l, p, c}. The forgetful F-functor U : T-Algw → C
creates w-limits of loose morphisms.
In its F-categorical formulation above this is a specialisation of Theorem 5.13
of [16] which characterises those limits created by the forgetful F-functors U :
T-Algw → C. However for w-limits of loose morphisms, as concern us, the result
goes back to [3] and [13].
Because lax limits are colax limits in Aco we can and will avoid them entirely. In
order to work with colax and pseudolimits simultaneously let us introduce a final
piece of notation. For w ∈ {l, p} we will use
¯Wf
A B
pf
 qf
11cccccccc
f
GGGoGoGoGoGoGoGoGoGo
λf 
to denote the w-limit of f and universal w-cone: so Cf and its colax cone when
w = l and Pf when w = p. When w = p the 2-cell λf should be interpreted as
invertible.
Example 12. In Cat the colax limit of a functor F : A → B is given by the
comma category B/F: this has objects (x, α : x → Fa, a) and morphisms (r, s) :
(x, α, a) → (y, β, b) given by pairs of arrows r : x → y ∈ B and s : a → b ∈ A
rendering commutative the square on the left.
x Fa
y Fb
α GG
β
GG
r

Fs

B/F
A B
p
||yyyyyyy q
44iiiiiii
F
GG
λ 
The projections p : B/F → A and q : B/F → B of the colax cone (p, λ, q) act on
a morphism (r, s) of B/F as p(r, s) = s : a → b and q(r, s) = r : x → y; the value
of λ : q ⇒ pF at (x, α, a) is simply the morphism α : x → Fa itself.
The pseudolimit of F is the full subcategory of B/F whose objects are those pairs
(α : x → Fa, a) with α invertible, whilst the lax limit of F is the comma category
F/B.
Example 13. It is illuminating to consider the colax limit of a lax monoidal
functor F = (F, f, f0) : A B. The forgetful F-functor U : MonCatl → Cat
creates these limits: to see how this goes first consider the colax limit of the functor
F, the comma category B/F equipped with its colax cone (p, λ, q) described above.
The crux of the argument is to show that this lifts uniquely to a colax cone in
MonCatl: that B/F admits a unique monoidal structure such that p and q become
strict monoidal and λ a monoidal transformation.
18 JOHN BOURKE
So consider two objects (x, α, a) and (y, β, b) of B/F: if p and q are to be strict
monoidal the tensor product (x, α, a) ⊗ (y, β, b) must certainly be of the form (x ⊗
y, θ, a⊗b); furthermore the tensor condition for λ to be a monoidal transformation
interpreted at this pair asserts precisely that (x, α, a) ⊗ (y, β, b) equals
x ⊗ y Fa ⊗ Fb F(a ⊗ b)
α⊗β
GG
fa,b
GG
Likewise the unit condition for a monoidal transformation forces us to define the
unit of B/F to be (f0 : iB → FiA, iA). For p and q to preserve tensor products of
morphisms we must define the tensor product as (r, s) ⊗ (r , s ) = (r ⊗ r , s ⊗ s )
at morphisms of B/F – to say the resulting pair is a morphism of B/F is then to
say that the following square is commutative.
x ⊗ y Fa ⊗ Fb F(a ⊗ b)
x ⊗ y Fa ⊗ Fb F(a ⊗ b )
α⊗β
GG
fa,b
GG
α ⊗β
GG
fa ,b
GG
r⊗r

F(s⊗s )

Fs⊗Fs

The left square trivially commutes and the right square commutes by naturality
of the fa,b. It remains to give the associator and the left and right unit constraints
for the monoidal structure on B/F – this is where the coherence axioms for a
lax monoidal functor finally come into play. Certainly if p and q are to be strict
monoidal they must preserve the associators strictly: this means that the associator
at a triple of objects ((x, α, a), (y, β, b), (z, γ, c)) of B/F must be given by
(λB
x,y,z, λA
a,b,c). To say that this is a morphism of B/F is equally to say that the
composite square
(x ⊗ y) ⊗ z (Fa ⊗ Fb) ⊗ Fc F(a ⊗ b) ⊗ Fc F((a ⊗ b) ⊗ c)
x ⊗ (y ⊗ z) Fa ⊗ (Fb ⊗ Fc) Fa ⊗ F(b ⊗ c) F(a ⊗ (b ⊗ c))
(α⊗β)⊗γ
GG
fa,b⊗1
GG
fa⊗b,c
GG
α⊗(β⊗γ)
GG
1⊗fb,c
GG
fa,b⊗c
GG
λB
x,y,z

λB
F a,F b,F c

FλA
a,b,c

is commutative. The left square commutes by naturality of the associators in
B with the right square asserting exactly the associativity condition for a lax
monoidal functor. Similarly the left and right unit constraints at (x, α, a) must
be given by (ρB
l x, ρA
l a) and (ρB
r x, ρA
r a) – that these lift to isomorphisms of B/F
likewise correspond to the left and right unit conditions for a lax monoidal functor.
Having given the monoidal structure for B/F it remains to check it verifies the
axioms for a monoidal category, but all of these clearly follow from the corresponding
axioms for A and B because p and q preserve the structure strictly and are
jointly faithful.
Finally one needs to verify that this uniquely lifted colax cone in MonCatl satisfies
the universal property of the colax limit of (F, f, f0) therein. That p and q jointly
detect tightness follows from the fact that they jointly reflect identity arrows –
from here it is straightforward to verify that B/F has the universal property of
the colax limit in MonCatl. For w ∈ {p, c} one constructs the ¯w-limit of a loose
morphism in MonCatw in an entirely similar way.
TWO-DIMENSIONAL MONADICITY 19
Observe that in lifting the colax cone (p, λ, q) to MonCatl we used all of the
coherence axioms for a lax monoidal functor, and indeed these generating coherence
axioms are required for the colax cone to lift. Thus while l-doctrinal adjunction
is related to the laxness – orientation and non-invertibility – of our lax monoidal
functors, colax limits of loose morphisms concern the coherence axioms these lax
morphisms must satisfy.
4.2. Loose morphisms as tight spans. For w ∈ {l, p} we suppose that A admits
¯w-limits of loose morphisms. Then given f : A B we have the commutative
triangle on the left below.
A
A B
1

f
111•1•1•1•1•
f
GGGoGoGoGoGoGoGoGoGo
=
A
¯Wf
A B
rf

y
y
y
1

f

6d 4˜ 2– 0” '‘ %‰
#‡
!…
„
‚

y
x
w
pf
 qf
11cccccccc
f
GGGoGoGoGoGoGoGoGoGo
λf 
By the universal property of ¯Wf we obtain a unique 1-cell rf : A ¯Wf satisfying
pf .rf = 1, qf .rf = f and λf .rf = 1 (4.1)
as expressed in the equality of pasting diagrams above. Since pf and qf jointly
detect tightness we also have that
rf is tight just when f is. (4.2)
Proposition 14. Given f : A B as above we have a w-reflection (1, pf rf , ηf )
where ηf : 1 ⇒ rf pf is the unique 2-cell satisfying
pf .ηf = 1 and qf .ηf = λf . (4.3)
Proof. Let us consider firstly the case w = l. We need to give a unit ηf : 1 ⇒ rf .pf .
To give such a 2-cell is, by the 2-dimensional universal property of Cf , equally
to give 2-cells θ1 : pf .(1) ⇒ pf .(rf .pf ) and θ2 : qf .(1) ⇒ qf .(rf .pf ) satisfying
θ1 ◦λf = (λf .rf .pf )◦θ2. We take θ1 to be the identity and θ2 to be λf : qf ⇒ f.pf ;
the required equality involving θ1 and θ2 is then the assertion that λf equals itself.
We thus obtain a unique ηf : 1 ⇒ rf .pf such pf .ηf = 1 and qf .ηf = λf . If the
identity 2-cell pf .qf = 1 is to be the counit of the adjunction then the triangle
equations become pf .ηf = 1 and ηf .rf = 1. So it remains to check that ηf .rf = 1
for which it suffices, again by the 2-dimensional universal property of Cf , to show
that pf .ηf .rf = 1 and qf .ηf .rf = 1. The first of these holds since pf .ηf = 1; the
second since qf .ηf = λf and λf .rf = 1.
The case w = p is essentially identical – the key point is that the 2-cells θ1 = 1
and θ2 = λf used above to construct ηf are now both invertible. That ηf is
itself invertible follows from the fact that pf and qf are jointly conservative – this
conservativity follows from the 2-dimensional universal property of Pf .
The above constructions have their genesis in the proof of Theorem 4.2 of [3],
in which F-categorical aspects of pseudolimits of arrows in T-Algp were used to
study establish properties of pseudomorphism classifiers. If we ignore F-categorical
20 JOHN BOURKE
aspects then the above constructions and resulting factorisations f = qf .rf have
appeared in other contexts too. In the pseudolimit case the factorisation is the
(trivial cofibration, fibration)-factorisation of the natural model structure on a 2category
[14]. In Cat the factorisation qf .rf : A → Cf → B of a functor f through
its colax limit coincides with its factorisation A → B/f → B through the comma
category B/f – this is the factorisation (Lf, Rf) of a natural weak factorisation
system on Cat described in [5].
Let us return to f : A B as in Proposition 14. By that result we have a span
of tight morphisms (a tight span) as below:
A ¯Wf B
pf rf
oo
qf
GG
whose left leg pf is equipped with the structure of a w-reflection (1, pf rf , ηf ).
More generally let us use the term w-span to refer to a tight span equipped with
the structure of a w-reflection on its left leg. Now let ¯Wf : A → B denote the
w-span just described. Given f : A B and g : B C we are going to show that
¯Wf : A → B and ¯Wg : B → C can be composed to give a w-span ¯Wg
¯Wf : A → C
and furthermore we will study the relationship between ¯Wg
¯Wf and ¯Wgf .
In order to consider composition of such spans we will need to consider tight
pullbacks. Given tight morphisms f : A → C and g : B → C in A the tight
pullback D of f and g
D A
B C
p
GG
g
GG
q
 f

is the pullback in the 2-category Aλ with, moreover, both projections p and q tight
and jointly detecting tightness. This is equally to say that D is a pullback in Aτ
which is preserved by the inclusion j : Aτ → Aλ.
Lemma 15. Let w ∈ {l, p} and A admit ¯w-limits of loose morphisms. At f : A
B consider the induced tight projection pf : ¯Wf → A from the limit. The tight
pullback of pf along any tight morphism g : C → A exists.
Proof. In fact the tight pullback is given by ¯Wfg, the w-limit of the composite
fg : C A. For observe that by the universal property of ¯Wf the w-cone
(g.pfg, λfg, qfg) induces a unique tight map t : ¯Wfg → ¯Wg such that the left
square below commutes
¯Wfg
¯Wf B
C A B
t GG
qf
GG
g
GG
f
GGGoGoGoGoGo
pfg

1

pf

λf
ÐØ 
and such that λf .t = λgf . Now the universal property of ¯Wfg implies that the left
square is a tight pullback.
TWO-DIMENSIONAL MONADICITY 21
Lemma 16. For w ∈ {l, p} consider a tight pullback square
A B
C D
r GG
s
GG
f1
 f2

and w-reflection (1, f2 g2, η2). There exists a unique w-reflection (1, f1 g1, η1)
such that (r, s) : f1 → f2 is a morphism of w-reflections.
Proof. First suppose that w = l. Then the left square below
C B
C D
g2.s
GGGoGoGo
s
GG
1
 f2

C D
A B
s GG
r
GG
g1
y
y g2
y
y
commutes and so induces a unique loose morphism g1 : C A such that f1.g1 = 1
and such that the right square commutes. These necessary commutativities will
ensure the claimed uniqueness. By the 2-dimensional universal property of the
pullback there exists a unique 2-cell η1 : 1 ⇒ g1.f1 such that r.η1 = η2.r and
f1.η1 = 1. The other triangle equation η1.f1 = 1 also follows from the universal
property of the pullback. Since (r, s) commutes with both adjoints and the units it
is a morphism of l-reflections. Note that if η2 is invertible then, since the pullback
projections are jointly conservative, so too is η1. This gives the case w = p.
Now given f : A B and g : B C we can form the composite span
A
¯Wf
B
¯Wg
C
¯Wg
¯Wf
pf rf

qf
11cccccccc
pg rg
 qg
11cccccccc
pg,f rg,f
 qg,f
11ccccccc
in which the central square is the tight pullback of pg along qf . (This pullback exists
by Lemma 15). By Lemma 16 there exists a unique w-reflection (1, pg,f rg,f , ηg,f )
such that
(qg,f , qf ) : pg,f → pg is a morphism of w-reflections. (4.4)
We can then compose the w-reflections pf rf and pg,f rg,f to obtain another
w-reflection pf .pg,f rg,f .rf , so that the outer span becomes a w-span
¯Wg
¯Wf : A → C.
22 JOHN BOURKE
Let us consider the relationship between ¯Wg
¯Wf and ¯Wgf . By the universal property
of ¯Wgf the w-cone left below
¯Wg
¯Wf
¯Wf
A B
pf
 qf
11cccccccc
f
GGGoGoGoGoGoGoGoGoGo
λf 
¯Wg
C
pg
 qg
11cccccccc
g
GGGoGoGoGoGoGoGoGoGo
λg 
pg,f
 qg,f
11ccccccc
=
¯Wg
¯Wf
¯Wf
A B
pf

f
GGGoGoGoGoGoGoGoGoGo
λgf 
¯Wg
C
qg
11cccccccc
g
GGGoGoGoGoGoGoGoGoGo
pg,f
 qg,f
11ccccccc
¯Wgf
kg,f

pgf
wwooooooooooooooo
qgf
99yyyyyyyyyyyyyyy
induces a unique tight arrow kg,f : ¯Wg
¯Wf → ¯Wgf satisfying the equations
pgf .kg,f = pf .pg,f , qgf .kg,f = qg.qg,f and λgf .kg,f = (g.λf .pg,f ) ◦ (λg.qg,f ) (4.5)
equally expressed in the equality of pasting diagrams above. Furthermore
Lemma 17. Let w ∈ {l, p}. In the span map
A
A ¯Wg
¯Wf
¯Wgf B
B
pf .pg,f
oo
pgf
oo
qg.qg,f
GG
qgf
GG
kg,f
1

1

the commuting square (kg,f , 1) : pf .pg,f → pgf is a morphism of w-reflections.
Proof. To show that (kg,f , 1) : pf .pg,f → pgf is a morphism of w-reflections it
suffices, by Lemma 6, to show that the mate of this square is an identity. Because
the w-reflection pf .pg,f rg,f .rf has identity counit the mate of (kg,f , 1) is simply
A
¯Wgf
¯Wg
¯Wf
¯Wgf
A A
kg,f
GG
1
GG
pf .pg,f

pgf

rg,f .rf
YYY{Y{Y{Y{Y{Y{
1
GG
1 GG
rgf
YYY{Y{Y{Y{Y{Y{
ηgf "0
RRR
RRR
Now the 2-dimensional universal property of ¯Wgf implies that the projections
pgf and qgf jointly reflect identity 2-cells: see Lemma 3.1 of [13] in the case of
the colax limit. Therefore it suffices to show that the composite of the above
2-cell with both pgf and qgf yields an identity. One of the triangle equations
for pgf rgf gives pgf .ηgf =1; thus it remains to show qgf .ηgf .kg,f .rg,f .rf = 1.
By (4.3) this equals λgf .kg,f .rg,f .rf and by definition of kg,f (as in (4.5)) we have
λgf .kg,f = (g.λf .pg,f ) ◦ (λg.qg,f ). Therefore it suffices to show that the 2-cells
(g.λf .pg,f ).rg,f .rf and (λg.qg,f ).rg,f .rf are identities separately. With regards the
former we have that λf .pg,f .rg,f .rf = λf .rf = 1 where we first use that pg,f .rg,f = 1
and then (4.1); for the other composite we have (λg.qg,f ).rg,f .rf = λg.rg.qf .rf = 1.
The first equation holds by (4.4) and the second equation by (4.1).
Remark 18. Given any F-category A one can define a bicategory Spanw(A) of
w-spans in A. Their composition extends that described for w-spans of the form
¯Wf : A → B above. To ensure composites exist one allows only those w-spans
TWO-DIMENSIONAL MONADICITY 23
whose left legs admit tight pullbacks along arbitrary tight maps. Now when A
admits w-limits of loose morphisms the assigment of ¯Wf : A → B to f can be
extended to an identity on objects lax functor from the underlying category of
Aλ to Spanw(A) with the span map kg,f : ¯Wg
¯Wf → ¯Wgf describing one of the
coherence constraints.
4.3. Representing 2-cells via span transformations. Lastly we consider how
to represent a 2-cell α : f ⇒ g by a span map ¯Wf → ¯Wg. Here the cases w = l
and w = p diverge.
4.3.1. The case w=l. Suppose that A admits colax limits of loose morphisms.
Given α : f ⇒ g the colax cone left below
Cf
A B
pf

qf
11cccccccccccc
f
AARt Qs Pr Pr Iq Hp Go Fn Em Dl Dl Ck Bj
α 
g
SSBj Ck Dl Dl Em Fn Go Hp Iq Pr Pr Qs Rt
λf  =
Cf
Cg
A B
cα
pf

qf

pg
{{wwwwwwwwww
qg
55qqqqqqqqqq
g
GGGoGoGoGoGoGoGoGoGoGoGoGoGo
λg 
induces a unique tight arrow cα : Cf → Cg satisfying
pf = cα.pg, qf = cα.qg and (α.pf ) ◦ λf = λg.cα (4.6)
In particular the 2-cell α is represented by a span map cα : Cf → Cg as below.
A
A Cf
Cg B
B
pf
oo
pg
oo
qf
GG
qg
GG
cα

1

1

Lemma 19. Let cα : Cf → Cg be as above and let mα denote the mate of the
square (cα, 1) : pf → pg through the adjunctions pf rf and pg rg.
A
A Cf
BCg
B
rf
GGGoGoGoGoGo
rg
GGGoGoGoGoGo
qf
GG
qg
GG
cα

mα1

1

The composite 2-cell qg.mα equals α.
Proof. Because the counit of pf rf is an identity the mate mα is simply given
by
A Cf
A Cg
A
Cg
cα

rg
GGGoGoGoGoGoGoGoGoGo
1

rf
GGGoGoGoGoGoGoGoGoGo
1

pf
wwoooooooooooooooo
pg
wwoooooooooooooooo
1

ηg
s{ oooo
24 JOHN BOURKE
Therefore qg.mα = qg.ηg.cα.rf = λg.cα.rf = (α.pf .rf ) ◦ (λf .rf ) = α ◦ 1 = α where
the second, third and fourth equalities use (4.3), (4.6) and (4.3) respectively.
4.3.2. The case w=p. Suppose that A admits pseudolimits of loose morphisms.
If α : f ⇒ g is invertible then we may construct a map Pf → Pg in essentially
the same way as for colax limits of loose morphisms. However this approach
does not work in general. The following lemma describes a representation that
works for non-invertible 2-cells. It is based upon the notion of a transformation
of anafunctors [19] – see also [21] and [1]. Anafunctors can be viewed as spans of
categories and functors in which the left leg is a surjective on objects equivalence.
If we take A = Cat and functors F, G : A → B then the associated spans PF , PG :
A → B are anafunctors. In this setting the 2-cell ρα described below specifies
precisely a transformation between the anafunctors PF and PG.
Lemma 20. Given α : f ⇒ g consider the tight pullback left below (existing by
Lemma 15)
Kf,g Pf
Pg A
sf,g
GG
pg
GG
tf,g

pf

uf,g
11cccccccccccc
A
Kf,g Pf
Pg A
vf,g
555™5™5™5™5™5™ rf
22
Fn Fn Em Dl Ck Bj Ai @h 9g 8f 7e 6d 4˜ 3—
rg
55
ƒ
„
!…
"†
#‡
$ˆ
&
'‘
)“ 0” 2– 4˜ 5™
sf,g
GG
qf

tf,g

qg
GG
ρα
{Ó 
with diagonal denoted uf,g.
(1) There is a unique p-reflection uf,g vf,g such that (sf,g, 1) : uf,g → pf and
(tf,g, 1) : uf,g → pg are morphisms of p-reflections. If f and g are tight so too
is vf,g.
(2) There exists a unique 2-cell ρα : qf .sf,g ⇒ qg.tf,g such that ρα.vf,g = α.
Proof. (1) Consider the diagram
A
Kf,g Pf
Pg A
vf,g
555™5™5™5™5™5™ rf
55
Dl Dl Ck Bj Bj Ai @h 9g 9g 8f 7e 6d 5™
rg
00
"†
#‡
#‡
$ˆ
$ˆ
%‰
&
&
'‘
(’ )“ )“ 0”
sf,g
GG
pf

tf,g

pg
GG
in which vf,g is the unique loose map satisfying sf,g.vf,g = rf and tf,g.vf,g = rg.
It follows that uf,g.vf,g = 1. To give an invertible 2-cell θf,g : 1 ∼= vf,g.uf,g
is, by the universal property of Kf,g, equally to give invertible 2-cells θ1 :
sf,g
∼= sf,g.vf,g.uf,g and θ2 : tf,g
∼= tf,g.vf,g.uf,g satisfying pf .θ1 = pg.θ2. We
set θ1 = ηf .sf,g and θ2 = ηg.tf,g noting that pf .ηf .sf,g = 1 = pg.ηg.sf,g. The
triangle equations for the p-reflection follow using the universal property of
the pullback Kf,g and that sf,g : uf,g → pf and tf,g : uf,g → pg are morphisms
of p-reflections follows from the construction of uf,g and θf,g.
TWO-DIMENSIONAL MONADICITY 25
If f and g are tight so too, by (4.2), are rf and rg. By the universal property
of the tight pullback Kf,g it then follows that vf,g is tight.
(2) From the first part we have sf,g.vf,g = rf and tf,g.vf,g = rg as in the two
triangles above. Now by (4.3) we have qf .rf = f and qg.rg = g. Therefore
we can write α : (qf .sf,g).vf,g ⇒ (qg.tf,g).vf,g. Since vf,g : A Kf,g is an
equivalence in Aλ the functor Aλ(vf,g, A) : Aλ(Kf,g, A) → Aλ(A, A) is an
equivalence of categories – using its fully faithfulness we obtain ρα : qf .sf,g ⇒
qg.tf,g.
5. Orthogonality
The following theorem is the crucial result of the paper. The monadicity theorems
of Section 6 follow easily from it. We note that both this theorem and the
corollary that follows it are independent of the formalism of 2-monads.
Theorem 21. Let w ∈ {l, p, c}. Consider an F-category A with ¯w-limits of loose
morphisms. Then the inclusion of tight morphisms j : Aτ → A is orthogonal to
each w-doctrinal F-functor.
Proof. Consider a commuting square in F-CAT
Aτ A
B C
j
GG
R
 S

H
GG
K
ww
in which H is w-doctrinal. We must show there exists a unique diagonal filler K.
We begin by noting that the cases w = c and w = l are dual since A satisfies
the c-criteria of the theorem just when Aco satisfies the l-criteria with, equally,
H c-doctrinal just when Hco is l-doctrinal. Therefore it will suffice to suppose
w ∈ {l, p}.
(1) Before constructing the diagonal we fix some notation and make some observations
about lifted adjunctions that will be repeatedly used in what follows.
Given a w-reflection (1, f g, η) ∈ A we obtain a w-reflection (1, Sf Sg, Sη)
in C with Sf = Hf) since f is tight. As H is w-doctrinal this lifts uniquely
along H to a w-reflection in B which we denote by (1, Rf g, η).
Next consider w-reflections (1, f1 g1, η1) and (1, f2 g2, η2) in A and a tight
commuting square (r, s) : f1 → f2 in A as left below
A C
B D
r GG
s
GG
f1
 f2

RA RC
RB RD
Rr GG
Rs
GG
Rf1
 Rf2

RB RD
RA RC
g1
y
y
g2
y
y
Rr
GG
Rs GG
Since F-functors preserves morphisms of w-reflections the square (Sr, Ss) :
Sf1 → Sf2 is one in C. Now the tight commutative square (Rr, Rs) : Rf1 →
Rf2 has image under H the morphism of w-reflections (Sr, Ss) : Sf1 → Sf2.
Because H is w-doctrinal it follows that (Rr, Rs) : Rf1 → Rf2 is a morphism
between the lifted w-reflections in B. In particular we obtain a commuting
square of right adjoints (Rs, Rr) : g1 → g2.
Finally consider a composable pair of tight left adjoints f1 : A → B and
26 JOHN BOURKE
f2 : B → C with associated w-reflections (1, f1 g1, η1) and (1, f2 g2, η2)
in A. We can form the w-reflections (1, Rf1 g1, η1) and (1, Rf2 g2, η2) in
B and compose these to obtain a further w-reflection f2.f1 g1.g2 in B. It is
clear that this is a lifting of the w-reflection S(f2.f1) S(g1.g2) so that, by
uniqueness of liftings, the w-reflections Rf2.Rf1 g1.g2 and R(f2.f1) g1.g2
coincide.
(2) Now to begin constructing K observe that for the left triangle to commute
we must define KA = RA for each A ∈ A.
(3) Given f : A B ∈ A we recall from (4.1) its factorisation as qf .rf : A
Cf → B where (1, pf rf , ηf ). Since pf is tight we have the lifted wreflection
(1, Rpf rf , ηf ) in B living over (1, Spf Srf , Sηf ). We define
Kf : RA RB as the composite Rqf .rf : RA R ¯Wf → RB.
(4) Observe that HKf = HRqf .Hrf = Sqf .Srf = S(qf .rf ) = Sf as required.
To see that K extends R observe that if f : A → B is tight then, by (4.2),
rf is tight too, so that we have a w-reflection (1, Rpf Rrf , Rηf ) living over
(1, HRpf Srf , ηf ); thus (1, Rpf Rrf , Rηf ) = (1, Rpf rf , ηf ) so that
Kf = Rqf .Rrf = Rf. Thus K coincides with R on tight morphisms.
(5) As K agrees with R on tight morphisms we already know that it preserves
identity 1-cells. To see that it preserves composition of 1-cells it will suffice
to show that all of the regions of the diagram below commute.
R ¯Wgf
RA RB
R ¯Wf
Rqf 44hhhhhhh
R ¯Wg
RC
Rqg 99yyyyyyyyyy
rg
```|`|`|`|
Rqgf
))YYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
R ¯Wg,f
Rkg,f
yy
rf
UUUwUwUwUwUwUw
rgf
eeeÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
eÑ
rg,f
```|`|`|
Rqg,f
44hhhhhh
The rightmost quadrilateral certainly commutes as it is the image of a commutative
diagram in Aτ , from (4.5), under R. To see that the central square
commutes recall from (4.4) the morphism of w-reflections (qg,f , qf ) : pg,f → pg
in A. By Part 1 (Rqg,f , Rqf ) : Rpg,f → pg,f is a morphism of w-reflections
in B: now commutativity of the central square simply asserts commutativity
with the right adjoints. With regards the leftmost quadrilateral recall from
Lemma 17 the morphism of w-reflections in A given by (kg,f , 1) : pf .pg,f →
pgf . By Part 1 we know that (Rkg,f , 1) : R(pf .pg,f ) → Rpgf is a morphism
of the lifted w-reflections so that, in particular, we have a commuting square
of right adjoints (1, Rkg,f ) : rg,f .rf → rgf in B. Again by Part 1 we have
rg,f .rf = rg,f rf and therefore the desired commutativity.
(6) We define K differently on 2-cells depending upon whether w = l or w = p.
Consider α : f ⇒ g and the case w = l. The commuting square (cα, 1) : pf →
pg of Lemma 19 has image (Rcα, 1) : Rpf → Rpg. We denote the mate of this
TWO-DIMENSIONAL MONADICITY 27
square through the adjunctions Rpf rf and Rpg rg by mα : Rcα.rf ⇒ rg.
RA
RA RCf
RBRCg
B
rf
GGGoGoGoGo
rg
GGGoGoGoGo
Rqf
GG
Rqg
GG
Rcα

mα1

1

Now we set Kα = Rqg.mα as depicted above. Note that we have mα =
ηg.Rcα.rf since the counit of the l-reflection Rpf rf is an identity.
Let us show that HKα = Sα. We have from Lemma 19 the decomposition in
A of α as qg.mα = qg.ηg.cα.rf and also have Kα = Rqg.ηg.Rcα.rf from above.
Thus HKα = Sqg.Sηg.Scα.Srf = S(qg.ηg.cα.rf ) = Sα.
To see that K extends R observe that if both f and g are tight then by Part
4 we have (1, Rpf rf , ηf ) = (1, Rpf Rrf , Rηf ) and likewise for g. Using
the same decomposition of α we have Rα = Rqg.Rηg.Rcα.Rrf = Kα.
For w = p consider the p-reflection uf,g vf,g and the morphisms of preflections
(sf,g, 1) : uf,g → pf and (tf,g, 1) : uf,g → pg of Lemma 20. By
Part 1 their images (Rsf,g, 1) : Ruf,g → Rpf and (Rtf,g, 1) : Ruf,g → Rpg are
morphisms of p-reflections in B so that the two triangles in the diagram below
RA
RKf,g RPf
RPg A
vf,g
555™5™5™5™5™5™
rf
55
Fn Em Dl Dl Ck Bj Ai @h 9g 8f 7e 6d 5™
rg
55
ƒ
„
!…
"†
#‡
$ˆ
&
'‘
)“ 0” 2– 4˜
Rsf,g
GG
Rqf

Rtf,g

Rqg
GG
Rρα
{Ó 
commute. We set Kα = Rρα.vf,g.
By Lemma 20 we have that α = ρα.vf,g. Therefore Sα = Sρα.Svf,g =
HRρα.Hvf,g = HKα. If now both f and g are tight then so is vf,g, again by
Lemma 20. It follows that the p-reflections Ruf,g vf,g and Ruf,g Rvf,g
coincide, and therefore that Rα = Rρα.Rvf,g = Kα.
(7) Treating the cases w = l and w = p together again observe that because H
is locally faithful the functoriality of K on 2-cells trivally follows from the
functoriality of HK = S on 2-cells. Thus K is an F-functor.
(8) For uniqueness let us observe that the definition of the filler is forced upon
us by the constraints. Consider any diagonal filler L. We must have Lf =
L(qf .rf ) = Lqf .Lrf = Rqf .Lrf . Since HL = S we would need that the
image of the adjunction (1, pf rf , ηf ) under L were a lifting of its image
(1, Spf Srf , Sηf ) under HL. But by uniqueness of liftings we would then
have (1, Lpf Lrf , Lηf ) = (1, Rpf rf , ηf ) so that Lf = Rqf .rf . Now for
w = l we have, by Lemma 19, the decomposition of α : f ⇒ g as qg.mα =
qg.ηg.cα.rf and so must have Lα = Rqg.ηg.Rcα.rf . The argument when w = p
is similar but uses the decomposition α = ρα.vf,g of Lemma 20 instead.
Let us note that Theorem 21 remains true even if we remove the assumption
that w-doctrinal F-functors are locally faithful. The only place that we used
28 JOHN BOURKE
this assumption was in establishing the functoriality of the diagonal K on 2-cells.
This can alternatively be established by carefully analysing the functoriality of the
assignment in Section 4.3 which represents a 2-cell by a span transformation. However
the proof becomes significantly longer and more technical, very significantly
when w = p. Since local faithfulness is true of those F-functors U : T-Algw → C
that we seek to characterise (see Theorem 25) and because the property is easily
verified in practice, we have chosen to include it in our definition of w-doctrinal
F-functor.
As an immediate consequence of Theorem 21 we have:
Corollary 22. Let w ∈ {l, p, c}. Consider an F-functor H : A → B to a 2-category
and suppose that A has ¯w-limits of loose morphisms and that H is w-doctrinal.
Then the decomposition in F-CAT
Aτ A B
j
GG H GG
of the 2-functor Hτ : Aτ → B is an orthogonal (⊥w-Doct, w-Doct)-decomposition.
Since orthogonal decompositions are essentially unique, this asserts that an Fcategory
satisfying some completeness properties and sitting over a base 2-category
in a certain way – such as MonCatw or T-Algw over Cat – is uniquely determined
by how its tight part sits over the base. This is the core idea behind our monadicity
results of Section 6.
5.1. A note on alternative hypotheses. There are other hypotheses upon A
which ensure that j : Aτ → A belongs to ⊥w-Doct. Let us focus on the lax case.
Say that A admits loose morphism classifiers if the inclusion j : Aτ → Aλ has a
left 2-adjoint Q and write pA : A QA and qA : QA → A for the unit and counit.
Each loose f : A B corresponds to a tight morphism f : QA → B such that
f .pA = f. One triangle equation gives qA.pA = 1; if for each A this happens to
be the counit of an l-reflection qA pA then each f is represented by an l-span
A QA B
qA pA
oo f
GG
and it can be easily shown that j : Aτ → A ∈⊥ l-Doct. Of course these hypotheses
are strong: it is not easy to check whether a given F-category admits loose
morphism classifiers.
6. Monadicity
In this section we give our monadicity theorems. We begin by extending EilenbergMoore
comparison 2-functors to F-functors. In Theorem 24 we show these comparisons
to be natural in w, in the sense of Diagram 2 of the Introduction. Our
main result on monadicity is Theorem 25.
6.1. Extending the Eilenberg-Moore comparison.
Theorem 23. Let H : A → B be an F-functor to a 2-category whose tight part Hτ :
Aτ → B has a left adjoint and consider the induced Eilenberg-Moore comparison
TWO-DIMENSIONAL MONADICITY 29
2-functor E left below.
Aτ
B
T-Algs
Hτ
''UUUUUUUUUUU
E GG
Us
ÓÓ§§§§§§§§§§§
Aτ A
T-Algs T-Algw
B
j
GG
jw
GG
H
66ttttttt
E

Ew
 U
XXtttttt
Let w ∈ {l, p, c} and suppose that A admits w-limits of loose morphisms. Then
E : Aτ → T-Algs admits a unique extension to an F-functor Ew : A → T-Algw
over B, as depicted on the right above.
Proof. The commutativity of the outside of the right diagram just says that UsE =
Hτ . By Corollary 9 and Theorem 21 we know that U : T-Algw → B is w-doctrinal
and that j : Aτ → A is orthogonal to each w-doctrinal F-functor, in particular U.
Therefore there exists a unique F-functor Ew : A → T-Algw satisfying the depicted
equations Ewj = jwE and UEw = H. These respectively assert that Ew extends
E and lives over the base B.
In order to understand the naturality in w of the above Eilenberg-Moore extensions
Ew : A → T-Algw it will be convenient to briefly consider F2-categories. An
F2-category consists of a 2-category equipped with three kinds of morphism: tight,
loose and very loose, all satisfying the expected axioms. For instance we have the
F2-category of monoidal categories, strict, strong and lax monoidal functors; likewise
of algebras together with strict, pseudo and lax morphisms for a 2-monad.
One presentation of an F2-category is as a triple on the left below
Aτ Aλ Aφ
j
GG j
GG Aτ,λ Aτ,φ
j
GG
in which each inclusion is the identity on objects, faithful and locally fully faithful.
Thus an F2-category has three associated F-categories Aτ,λ, Aλ,φ and Aτ,φ, and is
determined by the first and third of these together with the inclusion F-functor,
right above, which views tight and loose morphisms as tight and very loose respec-
tively.
We commonly encounter F2-categories sitting over a base 2-category as on the left
below. See how monoidal categories, strict, strong and lax monoidal functors sit
over Cat for instance.
Aτ Aλ Aφ
B
j
GG j
GG
Hτ
11cccccccccccc
Hλ

Hφ

Aτ,λ Aτ,φ
B
j
GG
Hτ,λ
%%QQQQQQQQQQ
Hτ,φ
ÕÕ
To give such a diagram is equally to give a commutative triangle in F-CAT as on
the right above – here the F-functors Hτ,λ and Hτ,φ agree as Hτ on tight parts,
and have loose parts Hλ and Hφ respectively.
30 JOHN BOURKE
Theorem 24. Let w ∈ {l, c}. Consider an F2-category over a 2-category as below
Aτ,λ Aτ,φ
B
j
GG
Hτ,λ
%%QQQQQQQQQQ
Hτ,φ
ÕÕ
Suppose that Hτ admits a left adjoint and that Aτ,λ and Aτ,φ satisfy the (p/w)
variants of the completeness criteria of Theorem 23 so that the Eilenberg-Moore
comparison 2-functor E : Aτ → T-Algs extends uniquely to F-functors Ep : Aτ,λ →
T-Algp and Ew : Aτ,φ → T-Algw over the base B.
When all of this holds the square
Aτ,λ Aτ,φ
T-Algp T-Algw
j
GG
Ew

Ep

j
GG
commutes.
Proof. Consider the diagram left below
Aτ Aτ,λ Aτ,φ
T-Algp T-Algw B
j
GG j
GG
Ew

Ep

j
GG
Uw
GG
Aτ Aτ,λ
T-Algw B
j
GG
Hτ,λ

jE
 {{
Uw
GG
Now Ep and Ew agree as E : Aτ → T-Algs on tight morphisms. Consequently
both paths of the square coincide upon precomposition with j : Aτ → Aτ,λ as
the composite jE : Aτ → T-Algs → T-Algw. Because both Eilenberg-Moore
extensions Ep and Ew lie over the base B we find that postcomposing both paths
of the square with Uw yields the common composite Hτ,λ : Aτ,λ → B. Therefore
both paths of the square are diagonal fillers for the square on the right above. Now
by Corollary 9 we know that Uw : T-Algw → B is p-doctrinal. But by Theorem 21
j : Aτ → Aτ,λ is orthogonal to each p-doctrinal F-functor so that both paths
coincide as the unique filler.
6.2. 2-categorical monadicity. We now turn to monadicity. Let us say that a
2-functor with a left adjoint is monadic if the induced Eilenberg-Moore comparison
is a 2-equivalence and strictly monadic if the comparison is an isomorphism.
Theorem 25. Let H : A → B be an F-functor to a 2-category B. Let w ∈ {l, p, c}
and suppose that
(1) Hτ : Aτ → B is monadic.
(2) A admits w-limits of loose morphisms.
(3) B admits w-limits of arrows.
(4) H is w-doctrinal (it suffices that H satisfies w-doctrinal adjunction, is locally
faithful and reflects identity 2-cells).
TWO-DIMENSIONAL MONADICITY 31
Then the 2-equivalence E : Aτ → T-Algs extends uniquely to an equivalence of
F-categories Ew : A → T-Algw over B. Moreover if Hτ is strictly monadic then
Ew : A → T-Algw is an isomorphism of F-categories.
Proof. As in Theorem 23 we have our extension Ew : A → T-Algw, unique in filling
the square
Aτ A
T-Algw B
T-Algs
E

jw

U
GG
j
GG
H

Ew

Recall that this filler exists because U is w-doctrinal and j orthogonal to such
F-functors.
We begin by proving the theorem in the case of strict monadicity and deduce the
general case from that – so suppose that E : Aτ → T-Algs is an isomorphism of 2categories.
By Proposition 11 U : T-Algw → B creates w-limits of loose morphisms
so that T-Algw admits them. Now Theorem 21 implies that the inclusion jw :
T-Algs → T-Algw is orthogonal to each w-doctrinal F-functor; since E : Aτ →
T-Algs is an isomorphism jwE : A → T-Algw is also orthogonal to w-doctrinal
F-functors. Now H is w-doctrinal by assumption so that the two outer paths of
the square are orthogonal decompositions of a common F-functor. Therefore the
unique filler Ew : A → T-Algw is an isomorphism.
Suppose now that E : Aτ → T-Algs is only an equivalence of 2-categories. The
problem now is that E will no longer be orthogonal to w-doctrinal F-functors. We
will rectify this by factoring the composite 2-functor jwE : Aτ → T-Algs → T-Algw
as the identity on objects followed by 2-fully faithful through a 2-category Aλ as
in the commutative square below.
Aτ Aλ
T-Algs T-Algw
j
GG
E

K
jw
GG
Then K is 2-fully faithful by construction but also essentially surjective on objects
since each of E, j and jw are; as such K is a 2-equivalence. Moreover the composite
Kj = jwE is both faithful and locally fully faithful since both jw and E are,
whilst K is 2-fully faithful. Therefore j is faithful and locally fully faithful too and
since it is identity on objects by construction it is the inclusion of an F-category
A : Aτ → Aλ. It now follows that the above commutative square, whose vertical
legs are 2-equivalences, exhibits L = (E, K) : A → T-Algw as an equivalence of
F-categories.
Consider the diagram defining the extension Ew again, now drawn on the left
32 JOHN BOURKE
below
Aτ A
T-Algw B
A
j

L

U
GG
j
GG
H

Ew

E
ww
Aτ A
T-Algw BA
j

L
GG
U
GG
j
GG
H

E
{{
with the left leg rewritten as the composite L ◦ j. Since L is F-fully faithful (2fully
faithful on tight and loose parts) it is orthogonal to each bijective on objects
F-functor, in particular j, so that we obtain a unique diagonal filler E : A → A
making the two leftmost triangles commute.
Our goal is to show that Ew is an equivalence of F-categories - but since L is an
equivalence this is, by 2 out of 3, equivalently to show that E is an equivalence of
F-categories. Now consider the square on the right. The bottom leg is w-doctrinal
as both of its components are, L by Proposition 10. Since the F-category A is
equivalent to T-Algw it has the same completeness properties, so that, using Theorem
21 again, the left leg is orthogonal to each w-doctrinal F-functor. Therefore
the right commutative square consists of two orthogonal decompositions of a common
F-functor and we conclude that E is an isomorphism.
Note that although Theorem 25, in each of its variants, only asks that A admits
certain limits it follows that H creates those limits: for U : T-Algw → B does
so by Proposition 11 and E : A → T-Algw is an equivalence of F-categories. In
applying the theorem this is often useful in that it tells us how these limits must
be constructed in A.
6.3. F-categorical monadicity. Our focus has been upon 2-monads but indeed
the above results extend in a routine way to cover F-categorical monadicity too.
Let us briefly explain how this goes. An F-monad [16] is a monad in the 2category
F-CAT and so consists of an F-functor T : A → A and two F-natural
transformations satisfying the usual equations. 2-monads are just F-monads on
2-categories viewed as F-categories. An F-monad T induces 2-monads Tτ and Tλ
and so we have strict Tτ and Tλ-algebra morphisms as in the two leftmost diagrams
below.
TA TB
A B
Tf
GG
b

a

f
GG
TA TB
A B
Tf
GGGoGoGoGo
b

a

f
GGGoGoGoGoGo
TA TB
A B
Tf
GGGoGoGoGo
b

a

f
GGGoGoGoGoGo
f
These are the tight and loose morphisms of the Eilenberg-Moore F-category which
is denoted by T-Algs. We also have pseudo, lax and colax Tλ-morphisms – a lax
Tλ-morphism is drawn above. These are the loose morphisms of the F-categories
T-Algw whose tight morphisms are the strict Tτ -algebra maps. Now for each
TWO-DIMENSIONAL MONADICITY 33
w ∈ {l, p, c} we have the F2-category whose tight and loose morphisms are the strict
Tτ and Tλ-morphisms and whose very loose morphisms are the w-Tλ-morphisms:
this is captured by the inclusion F-functor jw : T-Algs → T-Algw. We have the
evident forgetful F-functors Us : T-Algs → A and Uw : T-Algw → A commuting
with jw over the base. The key point for our applications is that these have the
same properties as in the 2-monads case: namely Us creates all limits, Uw creates
w-limits of loose morphisms (this follows from Theorem 5.13 of [16]) and Uw is
w-doctrinal. With these facts in place we can give our monadicity theorem for
F-monads – we leave it to the reader to formulate the naturality of the EilenbergMoore
extensions, which can be done using F3-categories.
Theorem 26. Consider an F2-category Aτ → Aλ → Aφ over an F-category B as
below
Aτ,λ Aτ,φ
B
j
GG
G
%%QQQQQQQQQQ
H
ÕÕ
(this means that Gτ = Hτ ). Suppose that G has a left F-adjoint so that we have
the Eilenberg-Moore comparison F-functor E : Aτ,λ → T-Algs over the base B.
Now let w ∈ {l, p, c} and suppose that both Aτ,λ and Aτ,φ admit w-limits of loose
morphisms. Then there exists a unique F-functor Ew : Aτ,φ → T-Algw extending
E and living over the base, as depicted by the following everywhere commutative
diagram.
Aτ,λ Aτ,φ
T-Algs T-Algw
B
j
GG
jw
GG
H
66tttttt
E

Ew
 Uw
XXtttttt
If G is F-monadic, B admits w-limits of loose morphisms and H is w-doctrinal
then Ew is an equivalence of F-categories, and an isomorphism of F-categories
whenever G is strictly monadic.
Proof. The outside of the diagram clearly commutes – since Hj = G and Uwjw =
Us this just amounts to the fact that the Eilenberg-Moore comparison E satisfies
UsE = G. Now Uw : T-Algw → B is w-doctrinal so that if we can show that the
inclusion j : Aτ,λ → Aτ,φ belongs to ⊥w-Doct then we will obtain Ew as the unique
filler. Now have a commutative triangle of inclusions
Aτ Aτ,λ
Aτ,φ
j
GG
j
77vvvvvvvvvvvv
j

34 JOHN BOURKE
in which the two j’s moving from left to right belong to ⊥w-Doct by Theorem 21;
thus by 2 out of 3 j : Aτ,λ → Aτ,φ ∈⊥ w-Doct. As such we obtain the EilenbergMoore
extension Ew as the unique filler. The remainder of the proof is a straightforward
modification of the proof of Theorem 25.
7. Examples and applications
We now turn to examples. We begin by completing our running example of
monoidal categories. Of course it is well known that monoidal categories, and
each flavour of morphism between them, can be described using 2-monads – see
Section 5.5 of [15] for an argument via colimit presentations – although this has
not previously been established by application of a monadicity theorem. We then
turn to more complex examples. Our final example, in 7.3, is new and typical of
the kind of result which cannot be established using techniques, such as colimit
presentations, that require explicit knowledge of a 2-monad.
7.1. Monoidal categories. Let us focus, as usual, on the lax monoidal functors
of MonCatl. From Example 5 we know that V : MonCatl → Cat satisfies
l-doctrinal adjunction. It is clearly locally faithful and reflects identity 2-cells.
From Example 13 we know that V : MonCatl → Cat creates colax limits of loose
morphisms. Therefore to apply Theorem 25 and establish monadicity it remains
to verify that the 2-functor Vs : MonCats → Cat is monadic. That Vs strictly
creates Vs-absolute coequalisers (in the enriched sense) is true by essentially the
same argument given for groups in 6.8 of [18]; by Beck’s theorem in the enriched
setting [4] it follows that Vs is strictly monadic so long as it has a left 2-adjoint.
Proposition 3.1 of [3] asserts that a 2-functor preserving cotensors with 2 admits a
left 2-adjoint just when its underlying functor admits a left adjoint. Now cotensors
with 2 are, in fact, just colax limits of identity arrows. That these are created by
Vs follows from the fact that V : MonCatl → Cat creates colax limits of loose
morphisms. It therefore remains to show that the underlying functor (Vs)0 admits
a left adjoint. Since this functor creates all limits it suffices to show that (Vs)0
satisfies the solution set condition. For this it suffices to show that given a small
category A each functor F : A → C = UC to a monoidal category factors as
ME : A → B → C with B monoidal, M strict monoidal and the cardinality of
the set of morphisms Mor(B) bounded by that of Mor(A). Here B will be the
monoidal subcategory of C generated by the image of F and M the inclusion of
this monoidal subcategory.
For the induced 2-monad T on Cat we now conclude, by Theorem 25, that the
isomorphism of 2-categories E : MonCats → T-Algs over Cat extends uniquely to
an isomorphism of F-categories El : MonCatl → T-Algl over Cat. Likewise one
can verify, in an entirely similar way, that V : MonCatw → Cat satisfies the conditions
of Theorem 25 in the cases w ∈ {p, c}. It follows that we have isomorphisms
Ew : MonCatw → T-Algw over Cat for each w ∈ {l, p, c}. By Theorem 24 these
isomorphisms are natural in p ≤ l and p ≤ c in the sense of Diagram 2 of the
Introduction.
7.2. Categories with structure and variants. Of course there was nothing
special about our taking monoidal categories in the preceding section. The same
TWO-DIMENSIONAL MONADICITY 35
arguments can be used to establish monadicity of categories with any kind of algebraically
specified structure and their various flavours of morphisms: categories
with chosen limits of some kind for instance, distributive categories and so forth.
All of these cases are well known to be monadic using colimit presentations of
2-monads, although it requires substantial and laborious calculation to use that
theory to establish monadicity in the detailed manner above. Such a detailed
treatment using colimit presentations, one expressed in terms of isomorphisms of
2-categories or F-categories, will not be found in the literature. Colimit presentations
are one of the two standard techniques for understanding the monadicity of
weaker kinds of morphisms; the other is direct calculation with a 2-monad known
to exist. As a representative example of this technique consider a small 2-category
J and the forgetful 2-functor U : [J, Cat] → [obJ, Cat] which restricts presheaves
to families along the inclusion obJ → J. U has a left 2-adjoint F given by left
Kan extension and is strictly monadic by Beck’s theorem; moreover the induced
2-monad T = UF admits a simple pointwise description: at X ∈ [obJ, Cat] we
have TX(j) = ΣiJ(i, j) × Xi. Using this formula (as in [3]) one directly calculates
that T-pseudomorphisms bijectively correspond to pseudonatural transformations
and so on, eventually deducing an isomorphism Ps(J, Cat) → T-Algp. It is in such
cases, more specifically when T is not so simple, that our results have most value.
An example of this kind was given in [10]. Given a complete and cocomplete symmetric
monoidal closed category V the authors take as base the 2-category V-Cat
of small V-categories. For a small class of weights Φ they consider the 2-category
Φ-Colim (we will write Φ-Colims) of V-categories with chosen Φ-weighted colimits,
V-functors preserving those colimits strictly and V-natural transformations.
This 2-category lives over V-Cat via a forgetful 2-functor Us : Φ-Colims → V-Cat.
One also has V-functors preserving colimits in the usual, up to isomorphism, sense:
these are the loose morphisms of the F-category Φ-Colimp : Φ-Colims → Φ-Colimp
which sits over V-Cat via a forgetful F-functor U : Φ-Colimp → V-Cat. The authors
show that Us : Φ-Colims → V-Cat is strictly monadic and then, by calculating
directly with the induced 2-monad T, show that one obtains an isomorphism
of 2-categories U : Φ-Colimp → T-Algp. Let us show how this can be deduced from
Theorem 25. Firstly observe that U satisfies p-doctrinal adjunction: this follows
from the fact that any equivalence of V-categories preserves colimits. Again since
U is locally fully faithful it is certainly locally faithful and reflects identity 2-cells.
It is not hard to see that Φ-Colimp admits pseudo-limits of loose morphisms –
indeed this is shown in Section 5 of [10]. It then follows immediately from Theorem
25 that the isomorphism of 2-categories E : Φ-Colims → T-Algs extends
uniquely to an isomorphism of F-categories E : Φ-Colimp → T-Algp over V-Cat.
Furthermore, because left adjoints preserve colimits, U : Φ-Colimp → V-Cat has
the additional property of satisfying c-doctrinal adjunction; since it is isomorphic
to T-Algp it also admits, by Theorem 2.6 of [3], lax limits of loose morphisms.
Therefore Theorem 25 ensures that the composite Φ-Colimp → T-Algp → T-Algc
is also an isomorphism, thus explaining why the colax and pseudo T-morphisms
coincide as those V-functors which preserve Φ-colimits. Again one easily applies
Theorem 25 to show that T-Algl is isomorphic to the F-category Φ-Coliml whose
loose morphisms are arbitary V-functors. This final isomorphism highlights the
36 JOHN BOURKE
fact that Ul : T-Algl → V-Cat is 2-fully faithful: 2-monads with this property are
called lax idempotent/Kock-Z¨oberlein and have been carefully studied in [9].
7.3. In a monoidal 2-category. In a monoidal category C one can consider the
category of monoids Mon(C) or of commutative monoids. If the forgetful functor
U : Mon(C) → C has a left adjoint then Beck’s theorem can be applied, with no
further information, to show that U is monadic. For our final example we study
the analogous situation in the context of a monoidal 2-category C, in which one
can consider monoids, pseudomonoids (generalising monoidal categories), braided
pseudomonoids and so on. We consider only the simplest case of monoids because,
in the absence of a suitable graphical calculus, it is difficult to encode diagrams
compactly. We have the 2-category of monoids, strict monoid morphisms and
monoid transformations Mon(C)s and a forgetful 2-functor Us : Mon(C)s → C.
Just as before, the enriched version of Beck’s theorem [4] can be applied to show
that if Us has a left 2-adjoint then it is monadic. However we now also have
(lax/pseudo/colax)-morphisms of monoids and we would like to understand that
these too are monadic in the appropriate sense: this is the content, under completeness
conditions on the base, of the present example.
By a monoidal 2-category C we will mean a monoidal V-category where V = Cat:
this satisfies the same axioms as a monoidal category with the exception that the
tensor product, the associator and the other data involved are now 2-functorial and
2-natural. In working with C we will write as though it were strict monoidal – this is
justified in the theorem that follows – and will use juxtaposition for the tensor product.
A monoid in C is just a monoid in the usual sense. Given monoids (X, mX, iX)
and (Y, mY , iY ) a lax monoid map (f, f, f0) : (X, mX, iX) → (Y, mY , iY ) consists
of an arrow f : X → Y and 2-cells as below
X2 Y 2
X Y
f2
GG
f
GG
mX

mY

f 
I
X Y
iX
ÒÒ¤¤¤¤¤¤ iY
((XXXXXX
f
GG
f0 
such that the equation
X3 XY 2 Y 3
X2 XY Y 2
X Y
1ff
GG f11
GG
1mX

1mY

1mY
1f 
1f
GG
f1
GG
f
GG
mX

mY

f 
=
X3 X2Y Y 3
X2 XY Y 2
X Y
11f
GG ff1
GG
mX 1

mX 1

mY 1
f1 
1f
GG
f1
GG
f
GG
mX

mY

f 
TWO-DIMENSIONAL MONADICITY 37
holds and such that both composite 2-cells
X X Y
X2 XY Y 2
X Y
1 GG f
GG
1iX

1iY

1iY

1f0 
1f
GG
f1
GG
f
GG
mX

mY

1f 
X Y Y
X2 XY Y 2
X Y
f
GG 1 GG
iX 1

iX 1

iY 1

f01 
1f
GG
f1
GG
f
GG
mX

mY

1f 
are identities. A monoid transformation α : (f, f, f0) ⇒ (g, g, g0) is a 2-cell α :
f ⇒ g satisfying the equations
X2 Y 2
X Y
mY

g
WW
mX

f2
88
g2
VV
g
α2

=
X2 Y 2
X Y
mY

f
77
g
WW
mX

f2
88
α 
f 
I
X Y
iX

iY

f
77
g
WW
f0 
α 
=
I
X Y
iX

iY

g
WW
g0

These are the 2-cells of the F-category Mon(C)l : Mon(C)s → Mon(C)l of monoids,
strict and lax monoid morphisms which sits over C via a forgetful F-functor U :
Mon(C)l → C. Likewise we have pseudo and colax monoid morphisms and forgetful
F-functors U : Mon(C)w → C for each w ∈ {l, p, c}.
For a simple statement we assume in the following result that C admits pie limits
[20]: these are a good class of limits containing w-limits of arrows for each w.
Theorem 27. Let C be a monoidal 2-category admitting pie limits and suppose that
Us : Mon(C)s → C has a left 2-adjoint. Let T be the induced 2-monad on C. Then
for each w ∈ {l, p, c} we have isomorphisms of F-categories Mon(C)w → T-Algw
over C and these are natural in w.
Proof. Let us begin by showing that it suffices to suppose C to be strict monoidal.
A straightforward extension of the usual argument for monoidal categories shows
that C is equivalent to a strict monoidal 2-category D via a strong monoidal 2equivalence
E : C → D.
Mon(C)w Mon(D)w
C D
E∗
GG
UD
UC

E
GG
Such an equivalence naturally lifts to a 2-equivalence E∗ : Mon(C)w → Mon(D)w
for each w and so induces a commuting square of F-categories and F-functors
as above with both horizontal legs equivalences of F-categories. Now to apply
Theorem 25 we must show that UC : Mon(C)w → C is w-doctrinal and that
Mon(C)w has w-limits of loose morphisms. So suppose that UD and Mon(D)w have
these properties and let us deduce from these the corresponding properties for C.
Certainly if UD were w-doctrinal then UC would be too; for both horizontal legs,
being equivalences, are w-doctrinal and such F-functors, being defined by lifting
38 JOHN BOURKE
properties (as in Section 3.3), are closed under 2 out of 3. Likewise any limits
existing in Mon(D)w exist in the F-equivalent Mon(C)w. Therefore it suffices to
suppose that C is strict monoidal.
Now certainly U : Mon(C)l → C is locally faithful and reflects identity 2-cells.
Moreover given a strict monoid map (f, f, f0) : X → Y and adjunction ( , f
g, η) ∈ C taking mates gives 2-cells
Y 2 X2
Y X
Y 2
X
mX

g
GG
mY

g2
GG
1
 f2
wwooooooooooooooo
f
wwoooooooooooooooo
1

2
s{ oooo
η
s{ oooo
I X
Y X
iX
GG
iY

g
GG
f
wwoooooooooooooooo
1

η
v~ tttttt
It is straightforward to see, by cancelling mates, that these give g the structure of a
lax monoid map (g, g, g0), with respect to which ( , f (g, g, g0), η) is an adjunction
in Mon(C)l. Uniqueness of the lifted adjunction follows from Proposition 3.1.
Therefore U satisfies l-doctrinal adjunction.
We will show that U : Mon(C)l → C creates colax limits of loose morphisms.
Consider a lax monoid map (f, f, f0) : X → Y and the colax limit C of f in C
with colax cone as below
C
X Y
p
ÒÒ¤¤¤¤¤¤ q
((XXXXXX
f
GG
λ 
By the universal property of C the composite colax cone left below induces a
unique map mC : CC → C such that p.mC = mX.p2, q.mC = mY .q2 and such
that the left equation below holds. Likewise we obtain a unique iC : I → C such
that p.iC = iX, q.iC = iY and satisfying λ.iC = f0 as on the right below.
C2
X2 Y 2
X Y
p2
 q2
11cccc
f2
GG
λ2

f
GG
mX

mY
f 
=
C2
X2 Y 2
C
X Y
p2
 q2
11cccc
p
 q
11ccccc
mC

λ 
f
GG
mX

mY

I
X Y
iX

iY

f0 
f
GG
=
I
C
X Y
iX

iY

p
 q
11ccccc
iC

λ 
f
GG
If we can show (C, mC, iC) to be a monoid then these combined equations will
assert exactly that p and q are strict monoid maps and λ : q ⇒ (f, f, f0).p a
monoid transformation. To show that mC.(mC1) = mC.(1mC) : C3 → C amounts
to showing that both paths coincide upon postcomposition with the components p,
q and λ of the universal colax cone. We have that p.mC.(mC1) = mX.p2.(mC1) =
mX.(mX1).p3 = mX.(1mX).p3 = mX.p2.(1mC) = p.mC.(1mC) and similarly for
q so that associativity of mC will follow if we can show that both paths coincide
TWO-DIMENSIONAL MONADICITY 39
upon postcomposition with λ. Consider the following series of equalities
C3
C2
C
X Y
1mC

mC

p
ÒÒ¤¤¤¤¤¤ q
((XXXXXX
f
GG
λ 
=
C3
X3 C2 Y 3
X2 Y 2
X Y
ppp
ÒÒ¥¥¥¥¥¥¥
qqq
((WWWWWWW
1mC

1mX

1mY

pp
ÒÒ¥¥¥¥¥¥¥
qq
((WWWWWWW
ff
GG
λλ 
f
GG
mX

mY
f 
=
C3
XC2 CY 2
X3
C2
Y 3
XC CY
X2 XY Y 2
X Y
p11
yyssss
1pp
{{xxxxx
p1
ss
yy
1qq
77uuuu
1q
uu
77
q11
55ppppp
1mC

1p
{{xxxxx 1q
55ppppp
1mY

p1
{{xxxxx q1
55ppppp
1λ  λ1 
1mX

1mC

1mY

1f
GG
f1
GG
f
GG
mX

mY

f 
=
C3
XC2 CY 2
X3 XY 2 Y 3
XC CY
X2 XY Y 2
X Y
p11
yyssss
1pp
{{xxxxx 1qq
55ppppp
1qq
77uuuu
p11
{{xxxxx q11
55ppppp
1mC

1p
{{xxxxx
1q
pp
55pp
1mY

p1
xx
{{xx
q1
55ppppp
1λ  λ1 
1mX

1mY

1mY

1f
GG
f1
GG
f
GG
mX

mY

f 
=
C3
XC2 CY 2
X3 X2Y XY 2 Y 3
X2 XY Y 2
X Y
p11
zzttt
1pp
wwoooooooo 1qq
44hhhhh
1qq
66ttt
p11
||zzzzz q11
22ffff
1λλ  λ11 
11f
GG
1f1
GG
f11
GG
1mX

1mY

1mY
1f 
1f
GG
f1
GG
f
GG
mX

mY

f 
The first holds by definition of mC, the second merely rewrites the tensor product
λλ and the third rewrites the commuting central diamond. The final equality
of pasting composites does two things at once: on its left side it again applies
the definition of mC postcomposed with λ; we rewrite the right hand side using
(λ1).(1mY ) = (1mY ).(λ11). By an entirely similar diagram chase we obtain the
equality
C3
C2
C
X Y
mC 1

mC

p
ÒÒ¤¤¤¤¤¤ q
((XXXXXX
f
GG
λ 
=
C3
X2C C2Y
X3 X2Y XY 2 Y 3
X2 XY Y 2
X Y
pp1
yyssss
11p
xxrrrrrr 11q
))````
11q
77uuuu
pp1
xxrrrrrr qq1
88vvvvvv
11λ  λλ1 
11f
GG
1f1
GG
f11
GG
mX 1

mX 1

mY 1
f1 
1f
GG
f1
GG
f
GG
mX

mY

f 
This final composite and the one above it each constitute λ3 sat atop either side
of the first equation for a lax monoid morphism: as such they agree and mC is
associative. Much smaller, though similar, diagram chases show that mC.(iC1) = 1
and that mC.(1iC) = 1; thus C is a monoid and the colax cone (p, λ : q ⇒ pf, q)
lifts (uniquely) to a colax cone in Mon(C)l.
That the lifted colax cone satisfies the universal property of the colax limit is
40 JOHN BOURKE
relatively straightforward and left to the reader. That p and q detect strict monoid
morphisms is a consequence of the fact that they jointly detect identity 2-cells in
C.
Now if Us : Mon(C)s → C has a left adjoint it is automatically strictly monadic by
the enriched version of Beck’s monadicity theorem [4]. Therefore, using the above,
Theorem 25 asserts that the isomorphism of 2-categories E : Mon(C)s → T-Algs
over C extends uniquely to an isomorphism of F-categories El : Mon(C)l → T-Algl
over C. In a similar way one verifies the conditions of Theorem 25 when w ∈ {p, c}
to obtain isomorphisms of F-categories Ew : Mon(C)w → T-Algw over C for each
w; by Theorem 24 these isomorphisms are natural in w.
7.4. A non-example. All of the examples we have seen are of the strictly monadic
variety and indeed this is the case whenever one studies structured objects over
some same base 2-category. Now Theorem 25 is general enough to cover ordinary
monadicity – up to equivalence of F-categories – but in fact there exist situations
of a weaker kind. Here is one such case. Let Catf ⊂ Cat be a full sub 2-category
of Cat whose objects form a skeleton of the finitely presentable categories (the
finitely presentable objects in Cat) and let [Cat, Cat]f ⊂ [Cat, Cat] be the full
sub 2-category consisting of those endo 2-functors preserving filtered colimits: this
is the tight part of the F-category Ps(Cat, Cat)f : [Cat, Cat]f → Ps(Cat, Cat)f
whose loose morphisms are pseudonatural transformations. Likewise we have an
F-category Ps(Catf , Cat) : [Catf , Cat] → Ps(Catf , Cat) and now restriction along
the inclusion Catf → Cat induces a forgetful F-functor R : Ps(Cat, Cat)f →
Ps(Catf , Cat). Further restricting along the inclusion obCatf → Catf gives a
commuting triangle
Ps(Cat, Cat)f Ps(Catf , Cat)
[obCatf , Cat]
R GG
SR
))YYYYYYYYYYY
S
ÑÑ£££££££££££
The composite Sτ Rτ : [Cat, Cat]f → [obCatf , Cat] is monadic though not strictly
so: for the induced 2-monad T we have T-Algp isomorphic to Ps(Catf , Cat) with
Rτ : [Cat, Cat]f → [Catf , Cat] the Eilenberg-Moore comparison 2-functor – that
this is a 2-equivalence follows from Cat’s being locally finitely presentable as a
2-category. Whilst Rτ is a 2-equivalence the 2-functor Rλ : Ps(Cat, Cat)f →
Ps(Catf , Cat) is not: indeed Ps(Catf , Cat) is locally small whereas Ps(Cat, Cat)f
is not. Yet Rλ turns out to be a biequivalence and R the uniquely induced Ffunctor
to the F-category of algebras.
References
[1] Bartels, T. Higher gauge theory 1:2-Bundles. PhD thesis, University of California, Riverside,
2006.
[2] Beck, J. Triples, algebras and cohomology. Rep. Theory Appl. Categ. 2 (2003), 1–59.
[3] Blackwell, R., Kelly, G. M., and Power, A. J. Two-dimensional monad theory. Journal
of Pure and Applied Algebra 59, 1 (1989), 1–41.
TWO-DIMENSIONAL MONADICITY 41
[4] Dubuc, E. J. Kan extensions in enriched category theory, vol. 145 of Lecture Notes in
Mathematics. Springer, 1970.
[5] Grandis, M., and Tholen, W. Natural weak factorization systems. Archivum Mathematicum
42, 4 (2006), 397–408.
[6] Kelly, G. M. Doctrinal adjunction. In Category Seminar (Sydney, 1972/1973), vol. 420 of
Lecture Notes in Mathematics. Springer, 1974, pp. 257–280.
[7] Kelly, G. M. On clubs and doctrines. In Category Seminar (Sydney, 1972/1973), vol. 420
of Lecture Notes in Mathematics. Springer, 1974, pp. 181–256.
[8] Kelly, G. M. Basic concepts of enriched category theory, vol. 64 of London Mathematical
Society Lecture Note Series. Cambridge University Press, 1982.
[9] Kelly, G. M., and Lack, S. On property-like structures. Theory and Applications of Categories
3, 9 (1997), 213–250.
[10] Kelly, G. M., and Lack, S. On the monadicity of categories with chosen colimits. Theory
and Applications of Categories 7, 7 (2000), 148–170.
[11] Kelly, G. M., and Power, A. J. Adjunctions whose counits are coequalizers, and presentations
of finitary enriched monads. Journal of Pure and Applied Algebra 89, 1–2 (1993),
163–179.
[12] Kelly, G. M., and Street, R. Review of the elements of 2-categories. In Category Seminar
(Sydney, 1972/1973), vol. 420 of Lecture Notes in Mathematics. Springer, 1974, pp. 75–103.
[13] Lack, S. Limits for lax morphisms. Appl. Categ. Structures 13, 3 (2002), 189–203.
[14] Lack, S. Homotopy-theoretic aspects of 2-monads. Journal of Homotopy and Related Structures
7, 2 (2007), 229–260.
[15] Lack, S. A 2-categories companion. In Towards higher categories, vol. 152 of IMA Vol. Math.
Appl. Springer, 2010, pp. 105–191.
[16] Lack, S., and Shulman, M. Enhanced 2-categories and limits for lax morphisms. Advances
in Mathematics 229, 1 (2011), 294–356.
[17] Le Creurer, I., Marmolejo, F., and Vitale, E. Beck’s theorem for pseudomonads.
Journal of Pure and Applied Algebra 173, 3 (2002), 293–313.
[18] Mac Lane, S. Categories for the Working Mathematician, vol. 5 of Graduate Texts in Mathematics.
Springer, 1971.
[19] Makkai, M. Avoiding the axiom of choice in general category theory. Journal of Pure and
Applied Algebra 108, 2 (1996), 109–173.
[20] Power, J., and Robinson, E. A characterization of pie limits. Mathematical Proceedings
of the Cambridge Philosophical Society 110, 1 (1991), 33–47.
[21] Roberts, D. Internal categories, anafunctors and localisations. Theory and Applications of
Categories 26, 29 (2012), 788–829.
[22] Shulman, M. Comparing composites of left and right derived functors. New York Journal
of Mathematics 17 (2011), 75–125.
Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 60000, Czech Republic
E-mail address: bourkej@math.muni.cz
Chapter 3
Algebraic weak factorisation
systems I: Accessible AWFS
This chapter contains the article Algebraic weak factorisation systems I: Accessible
AWFS by John Bourke and Richard Garner, published in the Journal
of Pure and Applied Algebra 220 (2016) 108–147. Both authors contributed
an equal share to the results of this article.
71
ALGEBRAIC WEAK FACTORISATION
SYSTEMS I: ACCESSIBLE AWFS
JOHN BOURKE AND RICHARD GARNER
Abstract. Algebraic weak factorisation systems (awfs) refine weak factorisation
systems by requiring that the assignations sending a map to its
first and second factors should underlie an interacting comonad–monad pair
on the arrow category. We provide a comprehensive treatment of the basic
theory of awfs—drawing on work of previous authors—and complete the
theory with two main new results. The first provides a characterisation of
awfs and their morphisms in terms of their double categories of left or right
maps. The second concerns a notion of cofibrant generation of an awfs by
a small double category; it states that, over a locally presentable base, any
small double category cofibrantly generates an awfs, and that the awfs so
arising are precisely those with accessible monad and comonad. Besides the
general theory, numerous applications of awfs are developed, emphasising
particularly those aspects which go beyond the non-algebraic situation.
1. Introduction
A weak factorisation system on a category C comprises two classes of maps L
and R, each closed under retracts in the arrow category, and obeying two axioms:
firstly, that each map f ∈ C admit a factorisation f = Rf · Lf with Lf ∈ L
and Rf ∈ R; and secondly, that each r ∈ R have the right lifting property with
respect to each ∈ L—meaning that, for every square as in the solid part of:
(1.1)
A

GG C
r

B GG
bb
D
there should exist a commuting diagonal filler as indicated. Weak factorisation
systems play a key role in Quillen model structures [42], which comprise two
intertwined weak factorisation systems on a category; but they also arise elsewhere,
for example in the categorical semantics of intensional type theory [4, 20].
Algebraic weak factorisation systems were introduced in [25]; they refine the
basic notion by requiring that the factorisation process f → (Lf, Rf) yield a
compatible comonad L and monad R on the arrow category of C. Given (L, R),
we re-find L and R as the retract-closures of the classes of maps admitting
L-coalgebra or R-algebra structure, so that one may define an algebraic weak
2000 Mathematics Subject Classification. Primary: 18A32, 55U35.
The first author acknowledges the support of the Grant agency of the Czech Republic, grant
number P201/12/G028. The second author acknowledges the support of an Australian Research
Council Discovery Project, grant number DP110102360.
1
2 JOHN BOURKE AND RICHARD GARNER
factorisation system (henceforth awfs) purely in terms of a comonad–monad
pair (L, R) satisfying suitable axioms; we recall these in Section 2 below.
As shown in [22], any cofibrantly generated weak factorisation system on a
well-behaved category may be realised as an awfs, so that the algebraic notions
are entirely appropriate for doing homotopy theory; this point of view has been
pushed by Riehl, who in [43, 44] gives definitions of algebraic model category and
algebraic monoidal model category, and in subsequent collaboration has used
these notions to obtain non-trivial homotopical results [6, 12, 15].
Yet awfs can do more than just serve as well-behaved realisations of their
underlying weak factorisation systems; by making serious use of the monad R and
comonad L, we may capture phenomena which are invisible in the non-algebraic
setting. For example, each awfs on C induces a cofibrant replacement comonad
on C by factorising the unique maps out of 0; and if we choose our awfs carefully,
then the Kleisli category of this comonad Q—whose maps A B are maps
QA → B in the original category—will equip C with a usable notion of weak map.
For instance, there is an awfs on the category of tricategories [24] and strict
morphisms (preserving all structure on the nose) for which Kl(Q) comprises the
tricategories and their trihomomorphisms (preserving all structure up to coherent
equivalence); this example and others were described in [23], and will be revisited
in the companion paper [13].
Probably the most important expressive advantage of awfs is that their left
and right classes can delineate kinds of map which mere weak factorisation
systems cannot—the reason being that we interpret the classes of an awfs as
being composed of the L-coalgebras and R-algebras, rather than the underlying
L-maps and R-maps. For example, here are some classes of map in Cat which
are not the R-maps of any weak factorisation system, but which—as shown in
Examples 28 below—may be described in terms of the possession of R-algebra
structure for a suitable awfs:
• The Grothendieck fibrations;
• The Grothendieck fibrations whose fibres are groupoids;
• The Grothendieck fibrations whose fibres have finite limits, and whose
reindexing functors preserve them;
• The left adjoint left inverse functors.
At a crude level, the reason that these kinds of map cannot be expressed as
classes of R-maps is that they are not retract-closed. The deeper explanation is
that being an R-map is a mere property, while being an R-algebra is a structure
involving choices of basic lifting operations—and this choice allows for necessary
equational axioms to be imposed between derived operations. As a further
demonstration of the power this affords, we mention the result of [27] that for
any monad T on a category C with finite products, there is an awfs on C whose
“algebraically fibrant objects”—R-algebras X → 1—are precisely the T-algebras.
The existence of awfs that reach beyond the scope of the non-algebraic theory
opens up intriguing vistas. In a projected sequel to this paper, we will consider
the theory of enrichment over monoidal awfs [44] and use it to develop an
abstract “homotopy coherent enriched category theory”. In fact, we touch on
this already in the current paper; Section 8 describes the enriched small object
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 3
argument and sketches some of its applications to two-dimensional category
theory, and to notions from the theory of quasicategories [32, 33, 40] such as
limits, Grothendieck fibrations and Kan extensions.
The role of these examples, and others like them, will be to illuminate and
justify the main contribution of this paper—that of giving a comprehensive
account of the theory of unenriched awfs. Parts of this theory can be found
developed across the papers [6, 22, 25, 43, 44]; our objective is to draw the most
important of these results together, and to complete them with two new theorems
that clarify and simplify both the theory and the practice of awfs.
In order to explain our two main theorems, we must first recall some doublecategorical
aspects of awfs. A double category A is an internal category in
CAT, as on the left below; we refer to objects and arrows of A0 as objects and
horizontal arrows, and to objects and arrow of A1 as vertical arrows and squares.
Internal functors and internal natural transformations between internal categories
in CAT will be called double functors and horizontal natural transformations;
they comprise a 2-category DBL.
A1×A0 A1
◦ GG A1
cod
GG
dom GG
A0idoo R-Alg×C R-Alg
◦ GG R-Alg
cod
GG
dom GG
C .idoo
To each awfs (L, R) on a category C we may associate a double category
R-Alg, as on the right above, whose objects and horizontal arrows are the objects
and arrows of C, whose vertical arrows are the R-algebras, and whose squares
are maps of R-algebras. The functor ◦: R-Alg ×C R-Alg → R-Alg encodes a
canonical composition law on R-algebras—recalled in Section 2.8 below—which
is an algebraic analogue of the fact that R-maps in a weak factorisation system
are closed under composition.
There is a forgetful double functor R-Alg → Sq(C) into the double category of
squares in C—wherein objects are those of C, vertical and horizontal arrows are
arrows of C, and squares are commuting squares in C. In [43, Lemma 6.9], it was
shown that the assignation sending an awfs (L, R) on C to the double functor
R-Alg → Sq(C) constitutes the action on objects of a 2-fully faithful 2-functor
(1.2) (–)-Alg: AWFSlax → DBL2
from the 2-category of awfs, lax awfs morphisms and awfs 2-cells—whose
definition we recall in Section 2.9 below—to the arrow 2-category DBL2
.
Our first main result, Theorem 6 below, gives an elementary characterisation of
the essential image of (1.2) and of its dual, the fully faithful coalgebra 2-functor
(–)-Coalg: AWFSoplax → DBL2
. In the theory of monads, a corresponding
characterisation of the strictly monadic functors—those in the essential image of
the 2-functor sending a monad T to the forgetful functor UT : T-Alg → C—is given
by Beck’s monadicity theorem; and so we term our result a Beck theorem for awfs.
Various aspects of this result are already in the literature—see [22, Appendix],
[23, Proposition 2.8] or [43, Theorem 2.24]—and during the preparation of this
paper, we became aware that Athorne had independently arrived at a similar
result as [3, Theorem 2.5.3]. We nonetheless provide a complete treatment here,
as this theorem is crucial to a smooth handling of awfs, in particular allowing
4 JOHN BOURKE AND RICHARD GARNER
them to be constructed simply by giving a double category of the correct form
to be one’s double category of R-algebras or L-coalgebras.
The second main result of this paper deals with the appropriate notion of
cofibrant generation for awfs. Recall that a weak factorisation system (L, R) is
cofibrantly generated if there is a mere set of maps J such that R is the precisely
the class of maps with the right lifting property against each j ∈ J. Cofibrantly
generated weak factorisation systems are commonplace due to Quillen’s small
object argument [42, §II.3.2], which ensures that for any set of maps J in a locally
presentable category [19], the awfs (L, R) cofibrantly generated by J exists.
In [22, Definition 3.9] is described a notion of cofibrant generation for awfs:
given a small category U : J → C2 over C2, the awfs cofibrantly generated by
J , if it exists, has as R-algebras, maps g equipped with choices of right lifting
against each map Uj, naturally with respect to maps of J . This notion is already
more permissive than the usual one—as witnessed by [45, Example 13.4.5], for
example—but we argue that it is still insufficient, since it excludes important
examples on awfs, such as the ones on Cat listed above, whose R-algebras are
not retract-closed.
To rectify this, we introduce in Section 6.2 the notion of an awfs being
cofibrantly generated by a small double category U : J → Sq(C). This condition
specifies the R-algebras as being maps equipped with liftings against Uj for each
vertical map j ∈ J, naturally with respect to squares of J, but now with the
extra requirement that, for each pair of composable vertical maps j : x → y
and k: y → z of J, the specified lifting against kj should be obtained by taking
specified lifts first against k and then against j.
This broader notion of cofibrant generation is permissive enough to capture
all our leading examples, including the ones on Cat listed above. Our second
main theorem justifies it at a theoretical level, by characterising the cofibrantly
generated awfs on locally presentable categories as being exactly the accessible
awfs—those (L, R) whose comonad L and monad R preserve κ-filtered colimits
for some regular cardinal κ. More precisely, we show for a locally presentable C
that every small J → Sq(C) cofibrantly generates an accessible awfs on C; and
that every accessible awfs on C is cofibrantly generated by a small J → Sq(C).
We now give a summary of the contents of the paper. We begin in Section 2
with a revision of the basic theory of awfs: the definition, the relation with weak
factorisation systems, double categories of algebras and coalgebras, morphisms of
awfs, and the fully faithful 2-functors (–)-Alg and (–)-Coalg. In Section 3, we give
our first main result, the Beck theorem for awfs described above, together with
two useful variants. Section 4 then uses the Beck theorem to give constructions
of a wide range of awfs; in particular, we discuss the awfs for split epis in
any category with binary coproducts; the awfs for lalis (left adjoint left inverse
functors) in any 2-category with oplax limits of arrows; and the construction of
injective and projective liftings of awfs.
Section 5 revisits the notion of cofibrant generation of awfs by small categories,
as introduced in [22], and uses the Beck theorem to give a simplified proof that
such awfs always exist in a locally presentable C. This prepares the way for
our second main result; in Section 6 we introduce cofibrant generation by small
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 5
double categories, and in Section 33, show that over a locally presentable base C,
such awfs always exist and are precisely the accessible awfs on C.
Finally in Section 8 we say a few words about enriched cofibrant generation. As
mentioned above, a sequel to this paper will deal with this in greater detail; here,
we content ourselves with giving the basic construction and a range of applications.
In particular, we will see how to express notions such as Grothendieck fibrations,
categories with limits, and Kan extensions in terms of algebras for suitable awfs,
and explain how to extend these constructions to the quasicategorical context.
2. Revision of algebraic weak factorisation systems
Algebraic weak factorisation systems were introduced in [25]—there called
natural weak factorisation systems—and their theory developed further in [2, 6,
22, 43, 44]. They are highly structured objects, and an undisciplined approach
runs the risk of foundering in a morass of calculations. The above papers, taken
together, show that a smoother presentation is possible; in this introductory
section, we draw together the parts of this presentation so as to give a concise
account of the basic aspects of the theory.
Before beginning, let us state our foundational assumptions. κ will be a
Grothendieck universe, and sets in κ will be called small, while general sets will
be called large. Set and SET are the categories of small and large sets; Cat and
CAT are the 2-categories of small categories (ones internal to Set) and of locally
small categories (ones enriched in Set). Throughout the paper, all categories will
be assumed to be locally small and all 2-categories will be assumed to be locally
small (=Cat-enriched) except for ones whose names, like SET or CAT, are in
capital letters.
2.1. Functorial factorisations. By a functorial factorisation on a category C, we
mean a functor F : C2 → C3 from the category of arrows to that of composable
pairs which is a section of the composition functor C3 → C2. We write F =
(L, E, R), to indicate that the value of F at an object f or morphism (h, k): f → g
of C2 is given as on the left and the right of:
X
Lf
−−→ Ef
Rf
−−→ Y
X
Lf
GG
h

Ef
E(h,k)

Rf
GG Y
k

W
Lg
GG Eg
Rg
GG Z .
Associated to (L, E, R) are the functors L, R: C2 → C2 with actions on objects
f → Lf and f → Rf, and the natural transformations : L ⇒ 1 and η: 1 ⇒ R
with components at f given by the commuting squares:
(2.1)
A
1 GG
Lf

A
f

Ef
Rf
GG B
and
A
Lf
GG
f

Ef
Rf

B
1 GG B .
Note that (L, E, R) is completely determined by either (L, ) or (R, η).
6 JOHN BOURKE AND RICHARD GARNER
2.2. Algebraic weak factorisation systems. An algebraic weak factorisation system
(awfs) on C comprises:
(i) A functorial factorisation (L, E, R) on C;
(ii) An extension of (L, ) to a comonad L = (L, , ∆);
(iii) An extension of (R, η) to a monad R = (R, η, µ);
(iv) A distributive law [9] of the comonad L over the monad R, whose underlying
transformation δ: LR ⇒ RL satisfies dom·δ = cod·∆ and cod·δ = dom·µ.
This definition involves less data than may be immediately apparent. The
counit and unit axioms L · ∆ = 1 and µ · ηR = 1 force the components of
∆: L → LL and µ: RR → R to be identities in their respective domain and
codomain components, so given by commuting squares as on the left and right of:
(2.2)
A
Lf

1 GG A
LLf

Ef
∆f
GG ELf
Ef
LRf

∆f
GG ELf
RLf

ERf
µf
GG Ef
ERf
µf
GG
RRf

Ef
Rf

B
1 GG B ;
while the conditions on δ in (iv) force its component at f to be given by the
central square. Thus the only additional data beyond the underlying functorial
factorisation (i) are maps ∆f : Ef → ELf and µf : ERf → Ef, satisfying
suitable axioms. Note that the comonad L and monad R between them contain
all of the data, so that we may unambiguously denote an awfs simply by (L, R).
By examining (2.1) and (2.2), we see that the comonad L of an awfs is
a comonad over the domain functor dom: C2 → C. By this we mean that
dom · L = dom and that dom · and dom · ∆ are identity natural transformations.
Dually, the monad R is a monad over the codomain functor.
2.3. Coalgebras and algebras. To an algebraic weak factorisation system (L, R),
we may associate the categories L-Coalg and R-Alg of coalgebras and algebras for
L and R; these are to be thought of as constituting the respective left and right
classes of the awfs. Because L is a comonad over the domain functor, a coalgebra
structure f → Lf on f : A → B necessarily has its domain component an identity,
and so is determined by a single map s: B → Ef; we write such a coalgebra as
f = (f, s): A → B. Dually, an R-algebra structure on g: C → D is determined
by a single map p: Ef → C, and will be denoted g = (g, p): C → D. If the base
category C has an initial object 0, then we may speak of algebraically cofibrant
objects, meaning L-coalgebras 0 → X; dually, if C has a terminal object 1, then
an algebraically fibrant object is an R-algebra X → 1.
2.4. Lifting of coalgebras against algebras. Given an L-coalgebra f = (f, s), an
R-algebra g = (g, p) and a commuting square as in the centre of:
(2.3)
A
f

a GG A
f

B
b
GG B
A
f

u GG C
g

B
bb
v
GG D
C
g

c GG C
g

D
d
GG D
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 7
there is a canonical diagonal filler Φf,g(u, v): B → C given by the composite
p · E(u, v) · s: B → Ef → Eg → C. These fillers are natural with respect
to morphisms of L-coalgebras and R-algebras; which is to say that if we have
commuting squares as on the left and right above which underlie, respectively,
an L-coalgebra morphism f → f and an R-algebra morphism g → g , then
composing (2.3) with these two squares preserves the canonical filler:
c · Φf,g(u, v) · b = Φf ,g (cua, dvb) .
Writing U : L-Coalg → C2 and V : R-Alg → C2 for the forgetful functors, this
naturality may be expressed by saying that the canonical liftings constitute the
components of a natural transformation
(2.4) Φ: C2
(U–, V ?) ⇒ C(cod U–, dom V ?): L-Coalgop
× R-Alg → Set .
2.5. Factorisations with universal properties. The two parts of the factorisation
f = Rf · Lf of a map underlie the cofree L-coalgebra Lf = (Lf, ∆f ): A → Ef
and the free R-algebra Rf = (Rf, µf ): Ef → B. The freeness of the latter says
that, for any R-algebra g = (g, p) and morphism (h, k): f → g in C2, there is a
unique arrow such that the left-hand diagram in
(2.5)
A
h GG
f

Lf
22
C
g

Ef
Rf

∃!
aa
B
1 GG B
k GG D
A
h GG
f

C
1 GG
Lg

C
g

Ef
Rg
33
B
∃!m
bb
k GG D
commutes and such that ( , k) is an algebra morphism Rf → g. The dual
universal property, as on the right above, describes the co-freeness of Lg with
respect to maps out of an L-coalgebra f = (f, s).
Observe also the following canonical liftings involving (co)free (co)algebras:
A
Lg

1 GG A
(g,p)

Eg
Rg
GG
p
bb
B
A
(f,s)

Lf
GG A
Rf

B
1
GG
s
bb
B
Ef
LRf

∆f
GG ELf
RLf

ERf µf
GG
∆f ·µf
YY
Ef .
(2.6) (2.7) (2.8)
Here, (2.6) implies that an R-algebra is uniquely determined by its liftings against
L-coalgebras, and dually in (2.7); while (2.8) expresses precisely the non-trivial
axiom of the distributive law δ: LR ⇒ RL.
2.6. Underlying weak factorisation system. By the preceding two sections, the
classes of maps in C that admit L-coalgebra or R-algebra structure satisfy all of
the axioms needed to be the two classes of a weak factorisation system, except
maybe for closure under retracts. On taking the retract-closures of these two
classes, we thus obtain a weak factorisation system (L, R), the underlying weak
factorisation system of (L, R).
8 JOHN BOURKE AND RICHARD GARNER
2.7. Algebras and coalgebras from liftings. A coalgebra lifting operation on a map
g: C → D is a function ϕ–g assigning to each L-coalgebra f and each commuting
square (u, v): f → g as in (2.3) a diagonal filler ϕf,g(u, v): B → C, naturally in
maps of L-coalgebras. Maps equipped with coalgebra lifting operations are the
objects of a category L-Coalg over C2—the nomenclature will be explained in
Section 5.1 below—whose maps (g, ϕ–g) → (g , ϕ–g ) are maps (c, d): g → g of
C2 for which c · ϕf,g(u, v) = ϕf,g (cu, dv). Any R-algebra g induces the lifting
operation Φ–g on its underlying map, and by (2.4) any algebra map respects
these liftings; so we have a functor ¯Φ: R-Alg → L-Coalg over C2.
Lemma 1. ¯Φ: R-Alg → L-Coalg is injective on objects and fully faithful, and has
in its image just those (g, ϕ–g) such that ϕLf,g(u, v)·µf = ϕLRf,g(ϕLf,g(u, v), v·µf )
for all maps (u, v): Lf → g:
(2.9)
A
Lf

1 GG A
Lf

u GG C
g

Ef
LRf

ERf µf
GG
WW
ϕ(u,v)·µf
Ef v
GG
hh
ϕ(u,v)
D
=
A
Lf

u GG C
g

Ef
LRf

WW
ϕ(u,v)
ERf v·µf
GG
ff
ϕ(ϕ(u,v),vµ
f)
D .
Proof. We observed in Section 2.5 that an R-algebra is determined by its liftings
against coalgebras, whence ¯Φ is injective on objects. For full fidelity, let g = (g, p)
and h = (h, q) be R-algebras and (c, d): (g, Φ–g) → (h, Φ–h) a map of underlying
lifting operations; we must show (c, d) is an R-algebra map. Consider the diagram
C
Lg

1 GG C
g

c GG D
h

Eg
Rg
GG
p
bb
C
d
GG C
=
C
Lg

c GG C
Lh

1 GG C
h

Eg
E(c,d)
GG Eh
Rh
GG
q
aa
D .
By (2.6) Φ–g assigns the filler p to the far left square; so as (c, d) respects liftings,
Φ–h assigns the filler c · p to the composite left rectangle. Likewise Φ–h assigns
the filler q to the far right square; so as the square to its left is a (cofree) map of
L-coalgebras, Φ–h assigns the filler q · E(c, d) to the composite right rectangle.
Thus c · p = q · E(c, d) and (c, d) is an R-algebra map as required.
We next show that each (g, Φ–g) in the image of ¯Φ satisfies (2.9). By universality
(2.5) and naturality (2.4), it suffices to take g = RLf and (u, v) = (LLf, 1),
and now
ΦLf,RLf (LLf, 1) · µf = ∆f · µf = ΦLRf,RLf (∆f , µf )
= ΦLRf,RLf (ΦLf,RLf (LLf, 1), µf )
by (2.6) and (2.8), as required. Finally, we show that ¯Φ is surjective onto those
pairs (g, ϕ–g) satisfying (2.9). Given such a pair, we define p = ϕLg,g(1, Rg);
now by universality (2.5) and naturality (2.4), the pair (g, ϕ–g) will be the image
under ¯Φ of g = (g, p) so long as (g, p) is in fact an R-algebra. The unit axiom is
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 9
already inherent in p’s being a lifting; as for the multiplication axiom, we have
p · µg = ϕLg,g(1, Rg) · µg = ϕLRg,g(ϕLg,g(1, Rg), Rg · µg)
= ϕLRg,g(p, RRg) = ϕLRg,g(p, Rg · E(p, 1))
= ϕLg,g(1, Rg) · E(p, 1) = p · E(p, 1)
by definition of p, (2.9), and naturality of ϕ–g with respect to the L-coalgebra
morphism (p, E(p, 1)): LRg → Lg.
It is not hard to see that the category L-Coalg is in fact isomorphic to the
category (R, η)-Alg of algebras for the mere pointed endofunctor (R, η) underlying
the monad R, with the functor ¯Φ corresponding under this identification to the
natural inclusion of R-Alg into (R, η)-Alg. Of course, all these results have a
dual form characterising coalgebras in terms of their liftings against algebras.
2.8. Double categories of algebras and coalgebras. In a weak factorisation system
the left and right classes of maps are closed under composition and contain the
identities. We now describe the analogue of this for awfs. For binary composition,
given R-algebras g: A → B and h: B → C we may define a coalgebra lifting
operation Φ–,h·g on the composite underlying map h·g, whose liftings are obtained
by first lifting against h and then against g, as on the left in:
(2.10) Φf,h·g(u, v) = Φf,g(u, Φf,h(gu, v)) Φf,1A
(u, v) = v .
It is easy to verify that this is a coalgebra lifting operation satisfying (2.9), so that
by Lemma 1, it is the canonical lifting operation associated to a unique R-algebra
h·g: A → C, the composite of g and h. As for nullary composition, each identity
map 1A bears a unique coalgebra lifting operation as on the right above, which
is easily seen to satisfy (2.9); so each identity map bears an R-algebra structure
1A which is in fact unique.
This composition law for algebras is associative and unital: to see associativity,
we check that k · (h · g) and (k · h) · g have the same lifting operations, and
apply Lemma 1; unitality is similar. We may also use Lemma 1 to verify that
each (a, a): 1A → 1A is a map of R-algebras, and that if (a, b): g → g and
(b, c): h → h are maps of R-algebras, then so too is (a, c): h · g → h · g .
Consequently, there is a double category R-Alg whose objects and horizontal
arrows are those of C, and whose vertical arrows and squares are the R-algebras
and the maps thereof. There is a forgetful double functor UR : R-Alg → Sq(C)
into the double category of commutative squares in C, which, displayed as an
internal functor between internal categories in CAT, is as on the left in:
R-Alg ×C R-Alg
UR×CUR
GG
m

p

q

C3
p

q

m

R-Alg
d

c

UR
GG C2
d

c

C
i
yy
1
GG C
i
yy
A1×A0 A1
V1×V0
V1
GG
m

p

q

C3
m

p

q

A1
d

c

V1
GG C2
d

c

A0
i
yy
V0=1
GG C .
i
yy
10 JOHN BOURKE AND RICHARD GARNER
Note that this double functor has object component an identity and arrow
component a faithful functor. By a concrete double category over C, we mean
a double functor V : A → Sq(C) (as on the right above) whose V has these
two properties. For example, the L-coalgebras also constitute a concrete double
category L-Coalg over C.
We note before continuing that the equality (2.9) may now be re-expressed as
saying that each square as on the left below is one of L-Coalg, and each square
on the right is one of R-Alg. These squares will be important in what follows.
(2.11)
A
Lf

1 GG A
Lf

Ef
LRf

ERf µf
GG Ef
Ef
Rf

∆f
GG ELf
RLf

Ef
Rf

B
1
GG B .
2.9. Morphisms of AWFS. Given awfs (L, R) and (L , R ) on categories C and
D, a lax morphism of awfs (F, α): (C, L, R) → (D, L , R ) comprises a functor
F : C → D and a natural family of maps αf rendering commutative each square
as on the left in:
(2.12)
FA
L Ff
||
FLf
44
E Ff
αf
GG
R Ff 44
FEf
FRf||
FB
E Ff
αf
GG
E (γA,γB)

FEf
γEf

E Gf
βf
GG GEf ,
and such that the induced (α, 1): R F2 ⇒ F2R and (1, α): L F2 ⇒ F2L are
respectively a lax monad morphism R → R and a lax comonad morphism L → L
over F2 : C2 → D2 (i.e., a monad functor and a comonad opfunctor in the
terminology of [48]). A transformation (F, α) ⇒ (G, β) between lax morphisms is
a natural transformation γ : F ⇒ G rendering commutative the square above right
for each f : A → B in C. Algebraic weak factorisation systems, lax morphisms
and transformations constitute a 2-category AWFSlax. Dually, an oplax morphism
of awfs involves maps αf with the opposing orientation to (2.12), and with the
induced (α, 1) and (1, α) now being oplax monad and comonad morphisms over
F2. With a similar adaptation on 2-cells, this yields a 2-category AWFSoplax.
2.10. Adjunctions of AWFS. Given awfs (L, R) and (L , R ) on categories C and
D, and an adjunction F G: C → D, there is a bijection between 2-cells α
exhibiting G as a lax awfs morphism and 2-cells β exhibiting F as oplax; this is
the doctrinal adjunction of [35]. From α we determine the components of β by
βf = FE f
FE (ηA,ηB)
−−−−−−−−→ FE GFf
FαF f
−−−−→ FGEFf
EF f
−−−→ EFf ,
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 11
so that β is the mate [38] of α under the adjunctions F G and F2 G2.
The functoriality of this correspondence is expressed through an identity-onobjects
isomorphism of 2-categories AWFSradj
coop ∼= AWFSladj, where AWFSradj
is defined identically to AWFSlax except that its 1-cells come equipped with
chosen left adjoints, and where AWFSladj is defined from AWFSoplax dually. By an
adjunction of awfs, we mean a morphism of one of these isomorphic 2-categories;
thus, a pair of a lax awfs morphism (G, α) and an oplax awfs morphism (F, β)
whose underlying functors are adjoint, and whose 2-cell data determine each
other by mateship. In this situation, an easy calculation shows that
(2.13) Φf,Gg(u, v) = ΦFf,g(¯u, ¯v)
for any L -coalgebra f, any R-algebra g, and any (u, v): f → Gg in D2 with
adjoint transpose (¯u, ¯v): Ff → g in C2.
2.11. Semantics 2-functors. A lax morphism of awfs (F, α): (C, L, R) → (C , L , R )
has an underlying lax monad morphism, yielding as in [30, Lemma 1] a lifted
functor as to the left in
(2.14)
R-Alg
UR

¯F GG R -Alg
UR

C2
F2
GG D2
R-Alg
UR

¯F GG R -Alg
UR

Sq(C)
Sq(F)
GG Sq(D) ,
whose action on algebras we will abusively denote by g → Fg. A short calculation
from the fact that (1, α) is a lax comonad morphism shows that FΦLf,g(u, v)·αf =
ΦL Ff,Fg(Fu, Fv · αf ) for each R-algebra g. From this and (2.10) it follows that
ΦL Ff,Fh·Fg(Fu, Fv · αhg) = ΦL Ff,F(h·g)(Fu, Fv · αhg) for all composable Ralgebras
g and h; now taking f = hg and (u, v) = (1, Rhg) and applying (2.6),
we conclude that Fh · Fg = F(h · g). Thus the lifted functor on the left
of (2.14) preserves algebra composition; it must also preserve the (unique)
algebra structures on identities, and so underlies a double functor as on the
right. Moreover, each 2-cell γ : (F, α) → (G, β) in AWFSlax has an underlying
lax monad transformation, so that γ : F → G may be lifted to a transformation
on categories of algebras, which—by concreteness—lifts further to a horizontal
transformation on double categories.
The preceding constructions have evident duals involving AWFSoplax and
coalgebras; and in this way, we obtain the left and right semantics 2-functors:
(2.15) (–)-Coalg: AWFSoplax → DBL2
(–)-Alg: AWFSlax → DBL2
,
where DBL denotes the 2-category of double categories, double functors and
horizontal transformations. The following result is now [43, Lemma 6.9]:
Proposition 2. The 2-functors (–)-Alg and (–)-Coalg are 2-fully faithful.
Proof. By duality, we need only deal with the case of (–)-Alg. Note that each
double functor Sq(C) → Sq(D) must be of the form Sq(F) for some F : C → D;
thus given (C, L, R) and (D, L , R ) in AWFSlax, a morphism between their images
in DBL2
amounts to a square as on the right of (2.14). Each such has its
12 JOHN BOURKE AND RICHARD GARNER
underlying action on vertical arrows and squares as on the left in (2.14), and so
must be induced by a unique lax monad morphism γ : R → R over F2. As R and
R are monads over the codomain functor, we must have γ = (α, 1) for natural
maps αf as in (2.12). It remains to show that (1, α) is a lax comonad morphism
L → L over F2. The compatibilities in (2.12) already show that (1, α) commutes
with counits; we need to establish the same for the comultiplications. Consider
the diagrams:
E Ff
R Ff

∆F f
GG E L Ff
R L Ff

E (1,αf )
GG E FLf
R FLf

αLf
GG FELf
FRLf

E Ff αf
GG
R Ff

FEf
FRf

1
GG FEf
FRf

FB
1
GG FB
1
GG FB
1
GG FB
E Ff
R Ff

αf
GG FEf
FRf

F∆f
GG FELf
FRLf

FEf
FRf

FB
1
GG FB
1
GG FB .
Every region here is a square of R -Alg: the leftmost as it is of the form (2.11),
the rightmost as it is the image under the lifted double functor of another such
square in R-Alg, and all the rest because (α, 1) is a lax monad morphism. So the
exterior squares are also squares in R -Alg; but as both have the same composite
FLLf with the unit morphism (L Ff, 1): Ff → R Ff, they must, by freeness of
R Ff, coincide; now the equality of their domain-components expresses precisely
the required compatibility with comultiplications.
This completes the proof of full fidelity on 1-cells; on 2-cells, it is easy to see
that in the commuting diagram of 2-functors
AWFSlax
(–)-Alg
GG
(C,L,R)→(C2,R)

DBL2
(–)1

MNDlax
(–)-Alg
GG CAT2
the left and bottom edges are locally fully faithful, and that the right edge is
locally fully faithful on the concrete double categories in the image of (–)-Alg;
whence the top edge is also locally fully faithful.
2.12. Orthogonal factorisation systems. Recall that an orthogonal factorisation
system [18] is a weak factorisation systems (L, R) wherein liftings (1.1) of L-maps
against R-maps are unique. The following result characterises those awfs arising
from orthogonal factorisation systems: it improves on Theorem 3.2 of [25] by
requiring idempotency of only one of L or R. This resolves the open question
posed in Remark 3.3(a) of ibid.
Proposition 3. Let (L, R) be an awfs on C. The following are equivalent:
(i) L-Coalg → C2 is fully faithful;
(ii) L is an idempotent comonad;
(iii) For each L-coalgebra f : A → B, there is a coalgebra map (f, 1): f → 1B;
(iv) R-Alg → C2 is fully faithful;
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 13
(v) R is an idempotent monad;
(vi) For each R-algebra g: C → D, there is an algebra map (1, g): 1C → g;
(vii) Liftings of L-coalgebras against R-algebras are unique;
(viii) The underlying weak factorisation system (L, R) is orthogonal.
Under these circumstances, moreover, the awfs (L, R) is determined up to isomorphism
by the underlying (L, R).
Proof. (i) ⇒ (ii) ⇒ (i) by standard properties of idempotent comonads, and
(i) ⇒ (iii) is trivial. We next prove (iii) ⇒ (vii). So given f ∈ L-Coalg and
g ∈ R-Alg, we must show that any diagonal filler j for a square (u, v): f → g, as
on the left below, is equal to the canonical filler Φf,g(u, v).
A
f

u GG C
g

B
j
bb
v
GG D
=
A
f

f
GG B
1

j
GG C
g

B
1
GG B v
GG D .
So factorise (u, v) as on the right above; by (iii), the left-hand square is a coalgebra
map f → 1B, whence by naturality (2.4) we have j = Φ1B,g(j, v) = Φf,g(u, v)
as required. We next prove (vii) ⇒ (iv); so for algebras g and h, we must
show that each (c, d): g → h underlies an algebra map g → h. By Lemma 1,
it suffices to show that (c, d) commutes with the coalgebra lifting functions;
in other words, that for each coalgebra f and each (u, v): f → g, we have
c · Φf,g(u, v) = Φf,h(cu, dv). This is so by (vii) since both these maps fill the
square (cu, dv): f → h.
Dual arguments now prove that (iv) ⇒ (v) ⇒ (iv) ⇒ (vi) ⇒ (vii) ⇒ (i); it
remains to show that (i)–(vii) are equivalent to (viii). Clearly (viii) ⇒ (vii);
on the other hand, if (i) and (iv) hold, then any retract of an L-coalgebra or
R-algebra is again a coalgebra or algebra—being given by the splitting of an
idempotent in R-Alg or L-Coalg—so that any map in L or R is the underlying
map of an L-coalgebra or R-algebra; whence by (vii) liftings of L-maps against
R-maps are unique.
Finally, if (i)–(viii) hold, then we can reconstruct L and R up to isomorphism
from R-Alg → C2 and L-Coalg → C2, and can reconstruct these in turn from L
and R as the full subcategories of C2 on the L-maps and the R-maps.
3. A Beck theorem for awfs
In this section, we give our first main result—Theorem 6 below—which provides
an elementary characterisation of the concrete double categories in the essential
image of the semantics 2-functors (2.15). This will allow us to construct an awfs
simply by exhibiting a double category of an appropriate form to be one’s double
category of algebras or coalgebras. As explained in the introduction, the essential
images of the corresponding semantics 2-functors for monads and comonads are
characterised by Beck’s (co)monadicity theorem, and so we term our result a
“Beck theorem” for algebraic weak factorisation systems.
14 JOHN BOURKE AND RICHARD GARNER
3.1. Reconstruction. There are two main aspects to the Beck theorem. The first
is the following reconstruction result; in the special case where R is idempotent,
this is [29, Proposition 5.9], while the general case is given as [6, Theorem 4.15];
for the sake of a self-contained presentation, we include a proof here, which
improves on that of [6] only in trivial ways.
Proposition 4. Let R: C2 → C2 be a monad over the codomain functor. The right
semantics 2-functor (2.15) induces a bijection between extensions of R to an
algebraic weak factorisation system (L, R) and extensions of the diagram
(3.1)
R-Alg
dUR

cUR

UR
GG C2
d

c

C
1 GG C
to a concrete double category over C.
Proof. To give an extension of (3.1) to a concrete double category over C is to
give a composition law h, g → h g on R-algebras which is associative, unital,
and compatible with R-algebra maps. For each such, we must exhibit a unique
(L, R) whose induced R-algebra composition is . As in Section 2.1, the monad
R determines the (L, ) underlying L, and so we need only give the maps ∆f
satisfying appropriate axioms. Consider the diagram on the left in:
A
LLf
GG
f

Lf
22
ELf
RLf

Ef
Rf

∆f
``
Ef
Rf

B
1 GG B
1 GG B
A
f
GG
f

Lf
22
B
1B

Ef
Rf

Rf
bb
B
1 GG B
1 GG B .
The outer square commutes, and the right edge bears the R-algebra structure
Rf RLf; now applying (2.5) yields the map ∆f , with the induced square
forming an R-algebra morphism Rf → Rf RLf. If did arise from an awfs,
this induced square would be exactly the right square of (2.11); whence these
∆f ’s are the unique possible choice for L’s comultiplication. By pasting together
appropriate squares as in the proof of Proposition 2—involving algebra squares
as on the left above, and also ones (Rf, 1): Rf → 1B induced by universality
as on the right above—we may now check that these ∆f ’s satisfy the comonad
and distributivity axioms; we thus have the desired (L, R), and it remains only
to show that the composition law it induces coincides with . So let g and h be
composable R-algebras, write f = hg for the composite of the underlying maps,
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 15
and consider the left-hand diagram in:
A
1
GG
Lf

LLf 55
A
g

ELf
RLf

∃!
``
B
h

Ef
Rf

1
GG Ef
Rf

∃!m
aa
C
1 GG C
1 GG C
Ef
Rf

∆f
GG ELf
RLf

GG A
g

Ef
m GG
Rf

B
h

C
1 GG C
1 GG C .
The dotted maps m and are obtained by successively applying (2.5) to the
maps (g, 1): f → h and (1, m): Lf → g; thus the two squares on the far right
above are maps of R-algebras RLf → g and Rf → h. The rectangle to the left
of these squares is a map of R-algebras Rf → Rf RLf by definition, and also
one Rf → Rf · RLf, by (2.11). Thus the composite square can be seen both
as a map Rf → RLf · Rf → h · g and as one Rf → RLf Rf → h g. Since
precomposing further with the unit f → Rf yields the identity square on f = hg,
we conclude by universality of Rf that ( ∆f , 1) is both the R-algebra structure
of h · g and that of h g, which thus coincide.
3.2. Monads over the codomain functor. What is missing from the last result is
a characterisation of when a monad on an arrow category is over the codomain
functor. Our next result provides this; it generalises the characterisation in [29,
Proposition 5.1] of idempotent monads over the codomain functor.
Proposition 5. A monad R on C2 is isomorphic to one over the codomain functor
if and only if:
(a) Each identity map has an R-algebra structure 1A;
(b) For each f : A → B, there is an algebra map (f, f): 1A → 1B;
(c) For each R-algebra g: A → B, there is an algebra map (g, 1): g → 1B.
Moreover, in this situation, the algebra structures on identity arrows are unique.
Proof. As the properties (a)–(c) are clearly invariant under monad isomorphism,
we may assume in the “only if” direction that R is a monad over the codomain
functor; now a short calculation shows that 1A = (1A, R1A): A → A is the unique
R-algebra structure on 1A, and that (b)–(c) are then satisfied. Conversely, let R
satisfy (a)–(c), and consider the unit map (r, s): f → Rf, as on the left in:
A
f

r GG C
Rf

B
s GG D
A
f

f
GG C
1

B
1 GG B
=
A
f

r GG C
Rf

t GG B
1

B
s GG D
u GG B .
We claim that s is invertible. Since Rf underlies the free R-algebra on f, and 1C
has by (a) the algebra structure 1C, the commuting square in the middle above
induces a unique algebra map (t, u): Rf → 1C making the rightmost diagram
commute. In particular us = 1, and it remains to show that su = 1. Using
(b), we have the algebra map (s, s) · (t, u): Rf → 1B → 1D, and using (c) we
16 JOHN BOURKE AND RICHARD GARNER
have (Rf, 1): Rf → 1D. These maps coincide on precomposition with the unit
(r, s): f → Rf, as on the left of:
A
f

r GG C
Rf

t GG B
1

s GG D
1

B
s GG D
u GG B
s GG D
=
A
f

r GG C
Rf

Rf
GG D
1

B
s GG D
1 GG D
C
Rf

1 GG C
R f

D
u GG B
So they must themselves agree; in particular, su = 1 and s and u are isomorphisms.
So defining R f = uRf, we have isomorphisms Rf ∼= R f for each f ∈ C2 as on
the right above; now successively transporting the functor R and the monad
structure thereon along these isomorphisms yields a monad R ∼= R over the
codomain functor, as required.
3.3. The Beck theorem. Combining the preceding two results, we obtain our
first main theorem, characterising the concrete double categories in the essential
image of the semantics 2-functor (–)-Alg. Of course, we have also the dual result,
which we do not trouble to state, characterising the essential image of (–)-Coalg.
Theorem 6. The 2-functor (–)-Alg: AWFSlax → DBL2
has in its essential image
exactly those concrete double categories V : A → Sq(C) such that:
(i) The functor V1 : A1 → C2 on vertical arrows and squares is strictly monadic;
(ii) For each vertical arrow f : A → B of A, the following is a square of A:
(3.2)
A
f
GG
f

B
1

B
1 GG B .
Proof. Any UR : R-Alg → Sq(C) clearly has property (i) while property (ii) follows
from Proposition 5. Suppose conversely that the concrete V : A → Sq(C) satisfies
(i) and (ii). As V1 is strictly monadic, A1 is isomorphic over C2 to the category
of algebras for the monad R induced by V1 and its left adjoint. This R satisfies
(a)–(c) of Proposition 5: (c) by virtue of (ii) above, and (a) and (b) using the
vertical identities of the double category A. Thus we have a monad R ∼= R
over the codomain functor; now transporting the double category structure of
V : A → Sq(C) along the isomorphisms A1
∼= R-Alg ∼= R -Alg and C2 yields
V : A → Sq(C) which, by Proposition 4, is in the image of (–)-Alg.
We will call a concrete double category right-connected if it satisfies (ii) above,
and monadic right-connected if it satisfies (i) and (ii). Now combining Theorem 6
with Proposition 4 and the remarks preceding it yields the following result; again,
there is a dual form which we do not state characterising AWFSoplax.
Corollary 7. The 2-category AWFSlax is equivalent to the full sub-2-category of
DBL2
on the monadic right-connected concrete double categories.
We conclude this section by describing two variations on our main result which
will come in useful from time to time.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 17
3.4. Discrete pullback-fibrations. Under mild conditions on the base category C,
we may rephrase Proposition 5 so as to obtain a more intuitive characterisation
of the concrete double categories over C in the essential image of the semantics
functor. Let us define a functor p: A → C2 to be a discrete pullback-fibration if,
for every g ∈ A over g ∈ C2 and every pullback square (h, k): f → g, there is a
unique arrow ϕ: f → g in A over (h, k), and this arrow is cartesian:
(3.3)
A
f

h GG C
g

∃!f
∃!ϕ
GG g 1 GG
B
k GG D .
If C has all pullbacks, then the codomain functor C2 → C is a fibration, with
the pullback squares in C2 as its cartesian arrows; now the fact that the lifts (3.3)
are cartesian implies that the composite functor cod · p: A → C2 → C is also a
Grothendieck fibration, with cartesian liftings preserved and reflected by p.
Proposition 8. If C has pullbacks, then a monad R on C2 is isomorphic to one
over the codomain functor if and only if UR : R-Alg → C2 is a discrete pullback-
fibration.
Under the hypotheses of this proposition, the composite cod · UR : R-Alg → C
is a Grothendieck fibration; which, loosely speaking, is the statement that “Ralgebra
structure is stable under pullback”.
Proof. First let R be over the codomain functor. Given an R-algebra g =
(g, p): C → D and pullback square as in (3.3), form the unique map q: Ef → A
with fq = Rf and hq = p·E(h, k). This easily yields an R-algebra (f, q): A → B
for which (h, k): (f, q) → (g, p) is a cartesian algebra map; moreover, any q
for which (h, k) were a map (f, q ) → (g, p) would have to satisfy the defining
conditions of q; whence q is unique as required. Conversely, suppose that UR is a
discrete pullback-fibration. Arguing as in Proposition 5, it suffices to show that
each unit-component (r, s): f → Rf has s invertible. Form the pullback of Rf
along s and induced map t as on the left in:
A
f

r
66t GG E
g

u GG C
Rf

B
1 GG B
s GG D
A
t GG
f

E
g

B
1 GG B
=
A
r GG
f

C
v GG
Rf

E
g

B
s GG D
w GG B .
As UR is a discrete pullback-fibration, there is a unique R-algebra structure g
on g making (u, s): g → Rf an algebra map; now applying freeness of Rf to
the central square above yields a unique algebra map (v, w): Rf → g making
the right-hand diagram commute. In particular, ws = 1; on the other hand, the
algebra map (uv, sw): Rf → g → Rf precomposes with the unit (r, s): f → Rf
to yield (r, s); whence (uv, sw) = (1, 1) and s is invertible as required.
The evident adaptation of the proof of Theorem 6 now yields:
18 JOHN BOURKE AND RICHARD GARNER
Theorem 9. If C admits pullbacks, then a concrete double category V : A → Sq(C)
is in the essential image of (–)-Alg: AWFSlax → DBL2
just when:
(i) V1 : A1 → C2 is strictly monadic;
(ii) V1 : A1 → C2 is a discrete pullback-fibration.
Remark 10. Note that for the “if” direction of this result, C does not need to have
all pullbacks; a closer examination of the proof of Proposition 8 shows that only
pullbacks along (the underlying maps of) R-algebras are needed. This is useful
in practice; categories with pullbacks of this restricted kind arise in the study of
stacks [47], or in the categorical foundations of Martin–L¨of type theory [20].
3.5. Intrinsic concreteness. Above, we have chosen to view the semantics 2functors
(2.15) as landing in the 2-category DBL2
. We have done this so as to
stress the analogy with the situation for monads and comonads, and also because
this is the most practically useful form of our results. Yet this presentation has
some redundancy, as we now explain.
In the presence of the remaining hypotheses, condition (ii) of Theorem 6
is easily seen to be equivalent to the requirement that the codomain functor
c: A1 → A0 of the double category A be a left adjoint left inverse for the identities
functor i: A0 → A1 (such lalis will appear again in Section 4.2 below). This is a
property of A, rather than extra structure; namely the property that
(3.4) each cf,iB : A1(f, iB) → A0(cf, B) is invertible.
But given only this, we may reconstruct the double functor V : A → Sq(A0).
First we reconstruct the components (3.2) of the unit η: 1 ⇒ ic using (3.4); now
we reconstruct V1 : A1 → A2
0 as the functor classifying the natural transformation
(3.5) A1
1
AA
ic
SSη
 A1
d GG A0 ;
finally, by pasting together unit squares and using (3.4), we verify that V = (id, V1)
preserves vertical composition, and so is a double functor. If W : B → Sq(B0) is
another concrete double category whose W is determined in this manner, then
any double functor F : A → B will by (3.4) satisfy F1 · ηA = ηB · F1, and so
Sq(F0) · V = W · F; that is, any double functor A → B is concrete.
Let us, therefore, define a double category A to be right-connected if it satisfies
(3.4), and monadic right-connected if in addition the induced functor
A1 → (A0)2 is strictly monadic. The preceding discussion now shows that:
Proposition 11. The assignation (C, L, R) → R-Alg gives an equivalence of 2categories
between AWFSlax and the full sub-2-category of DBL on the monadic
right-connected double categories.
4. Double categories at work
As a first illustration of the usefulness of the double categorical approach and
our Theorem 6, we use it to construct a variety of algebraic weak factorisation
systems, some well-known, and some new.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 19
4.1. Split epimorphisms. Given a category C, we write SplEpi(C) for the category
of split epimorphisms therein: objects are pairs of a map g: A → B of C together
with a section p of g, while morphisms (g, p) → (h, q) are serially commuting
diagrams as on the left in:
(4.1)
A
g

u GG C
h

B
p
yy
s
GG D
q
yy 1
m GG
1
00
0
e

me
00
1 m
GG 0 .
Split epimorphisms compose—by composing the sections—so that we have a
double category SplEpi(C) which is concrete over C and easily seen to be rightconnected.
It will thus be the double category of algebras of an awfs on C
whenever U : SplEpi(C) → C2 is strictly monadic.
Now, we may identify U with Cj : CS → C2, where S is the free split epimorphism,
drawn above right, and j : 2 → S is the evident inclusion. Thus U strictly
creates colimits, and so will be strictly monadic whenever it has a left adjoint:
which will be so whenever C is cocomplete enough to admit left Kan extensions
along j. Using the Kan extension formula one finds that only binary coproducts
are required; the free split epi Rf on f : A → B is f, 1 : A+B → B with section
ιB, while the unit f → Rf is given by:
(4.2)
A
f

ιA
GG A + B
f,1

B
1
GG B .
ιB
yy
The remaining structure of the awfs for split epis can be calculated using
Proposition 4. We find that each coproduct inclusion admits a L-coalgebra
structure; while if C is lextensive, then this coalgebra structure is unique, and
the category of L-coalgebras is precisely the category of coproduct injections
and pullback squares. This was proved in [25, Proposition 4.2], but see also
Example 12(iii) below.
Finally, let us observe an important property of the split epi awfs. Although
R = (R, η, µ) is a monad, its algebras—the split epis—are simultaneously the
algebras for its underlying pointed endofunctor (R, η); which is to say that the
monad R is algebraically-free [36, §22] on its underlying pointed endofunctor. We
will make use of this property in Proposition 16 below.
4.2. Lalis. In the terminology of Gray [26], a lali (left adjoint left inverse) in a
2-category C is a split epi (g, p): A → B with the extra property that g p with
identity counit. This property may be expressed either by requiring that for each
x: A → X and y: B → X in C, the function
(4.3) (–) · p: C(A, X)(x, yg) → C(B, X)(xp, y)
be invertible; or by requiring the provision of a—necessarily unique—unit 2-cell
η: 1 ⇒ pg satisfying gη = 1 and ηp = 1. Lalis in C form a category Lali(C),
20 JOHN BOURKE AND RICHARD GARNER
wherein a morphism is a commuting diagram as on the left of (4.1); it is automatic
by invertibility of (4.3) that such a morphism also commutes with the unit 2-cells.
Since split epis and adjoints compose, so too do lalis; thus—writing C0 for
the underlying category of C—lalis in C form a concrete sub-double category
Lali(C) of SplEpi(C0) which, since it is full on cells, inherits right-connectedness.
So Lali(C) will be the double category of algebras for an awfs on C0 whenever
U : Lali(C) → (C0)2 is strictly monadic. Now, we may identify U with restriction
2-CAT(j, C): 2-CAT(L, C) → 2-CAT(2, C)
along the inclusion j : 2 → L of 2 into the free lali L—which has the same
underlying category as the free split epimorphism S and a single non-trivial 2-cell
1 ⇒ me; so as before, monadicity obtains whenever C is cocomplete enough (as
a 2-category) to admit left Kan extensions along j. In this case, the colimits
needed are oplax colimits of arrows: given f : C → B, its oplax colimit is the
universal diagram as on the left of:
C
f
GG
00
α CQ
B
p

C
f
GG
f 11
1 CQ
B
1
A B .
Applying the one-dimensional aspect of this universality to the cocone on the
right yields a retraction g: A → B for p; now applying the two-dimensional
aspect shows that the pair (g, p) verifies invertibility in (4.3), and so is a lali, the
free lali on f, with as unit f → Rf the map ( , 1).
When C = Cat, we may characterise the corresponding category of L-coalgebras
as the subcategory of Cat2
whose objects are the categorical cofibrations—those
functors arising as pullbacks of the domain inclusion 1 → 2—and whose morphisms
are the pullback squares; see example (iv) in Section 4.4 below.1
On reversing or inverting the non-trivial 2-cell of L, it becomes the free rali
(right adjoint left inverse) Lco or the free retract equivalence Lg; from which we
obtain awfs whose algebras are ralis or retract equivalences on any 2-category
that admits lax colimits or pseudocolimits of arrows. When C = Cat, the
respective L-coalgebra structures on a morphism f are unique, and exist just
when f is a pullback of the codomain inclusion 1 → 2, respectively, when f is
injective on objects.
4.3. Via cocategories. The preceding examples fit into a common framework.
Let V be a suitable base for enriched category theory, and suppose that we are
given a cocategory object A in V-Cat:
A0 = I
d GG
c
GG A1ioo
p
GG
q
GGm GG A2
1This characterisation would extend to any suitably exact 2-category, except that there is no
account of the appropriate two-dimensional exactness notion; it appears to be closely related to
the one mentioned in the introduction to [49].
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 21
with object of co-objects the unit V-category, with d and c jointly bijective on
objects, and with i c with identity unit. For any V-category C, the image of A
under the limit-preserving V-CAT(–, C): V-Catop
→ CAT is an internal category
A(C) in CAT—a double category—which is right-connected and concrete over
C0, the underlying ordinary category of C. The induced comparison functor
A(C)1 → (C0)2 is the functor
(4.4) V-CAT(j, C): V-CAT(A1, C) → V-CAT(2, C)
where j : 2 → A1 is the functor from the free V-category on an arrow which
classifies the d-component of the unit η: 1 ⇒ ci: A1 → A1. This j is bijective on
objects, and so (4.4) will be monadic whenever C is sufficiently cocomplete that
each V-functor 2 → C admits a left Kan extension along j; whereupon Theorem 6
induces an awfs on C with A(C) as its double category of algebras. Note that
the assignation C → A(C) is clearly 2-functorial in C, so that by Proposition 11,
it underlies a 2-functor V-CAT → AWFSlax defined on all V-categories with
sufficient colimits and all V-functors (not necessarily cocontinuous) between them.
Of course, the preceding examples are instances of this framework on taking
V = Set with A1 = S (for split epis); or taking V = Cat with A1 = L, Lco or Lg
(for lalis, ralis or retract equivalences).
4.4. Stable classes of monics. Let C be a category with pullbacks, and consider a
class M of monics which contains the isomorphisms, is closed under composition,
and is stable under pullback along maps of C; we call this a stable class of monics.
The class M is said to be classified if the category Mpb of pullback squares
between M-maps has a terminal object, which we call a generic M-map. A
standard argument [31, p. 24] shows that the domain of a generic M-map must
be terminal in C.
Since M-maps and pullback squares compose, the category Mpb underlies a
double category Mpb concrete over C. The monicity of the M’s ensures that for
each m ∈ M, the square on the left below is a pullback and so in Mpb. Thus
Mpb is left-connected, and so will comprise the L-coalgebras of an awfs on C as
soon as the forgetful U : Mpb → C2 is comonadic.
A
1

1 GG A
m

A m
GG B
A1
f1
GG
a

B1
g1
GG
h1
GG
b

C1
c

A2
f2
GG B2
g2
GG
h2
GG C2
First we note that U always strictly creates equalisers. Indeed, if b c is a
parallel pair in Mpb as on the far right above, then on forming the equaliser
a → b in C2, as to the left, the equalising square will be a pullback—since c is
monic and both rows are equalisers—and so will lift uniquely to Mpb. Thus U
will be strictly comonadic as soon as it has a right adjoint.
For this, we must assume that the generic M-map p: 1 → Σ is exponentiable; it
follows that every M-map is exponentiable, since exponentiable maps are pullbackstable
in any category with pullbacks (see [50, Corollary 2.6], for example). In
particular, each map B×p: B → B×Σ is exponentiable, and so for any f : A → B
22 JOHN BOURKE AND RICHARD GARNER
of C, we may form the object ΠB×p(f) of C/(B × Σ); which has the universal
property that, in the category Pf whose objects are pullback diagrams as on the
left in
X

h GG A
f
GG B
B×p

Y
k
GG B × Σ
C
η

ε GG A
f
GG B
B×p

D
ΠB×p(f)
GG B × Σ
and whose maps (X, Y, h, k, ) → (X , Y , h , k , ) are maps X → X and Y → Y
satisfying the obvious equalities, there is a terminal object as on the right. Now,
an object of Pf determines and is determined by a pair of squares
X

h1
GG A
f
GG B
1

Y
k1
GG B
X

! GG 1
p

Y
k2
GG Σ
with the right-hand one a pullback. This ensures that —as a pullback of p—is
an M-map, whereupon by genericity of p, the map k2 is uniquely determined.
Thus we may identify Pf with the comma category U ↓ f, and the terminal
object therein provides the value η: C → D of the desired right adjoint at f.
For the (L, R) induced in this way, the restriction of the monad R to the slice
over B is the partial M-map classifier monad on C/B; see [46] and the references
therein. Note also that, in the terminology of [17], the ΠB×p(f) constructed
above is a pullback complement of f and B × p.
Examples 12. (i) Let E be an elementary topos, and M the stable class of all
monics. M is classified by : 1 → Ω, which—like any map in a topos—is
exponentiable; and so we have an awfs on E for which L-Coalg is the
category of monomorphisms and pullback squares. The corresponding
R-algebras are discussed in [39]. More generally, we can let E be any
quasitopos, and M the class of strong monomorphisms.
(ii) Let E be an elementary topos, j a Lawvere-Tierney topology on E, and
M the class of j-dense monomorphisms. This is a stable class of monics
classified by : 1 → J, where J is the equaliser of j, : Ω → Ω. So j-dense
monomorphisms and their pullbacks form the coalgebras of an awfs on
E. An algebraically fibrant object is a “weak sheaf”: an object equipped
with coherent, but not unique, choices of patchings for j-covers. An object
X is a sheaf if and only if both X → 1 and X → X × X admit R-algebra
structure.
(iii) Let C be a lextensive category and M the stable class of coproduct injections.
This is classified by ι1 : 1 → 1 + 1; which by extensivity is always
exponentiable, with Πι1 = (–) + 1: C → C/(1 + 1). More generally, the right
adjoint to pullback along B × ι1 is (–) + B : C/B → C/(B + B); whence the
awfs generated on C is that for split epis. This re-proves the fact that, in
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 23
any lextensive category, the coalgebras of the awfs for split epis are the
coproduct injections and pullback squares.
(iv) Consider, as in Section 4.2, the stable class of categorical cofibrations in
Cat. This is classified by the domain functor d: 1 → 2, which is exponentiable
in Cat, so that we have an awfs wherein L-coalgebras are the
categorical cofibrations and pullback squares. For any category B, the
functor ΠB×d : Cat/B → Cat/(B × 2) sends f ∈ Cat/B to the induced
map between the oplax colimits of f and of 1B—the latter being simply
B × 2—and it follows that this awfs is the one for lalis. This proves the
claim about its coalgebras made in Section 4.2 above.
(v) By a left cofibration of simplicial sets, we mean a pullback of the face
inclusion δ1 : ∆[0] → ∆[1]. The left cofibrations constitute a stable class of
monics, classified by δ1; and since SSet is a topos, this δ1 is exponentiable.
We thus have an awfs on simplicial sets whose L-coalgebras are left cofibrations
and pullbacks. The corresponding R-coalgebras we call the simplicial
lalis; we will have more to say about them in Section 8.2 below.
4.5. Projective and injective liftings. Our next example is a construction which,
under suitable circumstances, allows us to lift an awfs (D, L, R) along a functor
F : C → D to yield an awfs on C. There are two ways of doing this, well known
from the model category literature: we either lift the algebras (as in [16]) or the
coalgebras (as in [8]). In the former case, we form the pullback on the left in:
(4.5)
A GG
V

R-Alg
UR

Sq(C)
Sq(F)
GG Sq(D)
A1
GG
V1

R-Alg
UR

C2 F2
GG D2 .
The double category A therein, as a pullback of right-connected double categories,
is itself right-connected; while the component functor V1 of V as on the right
above is a pullback of the strictly monadic UR, whence easily seen to strictly
create coequalisers for V1-absolute coequaliser pairs. So whenever V1 admits a
left adjoint, A will by Theorem 6 constitute the algebra double category of an
awfs (L , R ) on C, the projective lifting of (L, R) along F. The left square of (4.5)
then amounts to a lax morphism of awfs (C, L , R ) → (D, L, R), which is easily
seen to be a cartesian lifting of F along the forgetful functor AWFSlax → CAT.
Dually, we may form the pullback of double categories on the left in
(4.6)
B GG
W

L-Coalg
UL

Sq(C)
Sq(F)
GG Sq(D)
B1
GG
W1

L-Coalg
UL

C2 F2
GG D2 ;
if on doing so, the component W1 of W as to the right has a right adjoint, then
B will be the coalgebra double category of an awfs on C, the injective lifting of
(L, R) along F. The left square of (4.6) now corresponds to an oplax map of awfs
providing a cartesian lifting of F with respect to the forgetful AWFSoplax → CAT.
24 JOHN BOURKE AND RICHARD GARNER
Sometimes, it may be that the required adjoints to V1 or W1 simply exist.
For example, the projective and injective liftings of an awfs (C, L, R) along a
forgetful functor C/X → C always exist, and coincide; the factorisations of the
resultant slice awfs are given by
A
f
GG
a 22
B
b~~
X
→
A
Lf
GG
a
22
Ef
Rf
GG
b·Rf

B .
b~~
X
More typically, we will appeal to a general result like the following, which
ensures that the desired adjoint exists without necessarily giving a closed formula
for it. As in the introduction, we call an awfs on a locally presentable category
accessible if its comonad L and monad R are so, in the sense of preserving κ-filtered
colimits for some regular cardinal κ; in fact, it is easy to see that accessibility of
the comonad implies that of the monad, and vice versa.
Proposition 13. Let C and D be locally presentable categories, and (L, R) an
accessible awfs on D.
(a) The projective lifting of (L, R) along any right adjoint F : C → D exists and
is accessible;
(b) The injective lifting of (L, R) along any left adjoint F : C → D exists and is
accessible.
Proof. For (a), let F : C → D be a right adjoint. To show the projective lifting
exists, we must prove that the functor V1 to the right of (4.5) has a left adjoint.
Now, C2 and D2 are locally presentable since C and D are, by [19, §7.2(h)], while
R-Alg is locally presentable since R is accessible, by [19, Satz 10.3]; moreover,
both F2 and UR are right adjoints. Since UR has the isomorphism-lifting property,
its pullback against F2 is also a bipullback [34]; but by [10, Theorem 2.18], the
2-category of locally presentable categories and right adjoint functors is closed
under bilimits in CAT, whence V1, like F2 and UR, lies in this 2-category; thus it
has a left adjoint K1 and is accessible by [19, Satz 14.6]. So the projective lifting
of (L, R) along F exists, and is accessible as its monad V1K1 is so.
For (b), let F : C → D be a left adjoint; we must show that W1 to the right
of (4.6) has a right adjoint. We now argue using [10, Theorem 3.15], which states
that the 2-category of locally presentable categories and left adjoint functors is
closed under bilimits in CAT. As before, F2 : C2 → D2 lies in this 2-category,
and UL : L-Coalg → D2 will so long as L-Coalg is in fact locally presentable. But
L is an accessible comonad on the accessible category D2; by [41, Theorem 5.1.6],
the 2-category of accessible categories and accessible functors is closed in CAT
under all bilimits, in particular under Eilenberg-Moore objects of comonads, and
so L-Coalg is accessible; it is also cocomplete (since UL creates colimits) and so
is locally presentable, as required. We thus conclude that the functor W1, like
UL and F2, is a left adjoint between locally presentable categories; while by [19,
Satz 14.6], its right adjoint G1 is accessible. So the projective lifting of (L, R)
along F exists, and is accessible as its comonad W1G1 is so.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 25
In fact, we may drop the requirement of local presentability from the first part
of the preceding proposition if we strengthen the hypotheses on U.
Proposition 14. Let D be a cocomplete category, let T be an accessible monad on
D and let (L, R) be an accessible awfs on D. Then the projective lifting of (L, R)
along the forgetful functor UT : T-Alg → D exists and is accessible.
Proof. As in (4.5), we form the pullback of double categories as to the left in
A GG
V

R-Alg
UR

Sq(T-Alg)
Sq(UT)
GG Sq(D)
A1
GG
V1

R-Alg

(T-Alg)2
(UT)2
GG D2
S oo σ
yy
ν
Ryy
!
T2 oo !
id
and must show that the underlying 1-component V1, as in the centre, has
a left adjoint. Now, the monad R on D2 is accessible by assumption, while
T2 is accessible since T is so; whence by [36, Theorem 27.1] the coproduct
S = T2 + R exists in MND(D2) and is accessible. This coproduct is equally
the pushout on the right above, and [36] guarantees that the functor (–)-Alg
sends this square to a pullback of categories, so that we may identify V1 with
ν∗ : S-Alg → T2-Alg = (T-Alg)2. Now as D2 is cocomplete and S is accessible,
this ν∗ has a left adjoint by [36, Theorem 25.4].
A standard application of the preceding results is to the construction of awfs
on diagram categories. If the cocomplete C bears an accessible awfs (L, R), and
I is a small category, then the functor U : CI → Cob I given by precomposition
with ob I → I has a left adjoint (given by left Kan extension) and is strictly
monadic; so that by Proposition 14 we may projectively lift the pointwise awfs
on Cob I along U to obtain the projective awfs on CI. If C is moreover complete,
then U also has a right adjoint given by right Kan extension; if it is moreover
locally presentable then we may apply Proposition 13(b) to injectively lift the
pointwise awfs along U, so obtaining the injective awfs on CI.
5. Cofibrant generation by a category
For our next application of Theorem 6, we use it to give a simplified treatment of
cofibrantly generated algebraic weak factorisation systems. An awfs is cofibrantly
generated when its R-algebras are precisely morphisms equipped with a choice
of liftings against some small category J → C2 of “generating cofibrations”.
This extends and tightens the familiar notion of cofibrant generation of a weak
factorisation system by a set of generating cofibrations; and the main result
of [22] extends and tightens Quillen’s small object argument to show that, under
commonly-satisfied conditions on C, the awfs cofibrantly generated by any small
J → C2 exists. The proof given there is quite involved; we will give an simpler
one exploiting Theorem 6. First we recall the necessary background.
5.1. Lifting operations. Given categories U : J → C2 and V : K → C2 over C2, a
(J , K)-lifting operation is a natural family of functions ϕj,k assigning to each
26 JOHN BOURKE AND RICHARD GARNER
j ∈ J and k ∈ K and each commuting square
dom Uj
Uj

u GG dom V k
V k

cod Uj
ϕj,k(u,v)
VV
v GG cod V k
a diagonal filler as indicated making both triangles commute. Naturality
here expresses that the ϕj,k’s are components of a natural transformation
ϕ: C2(U–, V ?) ⇒ C(cod U–, dom V ?): J op×K → Set. Thus Section 2.4 describes
the canonical (L-Coalg, R-Alg)-lifting operation associated to any awfs.
The assignation taking (J , K) to the collection of (J , K)-lifting operations is
easily made the object part of a functor Lift: (CAT/C2)op × (CAT/C2)op → SET;
we now have, as in [22, Proposition 3.8]:
Proposition 15. Each functor Lift(J , –) and Lift(–, K) is representable, so that
we induce an adjunction
(5.1) (CAT/C2)op
(–)
GG⊥ CAT/C2 .
(–)
oo
Proof. The category J → C2 representing Lift(J , –), has as objects, pairs of
g ∈ C2 and ϕ–g a (J , g)-lifting operation; and as maps (g, ϕ–g) → (h, ϕ–h), those
g → h of C2 which commute with the lifting functions. Dually, the representing
category K → C2 for Lift(–, K) comprises objects f ∈ C2 equipped with (f, K)lifting
operations, together with the maps of C2 between them that commute
with the lifting functions.
We have seen a particular instance of this result in Section 2.7 above; the
category L-Coalg described there is the representing object for Lift(L-Coalg, –),
and the functor ¯Φ: R-Alg → L-Coalg is the one induced by this representation
from the canonical lifting operation of Section 2.4.
5.2. Cofibrant generation. If the maps g: C → D and h: D → E of C are
equipped with lifting operations ϕ–g and ϕ–h against a category U : J → C2, then
their composite hg also bears a lifting operation ϕ–hg, defined as in (2.10) by
ϕj,hg(u, v) = ϕj,g(u, ϕj,h(gu, v)). This composition, together with the equipment
of an identity map with its unique lifting operation, provides the necessary
vertical structure to make J → C2 into a concrete double category J → Sq(C);
dually, we can make K → C2 into a concrete double category K → Sq(C).
We now define an awfs (L, R) on C to be cofibrantly generated by a small
J → C2 if R-Alg ∼= J over Sq(C). If this isomorphism is verified for a J which
is large, we say instead that C is class-cofibrantly generated by J → C2. There
are dual notions of fibrant or class-fibrant generation, involving an isomorphism
K ∼= L-Coalg over Sq(C); however, these are markedly less prevalent than their
duals in categories of mathematical interest.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 27
5.3. Existence of cofibrantly generated awfs. In [22, Definition 3.9] is given the
notion of an awfs being “algebraically-free” on U : J → C2: to which the notion
of cofibrant generation given above, though apparently different in form, is in
fact equivalent.2
Theorem 4.4 of ibid. thus guarantees, among other things, that
in a locally presentable category C, the awfs cofibrantly generated by any small
U : J → C2 exists. We now use Theorem 6 to give a shorter proof of this.
Proposition 16. If C is locally presentable then the awfs (L, R) cofibrantly generated
by any small U : J → C2 exists; its underlying monad R is algebraically-free
on a pointed endofunctor and accessible.
This result as stated is less general than [22, Theorem 4.4], which also deals
with certain kinds of non-locally presentable C. Though we have no need for
this extra generality here, let us note that reincorporating it would simply be a
matter of adapting the final paragraph of the following proof.
Proof. It is easy to see that V : J → Sq(C) is right-connected; so by Theorem 6,
it will comprise the double category of algebras for the desired awfs as soon as
V1 : J → C2 is strictly monadic. Now an object of J is equally a map g of C
together with a section ϕ–g of the natural transformation
(5.2) ψ–g : C(cod U–, dom g) ⇒ C2
(U–, g): J op
→ Set
whose j-component sends m: cod Uj → C to (m·Uj, gm): Uj → g; while a map
(g, ϕ–g) → (h, ϕ–h) of J is a map g → h of C2 for which the induced ψ–g → ψ–h
in [J op, Set]2 commutes with the sections. We thus have a pullback as on the
left in
(5.3)
J
V1

GG SplEpi([J op, Set])

C2 ψ
GG [J op, Set]2
Kψ
ε

Kηψ
GG KTψ

1
ρ
GG P .
We first show that ψ has a left adjoint. The composite cod·ψ: C2 → [J op, Set]
is the singular functor C2(U, 1) which has left adjoint F1 given by the left Kan
extension of U : J → C2 along the Yoneda embedding; while dom · ψ is the
functor C(cod · U, dom) ∼= C2(id · cod · U, 1) (as id dom: C2 → C) and so also
has a left adjoint F2. The natural transformation cod · ψ → dom · ψ induces one
α: F2 → F1, using which ψ admits a left adjoint K sending f : X → Y to the
pushout of F2f : F2X → F2Y along αX : F2X → F1X.
Now as we noted in Section 4.1 above, SplEpi([J op, Set]) may be identified
with the category of algebras for a pointed endofunctor (T, η) on [J op, Set]2 with
unit ηf : f → Tf given by (4.2). It follows that J is isomorphic over C2 to the
category of algebras for the pointed endofunctor (P, ρ) in the pushout square
above right (c.f. [53, Theorem 2.1]). An easy consequence of this identification
is that V1 strictly creates coequalisers of V1-absolute coequaliser pairs; so it
will be strictly monadic whenever it has a left adjoint—that is, whenever free
(P, ρ)-algebras exist. We show this existence using results of [36].
2This follows from Proposition 22 below.
28 JOHN BOURKE AND RICHARD GARNER
Since C is locally presentable, there is a regular cardinal κ such that the domain
and codomain of each Uj is κ-presentable; thus the domain and codomain of each
ψj– : C(cod Uj, dom –) ⇒ C2(Uj, –) preserves κ-filtered colimits, so that ψ does
so too. Now K, being a left adjoint, is cocontinuous, while T is so by inspection
of (4.2); whence the P of (5.3) is a pushout of functors preserving κ-filtered
colimits, and so also preserves them. Thus by [36, Theorem 22.3], free (P, ρ)algebras
exist, which is to say that V1 has a left adjoint F1 as required. Since P
preserves κ-filtered colimits, V1 creates them, and so the induced monad R = V1F1
preserves them; it is thus accessible. Finally, it follows from [36, Proposition 22.2]
that R is the algebraically-free monad on the pointed endofunctor (P, ρ).
This result provides a rich source of algebraic weak factorisation systems; in
particular, we may make any cofibrantly generated weak factorisation system on
a locally presentable category into an awfs by applying the result to its set J of
generating cofibrations, seen as a discrete category over C2. For applications that
make serious use of the algebraicity so obtained, see [21, 23]; for applications
utilising the extra generality of cofibrant generation by a small category, rather
than a small set of morphisms, see [5, 7].
5.4. Inadequacy of cofibrant generation by a category. Proposition 16 tells us
that the monad of an awfs cofibrantly generated by a small category has two
specific properties: it is accessible, and it is algebraically-free on a pointed
endofunctor. Our experience of monads suggests that the former property should
be more common than the latter, and in fact this is the case: many awfs of
practical interest verify the accessibility, but not the freeness. For example:
Proposition 17. The monad of the awfs for lalis on Cat is accessible, but
not algebraically-free on a pointed endofunctor; in particular, this awfs is not
cofibrantly generated by a small category.
Proof. The monad R at issue is given by left Kan extension and restriction
along the 2-functor j : 2 → L of Section 4.2; it is thus cocontinuous and in
particular, accessible. On the other hand, if it were algebraically-free on a pointed
endofunctor, then its category of algebras—Lali(Cat)—would be isomorphic to
the category of algebras for a pointed endofunctor, and as such would be retractclosed:
meaning that, for any lali f u and any retract g of f in Cat2
, there
would exist lali structure on g. Now, to equip the unique functor A → 1 with the
structure of a lali is to specify a terminal object of A; so it is enough to describe
a category A with a terminal object, and a retract B of A without one. To this
end, take B to be the free idempotent t2 = t as on the left in:
0t WW
1 GG
0t WW
! GG 1
1 GG 0t WW
and take A to be B with a terminal object 1 freely adjoined. The inclusion of B
into A admits a retraction, which fixes t and sends !: 0 → 1 to the identity on 0.
Yet B does not admit a terminal object.
In particular, this result tells us that, since lalis are not closed under retracts,
they cannot be characterised as the right class of maps for a mere weak
factorisation system; thus the algebraicity is, in this case, essential.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 29
6. Cofibrant generation by a double category
In light of Proposition 17, it is natural to ask whether there is a more refined
notion of cofibrant generation which encompasses such examples of awfs as
the one for lalis on Cat. In this section, we describe such a notion; it involves
lifting properties against a small double category, rather than a small category,
of generating cofibrations.
6.1. Double-categorical lifting operations. Let U : J → Sq(C) and V : K → Sq(C)
be double categories over Sq(C). We define a (J, K)-lifting operation to be a
(J1, K1)-lifting operation in the sense of Section 5.1 which is also compatible with
vertical composition in J and K, in the sense that
(6.1)
ϕj, ·k(u, v) = ϕj,k(u, ϕj, (V k · u, v))
and ϕj·i,k(u, v) = ϕj,k(ϕi,k(u, v · Uj), v)
for all vertically composable maps j · i: A → B → C and · k: D → E → F in J
and in K. For example, by virtue of (2.10) and its dual, the lifting operation (2.4)
associated to an awfs is an (L-Coalg, R-Alg)-lifting operation. As before, the
assignation sending J and K to the collection of (J, K)-lifting operations underlies
a functor Lift: (DBL/Sq(C))op × (DBL/Sq(C))op → SET; and also as before, we
have:
Proposition 18. Each functor Lift(J, –) and Lift(–, K) is representable, so that
the adjunction (5.1) extends to one
(6.2) (DBL/Sq(C))op
⊥
(–)
GG DBL/Sq(C) .
(–)
oo
Proof. Given U : J → Sq(C), we have as in Section 5.2 the concrete double
category J1 → Sq(C). The representing object J → Sq(C) for Lift(J, –) is
the sub-double category of J1 with the same objects and horizontal arrows,
and just those vertical arrows (g, ϕ–g) whose lifting operations respect vertical
composition in J, together with all cells between them. The representing object
K → Sq(C) for Lift(–, K) is defined dually.
Note that the 2-functor (–)1 : DBL → CAT sending a double category to its
vertical category is represented by the free vertical arrow 2v, and consequently
has a left adjoint, sending C to the product of 2v with the free horizontal
double category on C. This lifts to a left adjoint F for the induced functor
(–)1 : DBL/Sq(C) → CAT/C2 on slice categories; and as there are no non-trivial
vertical composites in FJ , we have that in fact J = (FJ ) ; thus the double
category structure on J , for which we offered no abstract justification in
Section 5.2 above, is now explained in terms of the adjunction (6.2).
6.2. Cofibrant generation by a double category. An algebraic weak factorisation
system (L, R) is said to be cofibrantly generated by a small double category
J → Sq(C) if R-Alg ∼= J over Sq(C); if we have this isomorphism for a large J,
then we call (L, R) class-cofibrantly generated by J → Sq(C). Once again, we have
the dual notions of fibrant and class-fibrant generation by a double category.
30 JOHN BOURKE AND RICHARD GARNER
If, as in the preceding section, we identify a category over C2 with the free
double category thereon, then this definition is a conservative extension of that
given in Section 5.2 above. However, by contrast with Proposition 17, we have:
Proposition 19. The awfs for lalis on Cat is cofibrantly generated by a small
double category.
Proof. We first claim that to equip a functor f : A → B with lali structure is
equally to give:
• A section u: ob B → ob A of the action of f on objects; and
• For each a ∈ A and b ∈ B, a section γa : B(fa, b) → A(a, ub) of the action
of f on morphisms; such that
• γa(β · fα) = γa (β) · α for all α: a → a in A and β : fa → b in B; and
• γub(1b) = 1ub for all b ∈ B.
Indeed, if f is part of a lali f p with unit η, then we obtain these data by taking
u to be the action of p on objects, and taking γa(β) = p(β) · ηa. Conversely,
given these data, we define a section p of f on objects by p(b) = u(b) and on
morphisms by p(β : b → b ) = γub(β): ub → ub , and define a unit η: 1 ⇒ pf
with components ηa = γa(1fa). This proves the claim.
We now define a small V : J → Sq(Cat) for which J ∼= Lali(Cat). The objects
of J are the ordinals 0, 1, 2 and 3, and its horizontal arrows are order-preserving
maps; on these, V acts as the identity. The vertical arrows of J are freely
generated by morphisms k: 0 → 1, : 1 → 2 and m: 2 → 3, which are sent by V
to the appropriate initial segment inclusions between ordinals. Its squares are
freely generated by the following four:
0
k

! GG
(a)
1

1
δ0
GG 2
1

δ0
GG
(b)
2
m

2
δ0
GG 3
1

δ1
GG
(c)
2
m

2
δ1
GG 3
0
k

id GG
(d)
0
k

1

2
!
GG 1 ,
where (following standard simplicial notation) δ0 and δ1 denote the orderpreserving
injections omitting 0 and 1 respectively. We claim that J → Sq(Cat)
cofibrantly generates the awfs for lalis; in other words, that J ∼= Lali(Cat) over
Sq(Cat). Indeed, given a functor f : A → B, we see that:
• To equip f with a V k-lifting operation is to give a section u of its action on
objects;
• To equip f with a V -lifting operation is to give, for every a ∈ A and
β : fa → b in B, a map γa(β): a → ¯b in A over β; to ask for the compatibility
(a) with the V k-lifting operation is to ask that, in fact, ¯b = ub.
• To equip f with a V m-lifting operation is to give, for every α: a → a in A
and β : fa → b, a map ψα(β): a → ¯b in A over β. The compatibility (b)
now forces ψα(β) = γa (β), while that in (c) forces ψα(β) · α = γa(β · fα);
so ψ is determined by γ, and we have γa(β · fα) = γa (β) · α.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 31
• Finally, the compatibility (d) forces γub(1b) = 1ub.
This verifies that vertical arrows of J are lalis in Cat; similar arguments show
that squares of J are ones preserving the lali structure, and that composition is
composition of lalis, as required.
6.3. Canonical class-cofibrant generation. As further evidence for the adequacy
of the notion of double-categorical cofibrant generation, we have the following
result, which tells us that any awfs is class-cofibrantly generated by its (typically
large) double category of coalgebras.
Proposition 20. Any awfs is class-cofibrantly generated by L-Coalg → Sq(C) and
class-fibrantly generated by R-Alg → Sq(C).
Proof. By duality, we need prove only the first statement. As above, (2.4) is
an (L-Coalg, R-Alg)-lifting operation, and so by Proposition 18 corresponds to a
concrete double functor Λ: R-Alg → L-Coalg over C. Of course, Λ is the identity
on objects and horizontal arrows; while its component Λ1 : R-Alg → (L-Coalg )1
on vertical arrows and squares postcomposes with the inclusion (L-Coalg )1 →
L-Coalg to yield the ¯Φ of Lemma 1. As ¯Φ is injective on objects and fully
faithful, so too is Λ1; while since the left square of (2.11) is one of L-Coalg, the
condition (6.1) defining the objects in (L-Coalg )1 implies that in (2.9), so that
Λ1 is also surjective on objects, and thus an isomorphism.
The fact of an isomorphism L-Coalg ∼= R-Alg is often a useful tool for
calculation. Many awfs that arise in practice are cofibrantly generated by
double categories in a natural way; as such, we have a concrete understanding of
the R-algebras. The above isomorphism offers one technique for obtaining the
corresponding L-coalgebras without explicit calculation of the values of L or R.
6.4. Freeness of cofibrantly generated awfs. Above, we have defined cofibrant
generation of an awfs in terms of a universal property of its double category
of algebras. We conclude this section by showing that this implies another
universal property: that its double category of coalgebras is freely generated
by the given J with respect to left adjoint functors between concrete double
categories. In the case of cofibrant generation by a mere category, this was shown
in [43, Theorem 6.22]; our result generalises this, and simplifies the proof.
The key step will be to extend the adjunction (6.2) to account for change of
base. To this end, we consider the 2-category DBL/Sq(– ladj), whose objects are
double functors J → Sq(C), whose 1-cells are squares as on the left of
(6.3)
J
U

¯F GG J
U

Sq(C)
Sq(F)
GG Sq(D)
J
U

¯F
AA
¯α
¯F
SS J
U

Sq(C)
Sq(F)
CC
Sq(α)
Sq(F )
QQ Sq(D)
together with a chosen right adjoint G for F, and whose 2-cells are diagrams as
on the right above. We define the 2-category DBL/Sq(– radj) dually.
32 JOHN BOURKE AND RICHARD GARNER
Proposition 21. The adjunction (6.2) extends to a 2-adjunction
(6.4) (DBL/Sq(– ladj))coop
⊥
(–)
GG DBL/Sq(– radj) ,
(–)
oo
with respect to which the canonical isomorphisms Λ: R-Alg → L-Coalg and
¯Λ: L-Coalg → R-Alg of Proposition 20 are 2-natural.
Proof. The 2-functor (–) is defined as before on objects, and as before on 1cells
(6.3) whose bottom edge is an identity. To complete the definition on 1-cells,
it thus suffices to consider squares whose top edge is an identity, as on the left in
(6.5)
J
U

1 GG J
Sq(F)·U

Sq(C)
Sq(F)
GG Sq(D)
(FJ)

GG J

Sq(D)
Sq(G)
GG Sq(C) .
Let FJ denote the double category J → Sq(D) over Sq(D) down the right edge
of this square, and let G be the chosen right adjoint for F. We claim that there
is in fact a pullback square as displayed right above: which we will take to be
the value of (–) at the left-hand square. For this, we must show that for each
g: C → D of D, the (FJ, g)-lifting operations ϕ–,g correspond functorially with
(J, Gg)-lifting operations ¯ϕ–,Gg. But such lifting operations are given by natural
transformations
ϕ–,g : C(cod FU–, dom g) ⇒ C2
(FU–, g): J op
→ Set
¯ϕ–,Gg : C(cod U–, dom Gg) ⇒ C2
(U–, Gg): J op
→ Set
satisfying axioms; so as F G, these are in a clear bijective correspondence,
under which the maps of lifting operations also correspond as required.
Finally, we must define (–) on 2-cells. Given a diagram as on the right of (6.3),
the 2-cell α: F ⇒ F therein transposes under adjunction to one β : G ⇒ G.
As the forgetful double functors (J ) → Sq(D) and J → Sq(C) are concrete,
there is a unique possible 2-cell ¯β : ( ¯F ) ⇒ ¯F lifting β; a short calculation
shows that this lifting indeed exists, so that we may take (α, ¯α) = (β, ¯β). This
completes the definition of (–) ; that of (–) is dual.
To verify that the extended (–) and (–) are 2-adjoint, we observe that for J
and K over Sq(C) and Sq(D), a map J → K in DBL/Sq(– ladj) is an adjunction
F G: D → C together with a (FJ, K)-lifting operation, while a map K → J in
DBL/Sq(– radj) is an adjunction F G together with a (J, GK)-lifting operation;
by the above argument, these data are in bijective correspondence. We argue
similarly for the bijection on 2-cells. Finally, the naturality of the isomorphisms
Λ and ¯Λ is exactly (2.13); their 2-naturality now follows by concreteness.
Using the result, we now give our promised generalisation of [43, Theorem 6.22].
Proposition 22. If (L, R) is (class-)cofibrantly generated by U : J → Sq(C), then
it provides a left 2-adjoint at J for (–)-Coalg: AWFSladj → DBL/Sq(– ladj).
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 33
Proof. We have natural isomorphisms
AWFSladj((L, R), (L , R )) ∼= AWFSradj((L , R ), (L, R))op
∼= DBL/Sq(– radj)(R -Alg, R-Alg)op
∼= DBL/Sq(– radj)(R -Alg, J )op
∼= DBL/Sq(– ladj)(J, R -Alg)
∼= DBL/Sq(– ladj)(J, L -Coalg) ,
by, respectively, the isomorphism AWFSladj
∼= AWFSradj
coop
; full fidelity of
(–)-Alg; the assumed isomorphism J ∼= R-Alg over Sq(C); adjointness (6.4); and
the natural isomorphisms of Proposition 20.
7. A characterisation of accessible awfs
We are now ready to give our second main result, Theorem 25, which gives
a characterisation of the accessible awfs on a locally presentable category. We
will show that, for a locally presentable C, any small double category over Sq(C)
cofibrantly generates an accessible awfs, and, moreover, that every accessible
awfs on C arises in this way.
7.1. Existence of cofibrantly generated AWFS. We begin by extending Proposition
16 from cofibrant generation by categories to that by double categories.
Proposition 23. If C is locally presentable, then the awfs cofibrantly generated
by any small double category U : J → Sq(C) exists and is accessible.
Proof. Write J2 = J1 ×J0 J1 for the category of vertically composable pairs in
J, and m: J2 → J1 for the vertical composition. Now the triangle on the left in
J2
m GG
U1m 22
J1
U1~~
C2
J1
m GG
33
J2
}}
Sq(C)
induces a morphism m of concrete double categories over C as on the right, which
on vertical arrows sends (g, ϕ–g): D → E to (g, ψ–g): D → E, with (J2, g)-lifting
operation ψ–g defined as on the left in
ψ(j,i),g(u, v) = ϕj·i,g(u, v) θ(j,i),g(u, v) = ϕj,g(ϕi,g(u, v · Uj), v) .
We also have the (J2, g) lifting operation θ–g as on the right above, and the
assignation (g, ϕ–g) → (g, θ–g) in fact gives the action on vertical arrows of a
further double functor δJ : J1 → J2 concrete over C. It is easy to see that we
now have an equaliser of concrete double categories over C as on the left in
(7.1)
J GG
77
J1

m GG
δJ
GG J2
xx
Sq(C)
J1
m GG
(δJ)1
GG
∼=

J2
∼=

R1-Alg
s∗
GG
t∗
GG R2-Alg .
34 JOHN BOURKE AND RICHARD GARNER
Since J1 and J2 are right-connected, so too will J be; whence by Theorem 6,
the awfs cofibrantly generated by J will exist if and only if (J )1 → C2 is strictly
monadic. As J1 and J2 are small, Proposition 16 ensures that there are awfs
(L1, R1) and (L2, R2) on C with J1
∼= R1-Alg and J2
∼= R2-Alg over C2; now by
full fidelity of the assignment Mnd(C2)op → CAT/C2, there are unique monad
maps s, t: R2 R1 rendering commutative the diagram above right. It follows
that (J )1 is isomorphic over C2 to the equaliser of the lower row. Since R1
and R2 are accessible, the parallel pair of monad morphisms s, t admits by [36,
Theorem 27.1] a coequaliser q: R1 → R which is again accessible, and for which
q∗ is the equaliser of s∗ and t∗. Thus (J )1
∼= R-Alg over C2 so that (J )1 → C2
is strictly monadic for an accessible monad, as required.
We now wish to show that every accessible awfs on a locally presentable
category arises in the manner of the preceding proposition. The key idea is to
show that any accessible awfs has a small dense subcategory of L-coalgebras,
and then deduce the result using Proposition 20 and the following lemma:
Lemma 24. Given a morphism of categories over C2
J
F GG
U 11
K ,
V
C2
the induced F : K → J is an isomorphism of categories whenever the identity
2-cell exhibits V as the pointwise Kan extension LanF U. In particular, this is
the case whenever F is dense and V preserves the colimits exhibiting this density.
Proof. As in the proof of Proposition 16, an element of K is a pair (g, ϕ–g) where
g: C → D in C and ϕ–g is a section of the ψ–g : C(cod V –, C) → C2(V –, g) of (5.2);
or equally (as cod id: C2 → C) a section of (1, g) · (–): C2(V –, 1C) → C2(V –, g).
In these terms, a map (g, ϕ–g) → (h, ϕ–h) of K is a map g → h of C2 for which
the induced C2(V –, g) → C2(V –, h) commutes with the given sections. The map
K → J is induced by the restriction functor [Fop, 1]: [Kop, Set] → [J op, Set],
and sends (g, ϕ–g) to the pair (g, ϕ ) with second component
ϕ = ϕ–gF : C2
(U–, g) → C2
(U–, 1C) .
So to prove F : K → J is an isomorphism, it will suffice to show that [Fop, 1]
becomes fully faithful when restricted to the full subcategory of [Kop, Set] on the
presheaves of the form C2(V –, g). Now, to say that the identity 2-cell exhibits V
as LanF U is equally to say that it exhibits V op as RanFop Uop; the continuous
C2(–, g): (C2)op → Set preserves this pointwise Kan extension, and so the identity
2-cell exhibits C2(V –, g) as the right Kan extension of C2(U–, g) along Fop. The
universal property of this right Kan extension now immediately implies that the
action of [Fop, 1] on morphisms
[Kop
, Set](C2
(V –, f), C2
(V –, g)) → [J op
, Set](C2
(U–, f), C2
(U–, g))
is invertible for any f and g in C2, as required. Finally, note that if F is dense,
then the identity 2-cell exhibits 1K as LanF F, and that if V preserves the colimits
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 35
exhibiting this density, then it preserves this left Kan extension, so that the
identity 2-cell exhibits V as LanF U.
Theorem 25. An awfs on a locally presentable category is accessible if and only
if it is cofibrantly generated by some small double category.
Proof. One direction is Proposition 23; for the other, let (L, R) be an accessible
awfs on the locally presentable C. By Proposition 20 we have an isomorphism
L-Coalg ∼= R-Alg over Sq(C); it will thus suffice to find a small sub-double
category Lλ-Coalg → L-Coalg with the induced L-Coalg → Lλ-Coalg invertible.
To this end, we consider the following diagram, whose left column is yet to be
defined:
Lλ-Coalg2
j2
GG
 
L-Coalg2
V2
GG
mL

pL

qL

C3
m

p

q

Lλ-Coalg
j1
GG
yy
 
L-Coalg
V1
GG
yy
iLdL

cL

C2
yy
id

c

Cλ
j0
GG C
1 GG C .
As colimits are pointwise in functor categories the right column is composed of
cocontinuous functors between cocomplete categories; since V1 creates colimits,
the same is true for L-Coalg and each of dL, iL and cL. Moreover dL has the
isomorphism-lifting property since V1 and d do, whence by [34] the pullback
L-Coalg2 of dL along cL is also a bipullback. Consequently, L-Coalg2 is cocomplete
and the projections qL and pL cocontinuous; so too is mL, since dL and cL jointly
create colimits; and similarly for V2. Thus the right two columns are composed
of cocomplete categories and cocontinuous functors.
Now C is locally presentable, whence also C2 and C3; and since L is accessible,
L-Coalg is locally presentable as in the proof of Proposition 13. Finally, L-Coalg2
is a bipullback of cocontinuous functors between locally presentable categories,
and so itself locally presentable by [10, Theorem 3.15]. So the right two columns
are composed of locally presentable categories and cocontinuous functors.
Any cocontinuous functor between locally presentable categories has an accessible
right adjoint [19, Satz 14.6]; thus there is a regular λ so that the right
two columns are composed of locally λ-presentable categories and functors with
λ-accessible right adjoints; it follows that each left adjoint preserves λ-presentable
objects. We now define Lλ-Coalg3
as the full sub-double category of L-Coalg with
as objects, isomorphism-class representatives of the λ-presentable objects of C.
This category is small, and the inclusion j0 : Cλ → C is dense; we claim that j1
and j2 are too. For j1, we consider the full dense subcategory (L-Coalg)λ of λpresentable
objects in L-Coalg. Because dL and cL preserve λ-presentability it follows
that each λ-presentable of L-Coalg has λ-presentable domain and codomain;
so the fully faithful dense functor (L-Coalg)λ → L-Coalg factors through the full
inclusion j1 : Lλ-Coalg → L-Coalg, whence, by [37, Theorem 5.13] j1 is dense. The
3Our notation Lλ-Coalg acknowledges the fact that this is equally the double category of
coalgebras associated to the restricted awfs on Cλ.
36 JOHN BOURKE AND RICHARD GARNER
density of j2 follows in the same manner, by considering the full dense subcategory
(L-Coalg2)λ of L-Coalg2 and arguing that it is contained in (Lλ-Coalg)2.
Since both j1 and j2 are dense, and since all of the functors to their right are
cocontinuous it follows from Lemma 24 that both j1 and j2 are isomorphisms
of categories; whence j1 and j2 as displayed in the right two columns of
L-Coalg GG
j

L-Coalg
j1

m GG
δ
GG (L-Coalg2)
j2

Lλ-Coalg GG Lλ-Coalg
m GG
δ
GG (Lλ-Coalg2)
are isomorphisms of double categories. But both rows are, as in (7.1), equalisers,
and it follows that the induced double functor j on the left is invertible as
claimed.
8. Fibre squares and enriched cofibrant generation
We conclude this paper with a further batch of examples based on a surprisingly
powerful modification of the small object argument of Proposition 16. Given a
well-behaved monoidal category V equipped with an awfs (L, R), a well-behaved
V-category C, and a small category U : J → (C0)2 over (C0)2, it constructs the
“free (L, R)-enriched awfs cofibrantly generated by J ”. This is an awfs on C0
whose algebras g: C → D are maps equipped with, for every Uj : A → B in
the image of U, a choice of R-map structure on the canonical comparison map
C(B, C) → C(B, D) ×C(A,D) C(A, C) in V, naturally in j.
The “enriched small object argument” which builds awfs of this kind is most
properly understood in the context of monoidal algebraic weak factorisation
systems [44]; and as indicated in the introduction, a comprehensive treatment
along those lines must await a further paper. Here, we content ourselves with
giving the construction and a range of applications.
8.1. Fibre squares. The cleanest approach to enriched cofibrant generation makes
use of the following fibre square construction. It associates to any awfs (L, R) on
a category C with pullbacks an awfs on the arrow category C2 whose algebras
are the R-fibre squares: maps (u, v): f → g of C2 as to the left of
A u
55
f
66
k 77
B ×D C GG

C
g

B v
GG D
FibreR(C) GG
V

R-Alg
UR

(C2)2
(u,v)→k
GG C2
equipped with an R-algebra structure on the comparison map k: A → B ×D C.
The R-fibre squares are the objects of a category FibreR(C), whose morphisms
are determined by way of the pullback square on the right above.
We may compose R-fibre squares: given (u, v): f → g with R-algebra structure
k: A → B ×D C and (s, t): g → h with structure : C → D ×F E, we can
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 37
pull back along v ×F E and use Proposition 8 to obtain an algebra structure
v∗ : B ×D C → B ×F E; the composite algebra v∗ · k: A → B ×D C → B ×F E
now equips (s, t) · (u, v): f → h with R-fibre square structure. In this way we
obtain a double category FibreR(C) → Sq(C2) which is concrete over C2 and
easily seen to be right-connected; so it will be the double category of algebras
for an awfs on C2 as soon as V : FibreR → (C2)2 is strictly monadic. As a
pullback of the strictly monadic UR : R-Alg → C2, this V will always create
V -absolute coequaliser pairs; so it suffices to show that it has a left adjoint.
Given a square in C as on the left below, we factor the induced k: A → B ×D C
as Rk · Lk: A → Ek → B ×D C, yielding the factorisation
(8.1)
A
u GG
f

C
g

=
A
f

Lk GG Ek
π2·Rk
GG
π1·Rk

C
g

B v
GG D B
1
GG B v
GG D
on the right. The rightmost square, when equipped with the algebra structure
Rk: Ek → B ×D C on the comparison map, now comprises the free R-fibre
square on (u, v). This shows that V has a left adjoint, whence by Theorem 6,
FibreR(C) → Sq(C2) is the double category of algebras for an awfs on C2; it is
not hard to calculate that the corresponding coalgebras are maps (u, v): f → g
for which u is equipped with L-coalgebra structure and v is invertible. Note that
if (L, R) is an accessible awfs on a locally presentable category, then the fibre
square awfs on C2 is also accessible, since its factorisations (8.1) are constructed
using the accessible L and R and finite limits in C.
8.2. Enriched cofibrant generation. Suppose now that V is a locally presentable
symmetric monoidal closed category: thus a suitable base for enriched category
theory. Let (L, R) be an accessible awfs on V, and let C be a V-category whose
underlying category C0 is locally presentable; finally, let U : J → (C0)2 be a small
category over (C0)2. By applying its monad and comonad pointwise, we may
lift the awfs (L, R) on V to an accessible awfs on [J op, V]; now applying the
fibre square construction of the preceding section to this yields an accessible
awfs on [J op, V]2. There is a right adjoint functor ˜U : C0 → [J op, V]2 sending
x ∈ C0 to the arrow C(cod U–, x) → C(dom U–, x) of [J op, V]; and we now define
the enriched awfs cofibrantly generated by J to be the projective lifting of the
fibre square awfs on [J op, V]2 along ˜U; note that the existence of this lifting is
guaranteed by Proposition 13(a). The double category of algebras of the awfs
(LJ , RJ ) so obtained is thus defined by a pullback square
RJ -Alg GG

Fibre[J op,V]([J op, V])

Sq(C0)
Sq( ˜U)
GG Sq([J op, V]2) ;
so that in particular, an algebra structure on a map g: C → D is given by the
choice, for each j ∈ J with image Uj : A → B in C2, of an R-algebra structure
38 JOHN BOURKE AND RICHARD GARNER
on the comparison map C(B, C) → C(A, C) ×C(A,D) C(B, D), naturally in j; we
may say that g is equipped with an enriched lifting operation against J .
8.3. Examples. The remainder of the paper will be given over to a range of
examples of enriched cofibrant generation. All these examples will in fact be
generated by a mere set of maps J, seen as a discrete subcategory J → (C0)2 of
a locally presentable C0.
Examples 26. (i) Let V = Set and (L, R) be the awfs for split epis thereon;
then the notion of enriched cofibrant generation reduces to the unenriched
one of Section 5.2. A full treatment of enriched cofibrant generation would
in fact generalise each aspect of the theory developed in Sections 5 and 6;
but as we have said above, this must await another paper.
(ii) More generally, let V be any locally presentable symmetric monoidal closed
category, and let (L, R) be the split epi awfs thereon. The enriched awfs
cofibrantly generated by J → C2 is precisely that obtained by the “enriched
small object argument” of [45, Proposition 13.4.2].
(iii) Let V = Set and let (L, R) be the initial awfs thereon; its algebra category
is the full subcategory of Set2
on the isomorphisms. In this case, the
algebra category of the enriched awfs (LJ , RJ ) generated by J → C2
may be identified with the full subcategory of C2 on those maps with the
unique right lifting property against each map in J. In particular, since
RJ -Alg → C2 is fully faithful, Proposition 3 applies, so that (LJ , RJ ) in fact
describes the orthogonal factorisation system whose left class is generated
by J [18, §2.2].
Examples 27. For our next examples of enriched cofibrant generation, we take
V = Cat equipped with the awfs (L, R) for retract equivalences of Section 4.2.
(i) Take C = Cat and let J comprise the single functor !: 0 → 1. It’s easy
to see that the enriched awfs generated by J is again that for retract
equivalences.
(ii) Take C = Cat and let J comprise the single functor : 1 → 2 picking out
the terminal object of 2. An algebra for the enriched awfs generated by J
is a functor G: C → D such that the induced C2 → D ↓ G bears retract
equivalence structure, or equivalently, is fully faithful and equipped with a
section on objects. Full fidelity corresponds to the requirement that every
arrow of C be cartesian over D, whereupon a section on objects amounts
to a choice of cartesian liftings: so an algebra is a cloven fibration whose
fibres are groupoids. We find further that maps of algebras are squares
strictly preserving the cleavage, and that composition of algebras is the
usual composition of fibrations.
(iii) Let (C, j) be a small site, let E = Hom(Cop, Cat) be the 2-category of
pseudofunctors, strong transformations and modifications Cop → Cat, and
let y: C → E be the Yoneda embedding. If (Ui → U : i ∈ I) is a covering
family in C, then the associated covering 2-sieve f : ϕ → yU in E is the
second half of the pointwise (bijective on objects, fully faithful) factorisation
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 39
i yUi → ϕ → yU; recall that an object X ∈ E is called a stack if
the restriction functor E(f, X): E(yU, X) → E(ϕ, X) is an equivalence of
categories for each covering 2-sieve.
Consider now the 2-category Es = Hom(Cop, Cat)s of pseudofunctors,
2-natural transformations and modifications. By the general theory of [11],
there is an accessible 2-monad T on Catob C
whose 2-category T-Algs of
strict algebras and strict algebra morphisms is Es—so that in particular,
Es is locally presentable—and whose 2-category T-Alg of strict algebras
and pseudomorphisms is E. It follows by [11, Theorem 3.13] that the
inclusion 2-functor Es → E has a left adjoint Q: Es → E. Thus we may
identify stacks with pseudofunctors X such that the restriction functor
Es(Qf, X): Es(QyU, X) → Es(Qϕ, X) is an equivalence for each covering
2-sieve f.
As Es is locally presentable, we may consider the enriched awfs thereon
generated by the set {Qf : f a covering 2-sieve in E}, and by the above,
we see that its algebraically fibrant objects are a slight strengthening of
the notion of stack; namely, they are pseudofunctors X : Cop → Cat such
that each E(f, X): E(yU, X) → E(ϕ, X) is given the structure of a retract
equivalence.
(iv) We may make arbitrary stacks into the algebraically fibrant objects of
an awfs by way of the following observations. Given F : C → D in Cat,
we may form its pseudolimit D ↓∼= F—the category whose objects are
triples (d ∈ D, c ∈ C, θ: d ∼= Fc)—and choices of equivalence section for
D ↓∼= F → D now correspond to choices of equivalence pseudoinverse for
F. It follows that if f : X → Y is a map in a finitely cocomplete 2-category
E, and ¯f : X → ¯Y is the injection of X into the pseudocolimit of f, then
for each F ∈ E, retract equivalence structures on E( ¯f, F) correspond with
equivalence structures on E(f, F).
In particular, if (C, j) is a small site and E and Es are as above, then
we may consider the enriched awfs on Es cofibrantly generated by the
set {Qf : f a covering 2-sieve in E}. Its algebraically fibrant objects are
now stacks in the usual sense; to be precise, they are pseudofunctors X
such that each E(yU, X) → E(ϕ, X) is provided with a chosen equivalence
pseudoinverse.
Examples 28. Our next collection of examples again take V = Cat, but now with
(L, R) the awfs for lalis of Section 4.2.
(i) Let C = Cat and let J comprise the single functor !: 0 → 1; then the
enriched awfs generated by J is simply that for lalis.
(ii) Let C = Cat and let J comprise the single functor : 1 → 2 picking out
the terminal object of 2. To equip G: C → D with algebra structure for
the enriched awfs this generates is to equip the induced C2 → D ↓ G with
a right adjoint right inverse; which by [26], is equally to equip G with the
structure of a cloven fibration. As before, maps of algebras are squares
strictly preserving the cleavage; and composition of algebras is the usual
composition of fibrations.
40 JOHN BOURKE AND RICHARD GARNER
(iii) Generalising the previous two examples, let Φ be some set of small categories,
and let J comprise the class of functors {A → A⊥ : A ∈ Φ} obtained by
freely adjoining an initial object to each A. To equip G: C → D with
algebra structure for the induced awfs is to give, for each A ∈ Φ, a right
adjoint section for the comparison functor [A⊥, C] → [A⊥, D] ×[A,D] [A, C].
Since a functor out of A⊥ is the same thing as a functor out of A together
with a cone over it, a short calculation shows that this amounts to giving, for
each A ∈ Φ, each diagram F : A → C, and each cone p: ∆V → GF in D, a
cone q: ∆W → F in C with Gq = p, such that for any cone q : ∆W → F,
each factorisation of Gq through p via a map k: GW → V lifts to a unique
factorisation of q through q.4
In particular, an algebraically fibrant object
for this awfs is a category equipped with a choice of limits for all diagrams
indexed by categories in Φ.
(iv) Generalising the preceding example, let C be any locally presentable 2category.
For any arrow f : A → B of C, let ¯f : A → ¯B denote the injection
into the lax colimit of the arrow f, and for any set J of arrows in C, let
¯J = { ¯f : f ∈ J}. By using the universal property of the colimit, we may
calculate that an algebra structure on a morphism g: C → D for the awfs
induced by ¯J is given by the choice, for every f : A → B in J and every
2-cell θ as on the left below, of a map k and 2-cell γ as in the centre with
gk = v and gγ = θ, and such that γ is initial over g; thus, for every diagram
as on the right with gα = θ · βf, there exists a unique 2-cell δ: ⇒ k with
gδ = β and α = γ · δf.
A
f

u GG
θ
uƒ
C
g

B v
GG D
A
f

u GG
γ
uƒ C
g

B v
GG
k
bb
D
A
f

u GG
α
uƒ C
g

B v
GG
bb
D .
β
In particular, the algebraically fibrant objects for this awfs are those C ∈ C
such that, for each f : A → B in J, the functor C(f, C): C(B, C) → C(A, C)
is equipped with a right adjoint; that is, such that each morphism A → C
in C is equipped with a chosen right Kan extension along f.
Of course, we may construct examples dual to the preceding ones—dealing
with ralis, opfibrations, final liftings of cocones, categories with colimits, and left
Kan extensions—by replacing the awfs for lalis in the above by that for ralis.
Example 29. Let V be the category SSet = [∆op, Set] of simplicial sets, equipped
with the awfs for trivial Kan fibrations, cofibrantly generated by the set of
boundary inclusions {∂∆[n] → ∆[n] : n ∈ N}. Given a set J of maps in
the locally presentable SSet-category C, the algebraically fibrant objects of the
enriched awfs cofibrantly generated by J are those X ∈ C such that, for each
f : A → B in J, the induced map C(B, X) → C(A, X) bears algebraic trivial
fibration structure. Most typically, one would consider this in the situation where
4Note that when A is a discrete category, this is precisely the notion of G-initial lifting [1,
Definition 10.57] of a discrete cone.
ALGEBRAIC WEAK FACTORISATION SYSTEMS I 41
C is a simplicial model category and the J’s are a class of cofibrations; then the
algebraically fibrant objects above which are also fibrant for the underlying model
structure are precisely the J-local objects [28, Definition 3.1.4]. The enriched
small object argument in this particular case was described in [14, §7].
Our final examples bear on the theory of quasicategories [32]. Quasicategories
are simplicial sets with fillers for all inner horns; they are a particular model for
(∞, 1)-categories—weak higher categories whose cells of dimension > 1 are weakly
invertible—and have a comprehensive theory [33, 40] paralleling the classical
theory of categories. In particular, there are good notions of Grothendieck
fibration, limit, and Kan extension for quasicategories; and by analogy with
Examples 27, we may hope to obtain these notions as fibrations or fibrant objects
for awfs constructed by enriched cofibrant generation, so long as we can find a
good quasicategorical analogue of the awfs for lalis.
For this we propose the awfs of Examples 12(v) above, whose R-algebras we
called the simplicial lalis. By using the explicit formulae of Section 4.4, we may
calculate the monad R at issue, and so obtain the following concrete description
of its algebras. First some notation: given a simplicial set X, we write x: a b
to denote a simplex x ∈ Xn+1 whose (n + 1)st face is a and whose final vertex is
b. Now by a simplicial lali, we mean a simplicial map f : A → B together with:
• A section u: B0 → A0 of the action of f on 0-simplices;
• For each a ∈ An and x: fa b in B, a simplex γa(x): a ub over x;
subject to the coherence conditions that:
• γa(x) · δi = γa·δi
(x · δi) and γa(x) · σi = γa·σi (x · σi);
• γub(b · σ0) = ub · σ0; and
• γγa(x)(x · σn+1) = γa(x) · σn+1.
By comparing with Proposition 19, we see that if A = NC and B = ND are
the nerves of small categories, then a simplicial lali A → B is the same thing as
a lali C → D. This, of course, does not in itself justify the notion of simplicial
lali; in order to do so, we will relate it to a notion introduced by Riehl and
Verity in [52, Example 4.4.7], which for the present purposes we shall refer to
as a quasicategorical lali. A simplicial map f : A → B is a quasicategorical lali
if equipped with a strict section u: B → A, a simplicial homotopy η: 1 ⇒ uf
satisfying fη = 1f (on the nose), and a 2-homotopy
u
1u
22
θ
u
1u
GG
ηu
bb
u
satisfying fθ = 11B . We have established the following result; its proof is
beyond the scope of this paper, but note that it is a quasicategorical analogue
of the correspondence between the two views of ordinary lalis described in
Proposition 19.
Proposition 30. Let f : A → B be a map of simplicial sets. If f is a simplicial
lali, then it is a quasicategorical lali; if it is a quasicategorical lali and an inner
Kan fibration, then it is a simplicial lali.
42 JOHN BOURKE AND RICHARD GARNER
Since Riehl and Verity are able to use quasicategorical lalis to describe various
aspects of the theory of quasicategories, the above proposition allows us to
describe these same aspects using simplicial lalis, and, hence, using the theory of
enriched awfs.
Examples 31. We consider enriched cofibrant generation in the case where V =
SSet and (L, R) is the awfs for simplicial lalis.
(i) Limits. Let A be a quasicategory, and let A⊥ denote the quasicategory
∆[0] ⊕ A obtained by adjoining an initial vertex to A. Riehl and Verity
show in [52, Corollary 5.2.19] that a quasicategory X admits all limits
of shape A—in the sense of [32, Definition 4.5]—just when the simplicial
map XA⊥ → XA bears quasicategorical lali structure. It follows that,
among the quasicategories, those equipped with a choice of all limits of
shape A may be realised as the algebraically fibrant objects of an awfs on
SSet; namely, the enriched awfs cofibrantly generated by the single map
A → A⊥. By enriched cofibrant generation with respect to the set of maps
J = {A → A⊥ : A a finite quasicategory} we may capture quasicategories
with all finite limits as algebraically fibrant objects.
(ii) Grothendieck fibrations. Riehl and Verity are in the process of analysing the
quasicategorical Grothendieck fibration—the “Cartesian fibrations” of [40,
§2.4]—and have shown [51] that an isofibration g: C → D of quasicategories
bears such a structure just when the comparison functor C2 → D ↓ g bears
quasicategorical lali structure. Here, C2 denotes the exponential C∆[1], while
D ↓ g is the pullback of g: C → D along Dδ0 : D∆[1] → D. Thus, among the
isofibrations of quasicategories, the Grothendieck fibrations can be realised
as the algebras of an awfs on SSet—namely, that obtained by enriched
cofibrant generation with respect to the single map δ0 : ∆[0] → ∆[1].
(iii) Right Kan extensions. By [52, Example 5.0.4 and Proposition 5.1.19], a
morphism of quasicategories f : C → D admits a right adjoint if and only if
the projection f ↓ D → D admits simplicial lali structure. So suppose that
J is a set of morphisms between quasicategories; for each f : A → B in J,
define ¯B to be the the pushout of f along δ1 × A: A → ∆[1] × A, and define
¯f : A → ¯B to be the composite of δ0 × A: A → ∆[1] × A with the pushout
injection ∆[1] × A → ¯B. Then a quasicategory X is an algebraically fibrant
object for the enriched awfs cofibrantly generated by { ¯f : f ∈ J} just when
each functor Xf : XB → XA has a right adjoint; that is, just when each
functor A → X admits a right Kan extension along f.
Of course, by considering the awfs for simplicial ralis in place of simplicial lalis,
we may capture notions such as colimits, opfibrations and left Kan extensions.
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ALGEBRAIC WEAK FACTORISATION SYSTEMS I 45
Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 60000, Czech Republic
E-mail address: bourkej@math.muni.cz
Department of Mathematics, Macquarie University, NSW 2109, Australia
E-mail address: richard.garner@mq.edu.au
Chapter 4
Skew structures in 2-category
theory and homotopy theory
This chapter contains the article Skew structures in 2-category theory and
homotopy theory by John Bourke, published in the Journal of Homotopy
and Related Structures 12 (2017) 31–81.
117
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY
THEORY
JOHN BOURKE
Abstract. We study Quillen model categories equipped with a monoidal
skew closed structure that descends to a genuine monoidal closed structure
on the homotopy category. Our examples are 2-categorical and include
permutative categories and bicategories. Using the skew framework, we adapt
Eilenberg and Kelly’s theorem relating monoidal and closed structure to
the homotopical setting. This is applied to the construction of monoidal
bicategories arising from the pseudo-commutative 2-monads of Hyland and
Power.
1. Introduction
The notion of a monoidal closed category captures the behaviour of the tensor
product and internal hom on classical categories such as those of sets and vector
spaces. Some of the basic facts about monoidal closed categories have an intuitive
meaning. For instance, the isomorphism
(1.1) C(A, B) ∼= C(I, [A, B])
says that elements of the internal hom [A, B] are the same thing as morphisms
A → B.
Recently some new variants have come to light. Firstly, the skew monoidal
categories of Szlach´anyi [36] in which the structure maps such as (A ⊗ B) ⊗ C →
A⊗(B⊗C) have a specified orientation and are not necessarily invertible. Shortly
afterwards the dual notion of a skew closed category was introduced by Street
[35]. Here one has a canonical map
(1.2) C(A, B) → C(I, [A, B])
but this need not be invertible. Intuitively, we might view this relaxation as
saying that [A, B] should contain the morphisms A → B as elements, but possibly
something else too.
In the present paper a connection is drawn between skew structures and homotopy
theory. We study examples of Quillen model categories C in which the
correct internal homs [A, B] have more general weak maps A B as elements.
By the above reasoning these examples are necessarily skew. These skew closed
categories form part of enveloping monoidal skew closed structures that descend
to the homotopy category Ho(C) where, in fact, they yield genuine monoidal
closed structures. The study of skew structures on a category that induce genuine
structures on the homotopy category is our main theme.
2000 Mathematics Subject Classification. Primary: 18D10, 55U35.
1
2 JOHN BOURKE
Our examples are 2-categorical in nature – most involve tweaking better known
weak 2-categorical structures to yield not strict, but skew, structures. For
example, we describe a monoidal skew closed structure on the 2-category of
permutative categories – symmetric strict monoidal categories – and strict maps.
This contains, on restricting to the cofibrant objects, a copy of the well known
monoidal bicategory of permutative categories and strong maps. More generally,
we describe a skew structure for each pseudo-commutative 2-monad T on Cat in
the sense of [13]. Other examples concern 2-categories and bicategories.
The theory developed in the present paper has a future goal, concerning Graycategories,
in mind. It was shown in [4] that there exists no homotopically well
behaved monoidal biclosed structure on the category of Gray-categories. The
plan is, in a future paper, to use the results developed here to understand the
correct enriching structure on the category of Gray-categories.
Let us now give an overview of the paper. Section 2 is mainly background on
skew monoidal, skew closed and monoidal skew closed categories. We recall
Street’s theorem describing the perfect correspondence between skew monoidal
structures (C, ⊗, I) and skew closed structures (C, [−, −], I) in the presence of
adjointness isomorphisms C(A ⊗ B, C) ∼= C(A, [B, C]). In Theorem 2.6 we reformulate
Eilenberg and Kelly’s theorem [8], relating monoidal and closed structure,
in the skew language. Finally, we introduce symmetric skew closed categories.
It turns out that all the examples of skew closed structures that we meet in the
present paper can be seen as arising from certain multicategories in a canonical
way. In Section 3 we describe the passage from such multicategories to skew
closed categories.
Using the multicategory approach where convenient, Section 4 gives concrete
examples of some of the skew closed structures that we are interested in. We describe
the examples of categories with limits, permutative categories, 2-categories
and bicategories.
Section 5 concerns the interaction between skew structures and Quillen model
structures that lies at the heart of the paper. We begin by describing how a
skew monoidal structure (C, ⊗, I) can be left derived to the homotopy category.
This is the skew version of Hovey’s construction [12]. We call (C, ⊗, I) homotopy
monoidal if the left derived structure (Ho(C), ⊗l, I) is genuinely monoidal. This
is complemented by an analysis of how skew closed structure can be right derived
to the homotopy category, and we obtain a corresponding notion of homotopy
closed category. Combining these cases Theorem 5.11 describes how monoidal
skew closed structure can be derived to the homotopy category. This is used to
prove Theorem 5.12, a homotopical analogue of Eilenberg and Kelly’s theorem,
which allows us to recognise homotopy monoidal structure in terms of homotopy
closed structure.
Section 6 returns to the examples of categories with limits and permutative
categories in the more general setting of pseudo-commutative 2-monads T on
Cat. We make minor modifications to Hyland and Power’s construction [13] of
a pseudo-closed structure on T-Alg to produce a skew closed structure on the
2-category T-Algs of algebras and strict morphisms. For accessible T this forms
part of an enveloping monoidal skew closed structure which, using Theorem 5.12,
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 3
we show to be homotopy monoidal. Using this, we give a complete construction of
the monoidal bicategory structure on T-Alg of Hyland and Power – thus solving
a problem of [13].
Section 7 consists of an in-depth analysis of the skew structure on the category
of bicategories and strict homomorphisms. Though not particularly interesting
in its own right, we regard this example as a preliminary to future work in higher
dimensions.
Acknowledgments. The author thanks the organisers of the Cambridge Category
Theory Seminar and of CT2015 in Aveiro for providing the opportunity to present
this work, and thanks Sofie Royeaerd for useful feedback on an early draft.
2. Skew monoidal and skew closed categories
2.1. Skew monoidal categories. Skew monoidal categories were introduced by
Szlach´anyi [36] in the study of bialgebroids over rings. There are left and right
versions (depending upon the orientation of the associativity and unit maps) and
it is the left handed case that is of interest to us.
Definition 2.1. A (left) skew monoidal category (C, ⊗, I, α, l, r) is a category C
together with a functor ⊗ : C × C → C, a unit object I ∈ C, and natural families
αA,B,C : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C), lA : I ⊗ A → A and rA : A → A ⊗ I
satisfying five axioms [36].
There is no need for us to reproduce these five axioms here as we will not use
them. We remark only that they are neatly labelled by the five words
abcd
iab aib abi
ii
of which the first refers to MacLane’s pentagon axiom.
Henceforth the term skew monoidal is taken to mean left skew monoidal. A
monoidal category is precisely a skew monoidal category in which the constraints
α, l and r are invertible.
2.2. Skew closed categories. In the modern treatment of monoidal closed categories
as a basis for enrichment [16] it is the monoidal structure that is typically
treated as primitive. Nonetheless, the first major treatment [8] emphasised
the closed structure, presumably because internal homs are often more easily
described than the corresponding tensor products. In the examples of interest to
us (see Section 4) this is certainly the case. These examples will not be closed in
the sense of ibid. but only skew closed.
Definition 2.2 (Street [35]). A (left) skew closed category (C, [−, −], I, L, i, j)
consists of a category C equipped with a bifunctor [−, −] : Cop × C → C and unit
object I together with
(1) components L = LA
B,C : [B, C] → [[A, B], [A, C]] natural in B, C and extranatural
in A,
(2) a natural transformation i = iA : [I, A] → A,
(3) components j = jA : I → [A, A] extranatural in A,
4 JOHN BOURKE
satisfying the following five axioms.
(C1)
[[A, C], [A, D]]
L
BB
[C, D]
L
TT
L

[[[A, B], [A, C]], [[A, B], [A, D]]]
[L,1]

[[B, C], [B, D]]
[1,L]
GG [[B, C], [[A, B], [A, D]]]
.
(C2) [[A, A], [A, C]] [I, [A, C]] (C3) [B, B] [[A, B], [A, B]]
[A, C] [A, C] I
1 GG
L
yy
i

[j,1]
GG L GG
j
——
j
aa
(C4) [B, C] [[I, B], [I, C]] (C5) I [I, I]
[[I, B], C] I
L GG j
GG
[i,1] 11 [1,i]{{ 1
44
i
||
(C, [−, −], I) is said to be left normal when the composite function
(2.1) C(A, B)
[A,−]
GG C([A, A], [A, B])
C(j,1)
GG C(I, [A, B])
is invertible, and right normal if i : [I, A] → A is invertible. A closed category is,
by definition, a skew closed category which is both left and right normal.1
Variants 2.3. We will regularly mention a couple of variants on the above definition
and we note them here.
(1) We will sometimes consider skew closed 2-categories: the Cat-enriched
version of the above concept. The difference is that C is now a 2-category,
[−, −] a 2-functor and each of the three transformations 2-natural in each
variable.
(2) We call a structure (C, [−, −], L) satisfying C1 but without unit a semi-closed
category.
2.3. The correspondence between skew monoidal and skew closed categories. A
monoidal category (C, ⊗, I) in which each functor − ⊗ A : C → C has a right
adjoint [A, −] naturally gives rise to the structure (C, [−, −], I) of a closed category.
Counterexamples to the converse statement are described in Section 3 of [6]:
no closed category axiom ensures the associativity of the corresponding tensor
1 The original definition of closed category [8] involved an underlying functor to Set. We are
using the modified definition of [33] (see also [26]) which eliminates the reference to Set.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 5
product. An appealing feature of the skew setting is that there is a perfect
correspondence between skew monoidal and skew closed structure.
Theorem 2.4 (Street [35]). Let C be a category equipped with an object I and a
pair of bifunctors ⊗ : C ×C → C and [−, −] : Cop ×C → C related by isomorphisms
ϕ : C(A ⊗ B, C) ∼= C(A, [B, C]) natural in each variable. There is a bijection
between extensions of (C, ⊗, I) to a skew monoidal structure and of (C, [−, −], I)
to a skew closed structure.
Our interest is primarily in the passage from the closed to the monoidal side
and, breaking the symmetry slightly, we describe it now: for the full symmetric
treatment see [35].
• l : I ⊗ A → A is the unique map such that the diagram
(2.2)
C(A, B)
v
AA
C(l,1)
GG C(I ⊗ A, B)
ϕ

C(I, [A, B])
commutes for all B. Here v = C(j, 1) ◦ [A, −] : C(A, B) → C(I, [A, B]) is the
morphism defining left normality. In particular l is invertible for each A
just when v is.
• r : A → A ⊗ I is the unique morphism such that the diagram
(2.3)
C(A ⊗ I, B)
ϕ

C(r,1)
GG C(A, B)
C(A, [I, B])
C(1,i)
SS
commutes for all B. In particular r is invertible for each A just when i is.
• Transposing the identity through the isomorphism ϕ : C(A ⊗ B, A ⊗ B) ∼=
C(A, [B, A ⊗ B]) yields a morphism u : A → [B, A ⊗ B] natural in each
variable. Write t : [A ⊗ B, C] → [A, [B, C]] for the composite
(2.4) [A ⊗ B, C]
L GG [[B, A ⊗ B], [B, C]]
[u,1]
GG [A, [B, C]]
which, we note, is natural in each variable. The constraint α : (A⊗B)⊗C →
A ⊗ (B ⊗ C) is the unique morphism rendering commutative the diagram
(2.5)
C(A ⊗ (B ⊗ C), D)
ϕ

C(α,1)
GG C((A ⊗ B) ⊗ C, D)
ϕ

C(A ⊗ B, [C, D])
ϕ

C(A, [B ⊗ C, D])
C(1,t)
GG C(A, [B, [C, D]])
6 JOHN BOURKE
for all D. In particular α is invertible just when t is.
Definition 2.5. A monoidal skew closed category consists of a skew monoidal
category (C, ⊗, I, α, l, r) and skew closed category (C, [−, −], I, L, i, j), together
with natural isomorphisms ϕ : C(A ⊗ B, C) ∼= C(A, [B, C]) all related by the
above equations.
Of course in the presence of the isomorphisms either bifunctor determines the
other. Accordingly a monoidal skew closed category is determined by either the
skew monoidal or closed structure together with the isomorphisms ϕ.
We remark that monoidal skew closed structures on the category of left R-modules
over a ring R that have R as unit correspond to left bialgebroids over R. This
was the reason for the introduction of skew monoidal categories in [36].
The following result – immediate from the above – is, minus the skew monoidal
terminology, contained within Chapter 2 and in particular Theorem 5.3 of [8].
Theorem 2.6 (Eilenberg-Kelly). Let (C, ⊗, [−, −], I) be a monoidal skew closed
category. Then (C, ⊗, I) is monoidal if and only if (C, [−, −], I) is closed and the
transformation t : [A ⊗ B, C] → [A, [B, C]] is an isomorphism for all A, B and C.
Eilenberg and Kelly’s theorem can be used to recognise monoidal structure
in terms of closed structure. However it can be difficult to determine whether
t : [A⊗B, C] → [A, [B, C]] is invertible. This difficulty disappears in the presence
of a suitable symmetry.
2.4. Symmetry. A symmetry on a skew closed category begins with a natural
isomorphism s : [A, [B, C]] ∼= [B, [A, C]]. If C is left normal the vertical maps
C(A, [B, C])
v

s0
GG C(B, [A, C])
v

C(I, [A, [B, C]])
C(I,s)
GG C(I, [B, [A, C]])
are isomorphisms, so that we obtain an isomorphism s0 by conjugating C(I, s).
If C underlies a monoidal skew closed category this in turn gives rise to a natural
isomorphism
C(A ⊗ B, C) ∼= C(B ⊗ A, C)
and so, by Yoneda, a natural isomorphism
(2.6) c : B ⊗ A ∼= A ⊗ B .
Our leading examples of skew closed categories do admit symmetries, but are not
left normal: accordingly, the symmetries are visible on the closed side but not
on the monoidal side. However they often reappear on the monoidal side upon
passing to the homotopy category – see Theorem 5.13.
Definition 2.7. A symmetric skew closed category consists of a skew closed
category (C, [−, −], I) together with a natural isomorphism s : [A, [B, C]] ∼=
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 7
[B, [A, C]] satisfying the following four equations.
(S1)
[A, [B, C]]
s
99
1 GG [A, [B, C]]
[B, [A, C]]
s
VV
(S2)
[A, [B, [C, D]]]
[1,s]

s GG [B, [A, [C, D]]]
[1,s]
GG [B, [C, [A, D]]]
s

[A, [C, [B, D]]]
s GG [C, [A, [B, D]]]
[1,s]
GG [C, [B, [A, D]]]
(S3)
[A, [B, C]]
s

L GG [[D, A], [D, [B, C]]]
[1,s]
GG [[D, A], [B, [D, C]]]
s

[B, [A, C]]
[1,L]
GG [B, [[D, A], [D, C]]]
(S4)
[A, B]
1
QQ
L GG [[A, A], [A, B]]
[j,1]
GG [I, [A, B]]
s GG [A, [I, B]]
[1,i]
GG [A, B]
C is said to be symmetric closed if its underlying skew closed category is closed.
Variants 2.8. As in Variants 2.3 there are evident notions of symmetric skew
closed 2-categories and symmetric semi-closed categories.
Remark 2.9. The notion of symmetric closed category described above coincides
with that of [6], though this may not be immediately apparent. Their invertible
unit map i : X → [I, X] points in the opposite direction to ours. Reversing
it, their (CC4) is clearly equivalent to our (S4). Their remaining axioms are a
proper subset of those above, with (C1), (C3), (C4) and (C5) omitted. But as
they point out in Proposition 1.3 any symmetric closed category in their sense is
a closed category and hence satisfies all four of these.
I first encountered a result close to the following one as Proposition 2.3 of
[6], which shows that a symmetric closed category C gives rise to a symmetric
promonoidal one by setting P(A, B, C) = C(A, [B, C]). This easily implies that
a symmetric closed category gives rise to a symmetric monoidal one on taking
adjoints. In a discussion about that result, Ross Street pointed out that a skew
monoidal category with an invertible natural isomorphism A ⊗ B ∼= B ⊗ A
satisfying the braid equation B5 of [14] is necessarily associative. The first part
of the following result essentially reformulates Street’s associativity argument in
terms of the closed structure. Diagram 4.8 of Chapter IV of De Schipper’s book
[7] – (2.7) below – proved helpful in making that reformulation.
8 JOHN BOURKE
Theorem 2.10 (Day-LaPlaza, Street). Let (C, ⊗, [−, −], I) be monoidal skew
closed.
(1) The transformation t : [A ⊗ B, C] → [A, [B, C]] is invertible if (C, [−, −], I)
is left normal and admits a natural isomorphism s : [A, [B, C]] ∼= [B, [A, C]]
satisfying S3. In particular, if (C, [−, −], I) is actually closed and admits
such a symmetry then (C, ⊗, I) is monoidal.
(2) If (C, [−, −], I, s) is symmetric closed then (C, ⊗, I, c) is symmetric monoidal.
Proof. For (1) we first prove that
(2.7)
[A ⊗ B, [C, D]]
t

s GG [C, [A ⊗ B, D]]
[1,t]

[A, [B, [C, D]]]
[1,s]
GG [A, [C, [B, D]]] s
GG [C, [A, [B, D]]]
commutes. From (2.4) we have t = [u, 1] ◦ L. So the upper path equals
[[1, [u, 1]] ◦ [1, L] ◦ s = [1, [u, 1]] ◦ s ◦ [1, s] ◦ L = s ◦ [1, s] ◦ [u, 1] ◦ L = s ◦ [1, s] ◦ t
by S3, naturality of s twice applied, and the definition of t.
By assumption each component v : C(A, B) → C(I, [A, B]) is invertible. Accordingly
a morphism f : [A, B] → [C, D] gives rise to a further morphism
f0 : C(A, B) → C(C, D) by conjugating C(I, f). At [A, g] : [A, B] → [A, C] we
obtain [A, g]0 = C(A, g). At L : [A, B] → [[C, A], [C, B]] an application of C3 establishes
that L0 = [C, −] : C(A, B) → C([C, A], [C, B]). Now (−)0, being defined
by conjugating through natural isomorphisms, preserves composition. Combining
the two last cases we find that t0 = [u, 1]0 ◦ L0 = C(u, 1) ◦ [B, −] = ϕ, the
adjointness isomorphism. Applying (−)0 to the above diagram, componentwise,
then gives the commutative diagram below.
C(A ⊗ B, [C, D])
ϕ

s0
GG C(C, [A ⊗ B, D])
C(C,t)

C(A, [B, [C, D]])
C(A,s)
GG C(A, [C, [B, D]]) s0
GG C(C, [A, [B, D]])
Since the left vertical path and both horizontal paths are isomorphisms, so is
C(C, t) for each C. Therefore t is itself an isomorphism. The remainder of (1)
now follows from Theorem 2.6.
As mentioned, Part 2 follows from Proposition 2.3 of [6]. We note an alternative
elementary argument. Having established the commutativity of (2.7) and that t
is an isomorphism, we are essentially in the presence of what De Schipper calls
a monoidal symmetric closed category.2
Theorem 6.2 of [7] establishes that a
monoidal symmetric closed category determines a symmetric monoidal one, as
required.
2 This is not exactly the case as De Schipper, following Eilenberg-Kelly, includes a basic
functor V : C → Set in his definition of symmetric closed category. However this basic functor
plays no role in the proof of the cited result.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 9
3. From multicategories to skew closed categories
Our examples of skew closed categories in Section 4 can be seen as arising from
closed multicategories equipped with further structure. In the present section we
describe how to pass from such multicategories to skew closed categories.
Multicategories were introduced in [25] and have objects A, B, C . . . together
with multimaps (A1, . . . , An) → B for each n ∈ N. These multimaps can be
composed and satisfy natural associativity and unit laws. We use boldface C for
a multicategory and C for its underlying category of unary maps. A symmetric
multicategory C comes equipped with actions of the symmetric group Sn on the
sets C(A1, . . . An; B) of n-ary multimaps. These actions must be compatible with
multimap composition. For a readable reference on the basics of multicategories
we refer to [27].
3.1. Closed multicategories. A multicategory C is said to be closed if for all
B, C ∈ C there exists an object [B, C] and evaluation multimap e : ([B, C], B) →
C with the universal property that the induced function
(3.1) C(A1, . . . , An; [B, C]) → C(A1, . . . , An, B; C)
is a bijection for all (A1, . . . , An) and n ∈ N. We can depict the multimap
e : ([B, C], B) → C as below.
(3.2) e
[B, C]
B
C
3.1.1. Semiclosed structure. Using the above universal property one obtains a
bifunctor [−, −] : Cop ×C → C. Given f : B → C the map [A, f] : [A, B] → [A, C]
is the unique one such that the two multimaps
(3.3) e
[A, f] [A, C][A, B]
A
C = e f C
B
A
[A, B]
coincide, whilst [f, A] is defined in a similar manner.
The natural bijections
C([B, C], [A, B], A; C) ∼= C([B, C], [A, B]; [A, C]) ∼= C([B, C], [[A, B], [A, C]])
10 JOHN BOURKE
induce a unique morphism L : [B, C] → [[A, B], [A, C]] such that the multimaps
(3.4)
e
e
L [[A, B], [A, C]]
[B, C]
[A, C][A, B]
A
C = e
e
B
[A, B]
A
[B, C]
C
coincide.
3.1.2. Symmetry. If C is a symmetric multicategory the natural bijections
C([A, [B, C]], B, A; C) ∼= C([A, [B, C]], B; [A, C]) ∼= C([A, [B, C]], [B, [A, C]])
induce a unique map s : [A, [B, C]] → [B, [A, C]] such that the multimaps
(3.5)
e
e
s
[B, [A, C]]
[A, [B, C]]
[A, C]B
A
C = e
e [B, C]
A
[A, [B, C]]
B C
coincide. The multimap above right depicts the image of e◦(e, 1) : ([A, [B, C]], A, B) →
([B, C], B) → C under the action
C([A, [B, C]], A, B; C) ∼= C([A, [B, C]], B, A; C)
of the symmetric group.
3.1.3. Nullary map classifiers and units.
Definition 3.1. A multicategory C has a nullary map classifier if there exists an
object I and multimap u : (−) → I such that the induced morphism
C(u; A) : C(I, A) → C(−; A)
is a bijection for each A.
Equivalently, if the functor C(−; ?) : C → Set sending an object A to the
set of nullary maps (−) → A is representable. In a closed multicategory a
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 11
nullary map classifier I enables the construction of morphisms i : [I, A] → A and
j : I → [A, A]. The former is given by
(3.6) u
e
I
[I, A]
A
For j observe that the identity 1 : A → A corresponds under the isomorphism
C(A, A) ∼= C(−; [A, A]) to a nullary map ˆ1 : (−) → [A, A]. Now j : I → [A, A] is
defined to be the unique map such that
(3.7) j ◦ u = ˆ1 .
In [29] the term unit for a closed multicategory means something stronger than
a nullary map classifier: it consists of a multimap u : (−) → I for which (3.6) is
invertible. By Remark 4.2 of ibid. a unit is a nullary map classifier.
3.2. The results. The following result is Proposition 4.3 of [29].
Theorem 3.2 (Manzyuk). If C is a closed multicategory with unit I then
(C, [−, −], I, L, i, j) is a closed category.
Since we are interested in constructing mere skew closed categories a nullary
map classifier suffices.
Theorem 3.3. If C is a closed multicategory with a nullary map classifier I then
(C, [−, −], I, L, i, j) is a skew closed category. Furthermore if C is a symmetric
multicategory then (C, [−, −], I, L, i, j, s) is symmetric skew closed.
Proof. We only outline the proof, which involves routine multicategorical diagram
chases best accomplished using string diagrams as in (3.2)–(3.6). (We note that
the deductions of C1 and C3 are given in the proof of Proposition 4.3 of [29].)
The axioms C1, C2 and C4 each assert the equality of two maps
X [Y1, . . . , [Yn−1, [Yn, Z]]..]
constructed using [−, −], L, i and j. These correspond to the equality of the
transposes
(X, Y1, . . . , Yn) Z
obtained by postcomposition with the evaluation multimaps. Since [−, −], L, i
and j are defined in terms of their interaction with the evaluation multimaps only
their definitions, together with the associativity and unit laws for a multicategory,
are required to verify these axioms. C3 and C5 each concern the equality of two
maps I A. Here one shows that the corresponding nullary maps (−) A
coincide. Again this is straightforward. The axioms S1-S4 are verified in a similar
fashion.
Theorem 3.3, as stated, will not apply to the examples of interest, none of
which quite has a nullary map classifier. What we need is a generalisation that
deals with combinations of strict and weak maps.
12 JOHN BOURKE
Definition 3.4. Let C be a multicategory equipped with a subcategory Cs ⊆ C
of strict morphisms containing all of the identities. We say that a multimap
f : (A1, . . . , An) → B is strict in i (or Ai abusing notation) if for all families of
multimaps {aj : (−) → Aj : j ∈ {1, . . . , i − 1, i + 1, . . . , n}} the unary map
f ◦ (a1, . . . ai−1, 1, ai+1, . . . an) : Ai → B
is strict.
Theorem 3.5. Let C be a closed multicategory equipped with a subcategory Cs → C
of strict maps containing the identities. Suppose further that
(1) A multimap (A1, . . . , An, B) → C is strict in Ai if and only if its transpose
(A1, . . . , An) → [B, C] is.
(2) There is a multimap u : (−) → I, precomposition with which induces a
bijection C(u, A) : Cs(I, A) → C(−; A) for each A.
Then (C, [−, −], L) is a semi-closed category. Moreover [−, −] and L restrict to
Cs where they form part of a skew closed structure (Cs, [−, −], I, L, i, j).
Furthermore if C is a symmetric multicategory then (C, [−, −], L, s) is symmetric
semi-closed and (Cs, [−, −], I, L, i, j, s) is symmetric skew closed.
Proof. We must show that these assumptions ensure that the bifunctor [−, −] :
Cop × C restricts to Cs and that the transformations L, i, j and s have strict
components. Beyond this point the proof is identical to that of Theorem 3.3.
A consequence of Definition 3.4 is that multimaps strict in a variable are closed
under composition: that is, given f : (A1, A2 . . . An) → Bk strict in Ai and
g : (B1, B2 . . . Bm) → C strict in Bk the composite multimap
(B1 . . . Bk−1, A1 . . . Ai . . . An, Bk+1 . . . Bm) → C
is strict in Ai. We use this fact freely in what follows.
Observe that since 1 : [A, B] → [A, B] is strict its transpose, the evaluation
multimap e : ([A, B], A) → B, is strict in [A, B]. It follows that if f : B → C is
strict then the composite multimap f ◦ e : ([A, B], A) → B → C of (3.3) is strict
in [A, B]. Accordingly its transpose [A, f] : [A, B] → [A, C] is strict. Likewise
[f, A] is strict if f is. Hence [−, −] restricts to Cs.
Since evaluation multimaps are strict in the first variable the composite e ◦ (1, e) :
([B, C], [A, B], A) → C of (3.4) is strict in [B, C]. Since transposing this twice
yields L : [B, C] → [[A, B], [A, C]] we conclude that L is strict. The composite
i = e ◦ (1, u) : [I, A] → A of (3.6) is strict as e : ([I, A], I) → A is strict in the
first variable. Clearly j is strict.
In a symmetric multicategory the actions of the symmetric group commute
with composition. It follows that if F : (A1, . . . An) → B is strict in Ai and
ϕ ∈ Sym(n) then ϕ(F) : (Aϕ(1), . . . , Aϕ(n)) → B is strict in Aϕ(i). Therefore
the composite ([A, [B, C]], B, A) → C on the right hand side of (3.5) is strict
in [A, [B, C]]. Since s : [A, [B, C]] ∼= [B, [A, C]] is obtained by transposing this
twice, it follows that s is strict.
In the next section we will encounter several examples of Cat-enriched multicategories,
hence 2-multicategories [13]. A 2-multicategory C has categories
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 13
C(A1, . . . , An; B) of multilinear maps and transformations between, and an extension
of multicategorical composition dealing with these transformations. There
is an evident notion of closed 2-multicategory, in which the bijection (3.1) is
replaced by an isomorphism, and of symmetric 2-multicategory. Theorem 3.5
generalises straightforwardly to 2-multicategories as we now record.
Theorem 3.6. Let C be a closed 2-multicategory equipped with a locally full sub
2-category Cs → C of strict maps containing the identities. Suppose further that
(1) A multimap (A1, . . . , An, B) → C is strict in Ai if and only if its transpose
(A1, . . . , An) → [B, C] is.
(2) There is a multimap u : (−) → I, precomposition with which induces an
isomorphism C(u, A) : Cs(I, A) → C(−; A) for each A.
Then (C, [−, −], L) is a semi-closed 2-category. Moreover [−, −] and L restrict to
Cs where they form part of a skew closed 2-category (Cs, [−, −], I, L, i, j).
Furthermore if C is a symmetric 2-multicategory then (C, [−, −], L, s) is symmetric
semi-closed and (Cs, [−, −], I, L, i, j, s) is a symmetric skew closed 2-category.
Remark 3.7. Theorem 5.1 of [29] shows that the notions of closed multicategory
with unit and closed category are, in a precise sense, equivalent. We do not know
whether skew closed categories are equivalent to some kind of multicategorical
structure.
4. Examples of skew closed structures
The goal of this section is to describe a few concrete examples of the kind of
skew closed structures that we are interested in. All can be seen to arise from
multicategories although sometimes it will be easier to describe the skew closed
structure directly.
In each case we meet a category, or 2-category, C of weak maps equipped with a
subcategory Cs of strict maps. The subcategory of strict maps is well behaved –
locally presentable, for instance – whereas C is not. The objects of the internal
hom [A, B] are the weak maps but these only form part of a skew closed structure
on the subcategory Cs of strict maps.
4.1. Categories with structure. The following examples can be understood as
arising from pseudo-commutative 2-monads in the sense of [13] – this more
abstract approach is described in Section 6.
4.1.1. Categories with specified limits. Let D be a set of small categories, thought
of as diagram types. There is a symmetric 2-multicategory D-Lim whose objects
A are categories A equipped with a choice of D-limits. The objects of the category
D-Lim(A1, . . . , An; B) are functors F : A1 × . . . × An → B preserving D-limits in
each variable, and the morphisms are just natural transformations. For the case
n = 0 we have D-Lim(−; B) = B.
The morphisms in the 2-category of unary maps D-Lim are the D-limit preserving
functors, amongst which we have the 2-category D-Lims of strict Dlimit
preserving functors and the inclusion j : D-Lims → D-Lim. Accordingly
a multimap F : (A1, . . . , An) → B is strict in Ai just when each functor
14 JOHN BOURKE
F(a1, . . . , ai−1, −, ai+1, . . . , an) : Ai → B preserves D-limits strictly.
The functor category [A, B] has a canonical choice of D-limits inherited pointwise
from B. Since D-limits commute with D-limits the full subcategory j :
D-Lim(A, B) → [A, B] is closed under their formation (although D-Lims(A, B) is
not!) and we write [A, B] for D-Lim(A, B) equipped with this choice of D-limits.
It is routine to verify that the objects [A, B] exhibit D-Lim as a closed 2multicategory
and moreover that a multimap F : (A1, . . . , An, B) → C is strict
in Ai just when its transpose (A1, . . . , An) → [B, C] is. With regards units, the
key point is that the forgetful 2-functor U : D-Lims → Cat has a left 2-adjoint
F. This follows from [3] but see also Section 6.2. Accordingly we have a natural
isomorphism D-Lim(−; A) ∼= Cat(1, A) ∼= D-Lims(F1, A). By Theorem 3.6 we
obtain the structure of a symmetric semi-closed 2-category (D-Lim, [−, −], L, s)
restricting to a symmetric skew closed 2-category (D-Lim, [−, −], F1, L, i, j, s).
The skew closed structure on D-Lims fails to extend to D-Lim because the unit
map j : F1 → [A, A] is only pseudo-natural in morphisms of D-Lim. (It does,
however, extend to a pseudo-closed structure on D-Lim in the sense of [13]). The
skew closed D-Lims is neither left nor right normal: for example, the canonical
functor D-Lims(A, B) → D-Lims(F1, [A, B]) is isomorphic to the inclusion
D-Lims(A, B) → D-Lim(A, B) and this is not in general invertible.
4.1.2. Permutative categories and so on. An example amenable to calculation
concerns symmetric strict monoidal – or permutative – categories. The symmetric
skew closed structure can be seen as arising from a symmetric 2-multicategory,
described in [9]. Because the relevant definition of multilinear map is rather long,
we treat the skew closed structure directly.
Let Perms and Perm denote the 2-categories of permutative categories with the
strict symmetric monoidal and strong symmetric monoidal functors between.
Using the symmetry of B the category Perm(A, B) inherits a pointwise structure
[A, B] ∈ Perm. Namely we set (F ⊗G)− = F −⊗G−. The structure isomorphism
(F ⊗ G)(a ⊗ b) ∼= (F ⊗ G)a ⊗ (F ⊗ G)b combines the structure isomorphisms
F(a ⊗ b) ∼= Fa ⊗ Fb and G(a ⊗ b) ∼= Ga ⊗ Gb with the symmetry as below:
F(a ⊗ b) ⊗ G(a ⊗ b) ∼= Fa ⊗ Fb ⊗ Ga ⊗ Gb ∼= Fa ⊗ Ga ⊗ Fb ⊗ Gb .
The structural isomorphism concerning monoidal units is obvious. The hom
objects [A, B] extend in the obvious way to a 2-functor [−, −] : Permop
×Perm →
Perm. Moreover, the functor [C, −] : Perm(A, B) → Perm([C, A], [C, B]) lifts
to a strict map L = [C, −] : [A, B] → [[C, A], [C, B]] because both domain and
codomain have structure inherited pointwise from B. We omits details of the
symmetry isomorphism s : [A, [B, C]] ∼= [B, [A, C]].
The unit F1 is the free permutative category on 1: the category of finite ordinals
and bijections. The unit map i : [F1, A] → A is given by evaluation at 1 whilst
j : F1 → [A, A] is the unique symmetric strict monoidal functor with j(1) = 1A.
Again the skew closed structure is neither left nor right normal.
This example can be generalised to deal with general symmetric monoidal categories.
A careful analysis of both tensor products and internal homs on the 2category
SMon of symmetric monoidal categories and strong symmetric monoidal
functors was given by Schmitt [31].
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 15
4.2. 2-categories and bicategories. Examples of skew closed structures not arising
from pseudo-commutative 2-monads, even in the extended sense of [28],
include 2-categories and bicategories. We focus upon the more complex case of
bicategories. Let Bicat denote the category of bicategories and homomorphisms
(also called pseudofunctors) and Bicats the subcategory of bicategories and strict
homomorphisms. We describe a symmetric skew closed structure on Bicats
with internal hom Hom(A, B) the bicategory of pseudofunctors, pseudonatural
transformations and modifications from A to B.
This skew closed structure arises from a closed symmetric multicategory. We
begin by briefly recalling the multicategory structure, which was introduced and
studied in depth in Section 1.3 of [37] by Verity, and to which we refer for further
details. The multicategory Bicat – denoted by Homs in ibid. – has bicategories
as objects. The multimaps are a variant of the cubical functors of [11]. More
precisely, a multimap F : (A1, . . . , An) → B consists of
• for each n-tuple (a1, . . . , an) an object F(a1, . . . , an) of B;
• for each 1 ≤ i ≤ n a homomorphism F(a1, . . . , ai−1, −, ai+1 . . . , an) extending
the above function on objects;
• for each pair 1 ≤ i < j ≤ n, n-tuple of objects (a1, . . . , an) and morphisms
f : ai → ai ∈ Ai and fj : aj → aj ∈ Aj, an invertible 2-cell:
F(ai, aj) F(ai, aj)
F(ai, aj) F(ai, aj)
F(fi,aj)
GG
F(fi,aj)
GG
F(ai,fj)

F(ai,fj)
F(fi,fj)
v~
where we have omitted to label the inactive parts of the n-tuple (a1, . . . , an)
under the action of F. These invertible 2-cells are required to form the
components of pseudonatural transformations both vertically – F(−i, fj) :
F(−i, aj) → F(−i, aj) – and horizontally – F(fi, −j) : F(ai, −j) → F(ai, −j)
– and satisfy a further cubical identity involving trios of morphisms.
A nullary morphism (−) → B is simply defined to be an object of B. Observe
that the category of unary maps of Bicat is simply Bicat. It is established in
Section 1.3 of ibid. – see Lemma 1.3.4 and the discussion that follows – that the
symmetric multicategory Bicat is closed, with hom-object given by the bicategory
Hom(A, B) of homomorphisms, pseudonatural transformations and modifications
from A to B.
A multimap F : (A1, . . . , An) → B is strict in Ai just when each homomorphism
F(a1, . . . , ai−1, −, ai+1 . . . , an) is strict. An inspection of the bijection of Lemma
1.3.4 of ibid. makes it clear that the natural bijection
Bicat(A1, . . . , An, B; C) ∼= Bicat(A1, . . . , An; Hom(B, C))
respects strictness in Ai.
Turning to the unit, recall that Bicat(−; A) = A0. The forgetful functor
(−)0 : Bicats → Set has a left adjoint F for general reasons – see Section 7
for more on this. It follows that we have a bijection Bicats(F1, A) ∼= Bicat(−; A)
16 JOHN BOURKE
where F1 is the free bicategory on 1. Concretely, F1 has a single object • and a
single generating 1-cell e : • → •. General morphisms are (non-empty) bracketed
copies of e such as ((ee)e), and two such morphisms are connected by a unique
2-cell, necessarily invertible.
By Theorem 3.5 we obtain a symmetric semi-closed category (Bicats, Hom, L)
which restricts to a symmetric skew closed structure (Bicats, Hom, F1, L, i, j).
As in the preceding examples the skew closed structure on Bicats is not closed
and fails to extend to Bicat.
In Section 7 we further analyse this symmetric skew closed structure. Accordingly
we describe a few aspects of it in more detail. Firstly, let us describe the action
of the functor Hom(−, −) : Bicatop
× Bicat → Bicat. From a homomorphism
f : A B the homomorphism Hom(f, 1) : Hom(B, C) → Hom(A, C) obtained
by precomposition is always strict, and straightforward to describe. The postcomposition
map Hom(1, f) = Hom(B, C) → Hom(B, D) induced by a strict
homomorphism f : C → D is equally straightforward.
Though not strictly required in what follows, for completness we mention the
slightly more complex case where f is non-strict. At η : g → h ∈ Hom(B, C) the
pseudonatural transformation fη : fg → fh has components fηa : fga → fha at
a ∈ B; at α : a → b the invertible 2-cell (fη)α:
fhα ◦ fηa
λf
CQ f(hα ◦ ηa)
fηα
CQ f(ηb ◦ gα)
λf
−1
CQ fηb ◦ fgα
conjugates fηα by the coherence constraints for f. The action of f∗ on 2cells
is straightforward. The coherence constraints f(η ◦ µ) ∼= f(η) ◦ f(µ) and
f(idg) ∼= id(fg) for f∗ are pointwise those for f.
The only knowledge required of L : Hom(B, C) → Hom(Hom(A, B), Hom(A, C))
is that it has underlying function
Hom(A, −) : Bicat(B, C) → Bicat(Hom(A, B), Hom(A, C)) .
The unit map i : Hom(F1, A) → A evaluates at the single object • of F1 whilst
j : F1 → Hom(A, A) is the unique strict homomorphism sending • to the identity
on A.
This example can be modified to deal with 2-categories. Let 2-Cat ⊂ Bicat
and 2-Cats ⊂ Bicats be the symmetric multicategory and category obtained by
restricting the objects from bicategories to 2-categories. Since Hom(A, B) is a
2-category if B is, we obtain a closed multicategory 2-Cat by restriction. In this
case we have a natural bijection 2-Cats(1, A) ∼= A0 = 2-Cat(−; A). It follows
that we obtain a symmetric skew closed structure (2-Cats, Hom, L, 1, i, j) with
the same semi-closed structure as before, but with the simpler unit 1.
4.3. Lax morphisms. Each of the above examples describes a symmetric skew
closed structure arising from a symmetric closed multicategory. In each case there
are non-symmetric variants dealing with lax structures, of which we mention
a few now. These have the same units but different internal homs. In D-Lims
the hom [A, B] is the functor category [A, B] equipped with D-limits pointwise
in B. In Perms the internal hom [A, B] consists of lax monoidal functors and
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 17
monoidal transformations. For Bicats one can take [A, B] to be the bicategory
of homomorphisms and lax natural transformations from A to B.
5. Skew structures descending to the homotopy category
In the present section we consider categories C equipped with a Quillen
model structure as well as a skew monoidal or skew closed structure. We
describe conditions under which the skew structures descend to the homotopy
category Ho(C) and call the skew monoidal/closed structures on C homotopy
monoidal/closed if the induced structures on Ho(C) are genuinely monoidal/closed.
Theorem 5.11 gives a complete description of how monoidal skew closed structure
descends to the homotopy category. Our analogue of Eilenberg and Kelly’s
theorem is Theorem 5.12: it allows us to recognise homotopy monoidal structure
in terms of homotopy closed structure.
We assume some familiarity with the basics of Quillen model categories, as
introduced in [30], and covered in Chapter 1 of [12]. Let us fix some terminology
and starting assumptions. We assume that all model categories C have functorial
factorisations. It follows that C is equipped with cofibrant and fibrant replacement
functors Q and R together with natural transformations p : Q → 1 and q : 1 → R
whose components are respectively trivial fibrations and trivial cofibrations. Let
j : Cc → C and j : Cf → C denote the full subcategories of cofibrant and fibrant
objects, through which Q and R respectively factor. The four functors preserve
weak equivalences and hence extend to the homotopy category. At that level we
obtain adjoint equivalences
Ho(Cc)
Ho(j)
EE
Ho(C)
Ho(Q)
mm Ho(Cf )
Ho(j)
EE
Ho(C)
Ho(R)
mm
with counit and unit given by Ho(p) and its inverse, and Ho(q) and its inverse
respectively. If a functor between model categories F : C → D preserves weak
equivalences between cofibrant objects we can form its left derived functor
Fl = Ho(FQ) : Ho(C) → Ho(D), equally Ho(Fj)Ho(Q) : Ho(C) → Ho(Cc) →
Ho(D). If G preserves weak equivalences between fibrant objects then Gr =
Ho(GR) = Ho(Gj)Ho(R) is its right derived functor.
5.1. Skew monoidal structure on the homotopy category. Let C be a model
category equipped with a skew monoidal structure (C, ⊗, I, α, l, r). Our interest
is in left deriving this to a skew monoidal structure on Ho(C). In the monoidal
setting this was done in [12] and the construction in the skew setting, described
below, is essentially identical.
Axiom M. ⊗ : C × C → C preserves cofibrant objects and weak equivalences
between them and the unit I is cofibrant.
The above assumption ensures that the skew monoidal structure on C restricts
to one on Cc and that the restricted functor ⊗ : Cc×Cc → Cc preserves weak equivalences.
Accordingly we obtain a skew monoidal structure (Ho(Cc), Ho(⊗), I) with
18 JOHN BOURKE
the same components as before. Transporting this along the adjoint equivalence
Ho(j) : Ho(Cc) Ho(C) : Ho(Q) yields a skew monoidal structure
(Ho(C), ⊗l, I, αl, ll, rl)
on Ho(C). We will often refer to (Ho(C), ⊗l, I) as the left-derived skew monoidal
structure since ⊗l is the left derived functor of ⊗. On objects we have A ⊗l B =
QA⊗QB and an easy calculation shows that the constraints for the skew monoidal
structure are given by the following maps in Ho(C).
(5.1)
Q(QA ⊗ QB) ⊗ QC
p⊗1
GG (QA ⊗ QB) ⊗ QC
α

QA ⊗ (QB ⊗ QC)
(1⊗p)−1
GG QA ⊗ Q(QB ⊗ QC)
(5.2) QI ⊗ QA
p⊗1
GG I ⊗ QA
l GG QA
p
GG A
(5.3) A
p−1
GG QA
r GG QA ⊗ I
(1⊗p)−1
GG QA ⊗ QI
Definition 5.1. Let (C, ⊗, I) be a skew monoidal structure on a model category
C satisfying Axiom M. We say that C is homotopy monoidal if (Ho(C), ⊗l, I) is
genuinely monoidal.
Proposition 5.2. Let (C, ⊗, I) be a skew monoidal category with a model structure
satisfying Axiom M. The following are equivalent.
(1) (C, ⊗, I) is homotopy monoidal.
(2) For all cofibrant X, Y, Z the map α : (X ⊗ Y ) ⊗ Z → X ⊗ (Y ⊗ Z) is a
weak equivalence, and for all cofibrant X both maps r : X → X ⊗ I and
l : I ⊗ X → X are weak equivalences.
Proof. Observe that the constraints (5.1),(5.2) and (5.3) are αQA,QB,QC, lQA
and rQA conjugated by isomorphisms in Ho(C). It follows that (Ho(C), ⊗l, I)
is genuine monoidal just when for αQA,QB,QC, lQA and rQA are isomorphisms
in Ho(C) for all A, B and C. Axiom M ensures for that cofibrant A, B, C that
we have isomorphisms αA,B,C
∼= αQA,QB,QC, lA
∼= lQA and rA
∼= rQA in Ho(C)2
so that the former maps are isomorphisms just when the latter ones are. This
proves the claim.
Notation 5.3. We call (C, ⊗, I) homotopy symmetric monoidal if (Ho(C), ⊗l, I)
admits the further structure of a symmetric monoidal category, but emphasise
that this refers to a symmetry on Ho(C) not necessarily arising from a symmetry
on C itself.
5.2. Skew closed structure on the homotopy category. Let C be a model category
equipped with a skew closed structure (C, [−, −], I, L, i, j). Our intention is to
right derive the skew closed structure to the homotopy category. This construction,
more complex than its monoidal counterpart, is closely related to the construction
of a skew closed category (C, [Q−, −], I) from a closed comonad Q [35].
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 19
Axiom C. For cofibrant X the functor [X, −] preserves fibrant objects and trivial
fibrations. For fibrant Y the functor [−, Y ] preserves weak equivalences between
cofibrant objects. The unit I is cofibrant.3
It follows from Axiom C that if A is cofibrant then [1, pB] : [A, QB] → [A, B]
is a trival fibration. Accordingly we obtain a lifting kA,B as below.
(5.4)
Q[A, B]
p[A,B]
66
kA,B
GG [A, QB]
[1,pB]zz
[A, B]
Because I is cofibrant we also have a lifting e as below.
(5.5)
I
1
00
e GG QI
pI
~~
I
Lemma 5.4. Let (C, [−, −], I) satisfy Axiom C. Then each of the following four
diagrams
(5.6)
Q[QB,C]
k

QL
GG Q[[QA,QB],[QA,C]]
Q[k,1]
GG Q[Q[QA,B],[QA,C]]
k GG [Q[QA,B],Q[QA,C]]
[1,k]

[QB,QC]
L
GG [[QA,QB],[QA,QC]]
[k,1]
GG [Q[QA,B],[QA,QC]]
(5.7)
Q[I, B]
Qi 55
k GG [I, QB]
i{{
QB
(5.8)
I
j
RR
e GG QI
Qj
GG Q[A, A]
Q[p,1]
GG Q[QA, A]
k GG [QA, QA]
(5.9)
Q[B, C]
Q[f,g]

kB,C
GG [B, QC]
[f,Qg]

Q[A, D]
kA.D
GG [A, QD]
3 We could weaken Axiom C by requiring that [X, −] preserves only weak equivalences
between fibrant objects, rather than all trivial fibrations. This is still enough to construct the
skew closed structure of Theorem 5.5 though the proof becomes slightly longer. Because we
need the stronger Axiom MC in the crucial monoidal skew closed case anyway, we emphasise
the convenient Axiom C.
20 JOHN BOURKE
commutes up to left homotopy. Moreover, if X is fibrant then the image of each
diagram under [−, X] commutes in Ho(C).
Note that in (5.9) A and B are cofibrant and the morphisms f : B → A and
g : C → D are arbitrary.
Proof. In each case we are presented with a pair of maps f, g : U V with U
cofibrant. To prove that f and g are left homotopic it suffices, by Proposition
1.2.5(iv) of [12], to show that there exists a trivial fibration h : V → W with
h ◦ f = h ◦ g. We take the trivial fibrations [1, p[QA,QC]], pB, [1, pA] and [1, pD]
respectively. Each diagram, postcomposed with the relevant trivial fibration, is
easily seen to commute.
For the second point observe that any functor C → D sending weak equivalences
between cofibrant objects to isomorphisms identifies left homotopic maps - this
follows the proof of Corollary 1.2.9 of ibid. Applying this to the composite of
[−, X] : C → Cop and Cop → Ho(C)op gives the result.
Axiom C ensures that the right derived functor
[−, −]r : Ho(C)op
× Ho(C) → Ho(C)
exists with value [A, B]r = [QA, RB]. The unit for the skew closed structure will
be I. Using Axiom C we form transformations Lr, ir and jr on Ho(C) as below.
(5.10)
[QA, RB]
[Qq,1]−1

[Q[QC, RA], R[QC, RB]]
[QRA, RB]
L GG [[QC, QRA], [QC, RB]]
[k,1]
GG [Q[QC, RA], [QC, RB]]
[1,q]
yy
(5.11) [QI, RA]
[e,1]
GG [I, RA]
i GG RA
q−1
GG A
(5.12) I
j
GG [A, A]
[p,1]
GG [QA, A]
[1,q]
GG [QA, RA]
Theorem 5.5. Let C be a model category equipped with a skew closed structure
(C, [−, −], I) satisfying Axiom C. Then Ho(C) admits a skew closed structure
(Ho(C), [−, −]r, I) with constraints as above.
Proof. In order to keep the calculations relatively short we will first describe
a slightly simpler skew closed structure on Ho(Cf ). We then obtain the skew
closed structure on Ho(C) by transport of structure.
So our main task is to construct a suitable skew closed structure on Ho(Cf ). Now
Axiom C ensures that [QA, B] is fibrant whenever B is. The restricted bifunctor
[Q−, −] : Cop
f × Cf → Cf then preserves weak equivalences in each variable and
so extends to a bifunctor Ho([Q−, −]) on Ho(Cf ). For the unit on Ho(Cf ) we
take RI.
The constraints are given by the following three maps.
(5.13) [QA, B]
L GG [[QC, QA], [QC, B]]
[k,1]
GG [Q[QC, A], [QC, B]]
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 21
(5.14) [QRI, A]
[Qq,1]
GG [QI, A]
[e,1]
GG [I, A]
i GG A
(5.15) RI
q−1
GG I
j
GG [A, A]
[p,1]
GG [QA, A]
We should explain why the above components are natural on Ho(Cf ) – in the
appropriate variance – since consideration of extraordinary naturality is perhaps
non-standard.
Given F, G : A B and a family of maps {ηA : FA → GA : A ∈ A} we can
consider the class of morphisms Nat(η) ⊆ Mor(A) with respect to which η is
natural. Nat(η) is closed under composition and inverses in A. If (A, W) is a
category equipped with a collection of weak equivalences W then each arrow of
Ho(A) is composed of morphisms in A together with formal inverses w−1 where
w ∈ W. It follows that the family {ηA : FA → GA : A ∈ A} is natural in Ho(A)
just when it is natural where restricted to A. Similarly given S : Aop × A → B
and a family of morphisms {θA : X → S(A, A) : A ∈ A} we can consider the
class Ex(θ) ⊆ Mor(A) with respect to which θ is extranatural. This has the
same closure properties as before. It follows that the family θA : X → S(A, A) is
extranatural in Ho(A) just when it is extranatural when restricted to maps in A.
Using this reasoning we deduce that (5.14) and (5.15) are natural. We likewise
obtain the naturality of the L-component of (5.13) in each variable. So it suffices
to show that (kC,A, 1) : [[QC, QA], [QC, B]] → [Q[QC, A], [QC, B]] is natural in
each variable. This follows from Diagram (5.9) of Lemma 5.4.
We verify the diagrams (C1-C5) below. Each involves an instance of the corresponding
diagram (C1-C5) for the skew closed structure on C itself, an application
of Lemma 5.4 and straightforward applications of naturality.
(C2)
[Q[QA,A],[QA,C]]
[Q[p,1],1]
GG [Q[A,A],[QA,C]]
[Qj,1]
GG [QI,[QA,C]]
1
99
[Qq,1]−1
GG [QRI,[QA,C]]
[Qq,1]

[QI,[QA,C]]
[e,1]

[[QA,QA],[QA,C]]
[k,1]
yy
[j,1]
GG [I,[QA,C]]
i

[QA,C]
L
yy
1
GG [QA,C]
(C3)
RI
q−1
GG I
j
77
j
GG [B,B]
L

[p,1]
GG [QB,B]
L

[[QA,B],[QA,B]]
[p,1] AA
[[1,p],1]
GG [[QA,QB],[QA,B]]
[k,1]

[Q[QA,B],[QA,B]]
(C4)
[[QRI,QB],[QRI,C]]
[k,1]
GG
[1,[Qq,1]] AA
[Q[QRI,B],[QRI,C]]
[1,[Qq,1]]
AA
[[QRI,QB],[QI,C]]
[k,1]
GG [Q[QRI,B],[QI,C]]
[1,[e,1]]

[QB,C]
L
TT
[Qi,1]

[i,1]
88
L
99
L GG [[QI,QB],[QI,C]]
[[Qq,1],1]
SS
[1,[e,1]]

[k,1]
GG [Q[QI,B],[QI,C]]
[1,[e,1]]

[Q[Qq,1],1]
SS
[[I,QB],[I,C]]
[1,i]

[[e,1],1]
GG [[QI,QB],[I,C]]
[1,i]

[k,1]
GG [Q[QI,B],[I,C]]
[1,i]

[Q[Qq,1],1]
GG [Q[QRI,B],[I,C]]
[1,i]

[[I,QB],C]
[k,1]
ww
[[e,1],1]
GG [[QI,QB],C]
[k,1]
GG [Q[QI,B],C]
[Q[Qq,1],1]
GG [Q[QRI,B],C]
[Q[I,B],C] [Q[e,1],1]
RR
(C5)
RI
q−1
GG I
1
11
j

j
GG [RI,RI]
[q,1]

[p,1]
GG [QRI,RI]
[Qq,1]

[I,I]
i

[1,q]
GG [I,RI]
i
 1
77
[p,1]
GG [QI,RI]
[e,1]

I q
GG RI [I,RI]
i
oo
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 23
(C1)
[Q[QA,C],[QA,D]]
L
GG[[Q[QA,B],Q[QA,C]],[Q[QA,B],[QA,D]]]
[k,1]

[QC,D]
L

L
GG[[QA,QC],[QA,D]]
L

L
GG
[k,1]
TT
[[[Q[QA,B],[QA,QC]],[Q[QA,B],[QA,D]]]
[[1,k],1]
SS
[[k,1],1]

[Q[Q[QA,B],[QA,C]],[Q[QA,B],[QA,D]]]
[Q[k,1],1]

[[[QA,QB],[QA,QC]],[[QA,QB],[QA,D]]]
[1,[k,1]]
GG
[L,1]

[[[QA,QB],[QA,QC]],[Q[QA,B],[QA,D]]]
[L,1]

[[QB,QC],[QB,D]]
[k,1]

[1,L]
GG[[QB,QC],[[QA,QB],[QA,D]]]
[k,1]
@@
[1,[k,1]]
GG[[QB,QC],[Q[QA,B],[QA,D]]]
[k,1]
AA
[Q[[QA,QB],[QA,C]],[Q[QA,B],[QA,D]]]
[QL,1]

[Q[QB,C],[QB,D]]
[1,L]
GG[Q[QB,C],[[QA,QB],[QA,D]]]
[1,[k,1]]
GG[Q[QB,C],[Q[QA,B],[QA,D]]]
24 JOHN BOURKE
We now transport the skew closed structure along the adjoint equivalence Ho(j) :
Ho(Cf ) Ho(C) : Ho(R). The skew closed structure obtained in this way has
bifunctor
Ho(C)op×Ho(C)
Ho(R)op×Ho(R)
GG Ho(Cf )op×Ho(Cf )
Ho([Q−,−])
GG Ho(Cf )
Ho(j)
GG Ho(C)
and unit given by
I
RI GG Ho(Cf )
Ho(j)
GG Ho(C) .
So Ho([QR−, R−]) and RI respectively. Neither is quite as claimed. We finally
obtain the skew closed structure stated in the theorem by transferring
this last skew closed structure along the isomorphisms of bifunctors [Qq, 1] :
Ho([QR−, R−]) → Ho([Q−, R−]) and of units q−1 : RI → I.
We often refer to (Ho(C), [−, −]r, I) as the right derived skew closed structure
since [−, −]r is the right derived functor of [−, −].
Definition 5.6. Let C be a model category with a skew closed structure (C, [−, −], I)
satisfying Axiom C. We say that (C, [−, −], I) is homotopy closed if the right
derived skew closed structure (Ho(C), [−, −]r, I) is genuinely closed.
Proposition 5.7. Let (C, [−, −], I) be a skew closed category satisfying Axiom C.
Then (C, [−, −], I) is homotopy closed if and only if the following two conditions
are met.
(1) For all cofibrant A and fibrant B the map
v = C(j, 1) ◦ [A, −] : C(A, B) → C(I, [A, B])
is a bijection on homotopy classes of maps.
(2) For all fibrant A the map i : [I, A] → A is a weak equivalence.
Proof. We will show that (1) and (2) amount to left and right normality of
(Ho(C), [−, −]r, I) respectively. Now Ho(C) is left normal just when
Ho(C)(A, B)
Ho(C)(jr,1)◦[A,−]r
GG Ho(C)(I, [A, B]r)
is a bijection for all A and B. As in any skew closed category this map is natural
in both variables. Since we have isomorphisms QA → A and B → RB in Ho(C)
the above map will be an isomorphism for all A, B just when it is so for all
cofibrant A and fibrant B. For such A and B we consider the diagram
C(A, B)
C(j,1)◦C([p,q],1)◦[QA,R−]
GG

C(I, [QA, RB])

Ho(C)(A, B)
[Ho(C)(jr,1)◦[A,−]r
GG Ho(C)(I, [QA, RB])
which is commutative by definition of [A, −]r and jr. The left and right vertical
morphisms are surjective and identify precisely the homotopic maps. It follows
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 25
that the bottom row is invertible just when the top row induces a bijection on
homotopy classes. By naturality of p and q we can rewrite the top row as
C(A, B)
C(j,1)◦[A,−]
GG C(I, [A, B])
C(1,[p,q])
GG C(I, [QA, RB]) .
Axiom C ensures that [p, q] : [A, B] → [QA, RB] is a weak equivalence between
fibrant objects. Since I is cofibrant it follows that C(1, [p, q]) is a bijection on
homotopy classes. Therefore the above composite is a bijection on homotopy
classes just when its left component is.
Now Ho(C) is right normal just when ir = q−1◦iRA◦[e, 1] : [QI, RA] → [I, RA] →
RA → A is invertible; equally just when iRA : [I, RA] → RA is invertible for
each A. For fibrant A we have iA
∼= iRA in Ho(C)2 and the result follows.
5.2.1. The symmetric case. Consider a skew closed structure (C, [−, −], I) satisfying
Axiom C. By Theorem 5.5 we may form the right derived skew closed structure
(Ho(C), [−, −]r, I). Then from a symmetry isomorphism s : [A, [B, C]] ∼=
[B, [A, C]] we can define a symmetry isomorphism sr : [A, [B, C]r]r
∼= [B, [A, C]r]r
on the right derived internal hom as below.
(5.16) [QA,R[QB,RC]]
[1,q]−1
GG [QA,[QB,RC]]
s GG [QB,[QA,RC]]
[1,q]
GG [QB,R[QA,RC]]
Proposition 5.8. Consider (C, [−, −], I) satisfying Axiom C, so we have the skew
closed structure (Ho(C), [−, −]r, I) of Theorem 5.5.
If a natural isomorphism s : [A, [B, C]] ∼= [B, [A, C]] satisfies any of S1 − S4
then so does sr : [A, [B, C]r]r
∼= [B, [A, C]r]r. In particular, if (C, [−, −], I, s) is
symmetric skew closed then so is (Ho(C), [−, −]r, I, sr).
Proof. As in the proof of Theorem 5.5 we transport the structure from Ho(Cf ).
We can extend the skew closed structure (Ho(Cf ), Ho[Q−, −], RI) described
therein by a symmetry transformation sr whose component at (A, B, C) is
[QA, [QB, C]]
sQA,QB,C
GG [QB, [QA, C]]
Since the components of sr are just those of s it follows that S1 and S2 hold in
Ho(Cf ) if they do so in C. The diagrams for S3 and S4 are below.
(S3)
[QA,[QB,C]]
s

L GG [[QD,QA],[QD,[QB,C]]]
[1,s]

[k,1]
GG [Q[QD,A],[QD,[QB,C]]]
[1,s]

[[QD,QA],[QB,[QD,C]]]
s

[k,1]
GG [Q[QD,A],[QB,[QD,C]]]
s

[QB,[QA,C]]
[1,L]
GG [QB,[[QD,QA],[QD,C]]]
[1,[k,1]]
GG [QB,[Q[QD,A],[QD,C]]]
26 JOHN BOURKE
(S4)
[QA,B]
1

L GG [[QA,QA],[QA,B]]
[j,1]

[k,1]
GG [Q[QA,A],[QA,B]]
[Q[p,1],1]
GG [Q[A,A],[QA,B]]
[Qj,1]

[QI,[QA,B]]
1
vv
[Qq,1]−1

[I,[QA,B]]
s

[QI,[QA,B]]
s

[e,1]
oo [QRI,[QA,B]]
[Qq,1]
oo
s

[QA,B] [QA,[I,B]]
[1,i]
oo [QA,[QI,B]]
[1,[e,1]]
oo [QA,[QRI,B]]
[1,[Qq,1]]
oo
Therefore (Ho(Cf ), Ho[Q−, −], RI, sr) satisfies any of S1−S4 when (C, [−, −], I, s)
does so. The desired structure on Ho(C) is obtained by transporting from the
equivalent Ho(Cf ) as in Theorem 5.5.
5.3. Monoidal skew closed structure on the homotopy category and the homotopical
version of Eilenberg and Kelly’s theorem. Let C be a model category
equipped with a monoidal skew closed structure (C, ⊗, [−, −], I). In order to
derive the tensor-hom adjunctions − ⊗ QA [QA, −] to the homotopy category,
we will make use of the concept of a Quillen adjunction.
An adjunction F : C D : U of model categories is said to be a Quillen
adjunction if the right adjoint U : D → C preserves fibrations and trivial fibrations.
One says that U is right Quillen. This is equivalent to asking that F
preserves cofibrations and trivial cofibrations, in which case F is said to be left
Quillen. The derived functors Fl and Ur then exist and form an adjoint pair
Fl : Ho(C) Ho(D) : Ur. This works as follows. At cofibrant A and fibrant B
the isomorphisms ϕA,B : D(FA, B) ∼= C(A, UB) are well defined on homotopy
classes and so give natural bijections
ϕA,B : Ho(D)(FA, B) ∼= Ho(C)(A, UB)
– we use the same labelling. At A, B ∈ Ho(C) the hom-set bijections
ϕd
A,B : Ho(D)(FlA, B) ∼= Ho(C)(A, UrB)
are then given by conjugating the ϕA,B as below:
Ho(D)(FQA,B)
(1,q)
GG Ho(D)(FQA,RB)
ϕA,B
GG Ho(C)(QA,URB)
(p,1)−1
GG Ho(C)(A,URB)
It follows that the unit of the derived adjunction – the derived unit – is given by
A
p−1
A
GG QA
ηQA
GG UFQA
UqF QA
GG URFQA
whilst the derived counit admits a dual description.
Proposition 5.9. For C a model category and (C, ⊗, [−, −], I) monoidal skew closed
the following are equivalent.
(1) For cofibrant X the functor − ⊗ X is left Quillen and for each cofibrant Y
the functor Y ⊗ − preserves weak equivalences between cofibrant objects.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 27
(2) For cofibrant X the functor [X, −] is right Quillen and for fibrant Y the
functor [−, Y ] preserves weak equivalences between cofibrant objects.
Proof. Certainly − ⊗ X is left Quillen just when [X, −] is right Quillen. Let
f : A → B be a weak equivalence between cofibrant objects. Then the natural
transformation of left Quillen functors − ⊗ f : − ⊗ A → − ⊗ B and the natural
transformation of right Quillen functors [f, −] : [B, −] → [A, −] are mates. By
Corollary 1.4.4(b) of [12] it follows that Y ⊗ f is a weak equivalence for all
cofibrant Y if and only if [f, Y ] is a weak equivalence for all fibrant Y .
Axiom MC. (C, ⊗, [−, −], I) satisfies either of the equivalent conditions of Proposition
5.9 and the unit I is cofibrant.
Proposition 5.10. Axiom MC implies Axioms M and C.
Proof. When Axiom MC holds Ken Brown’s lemma (1.1.12 of [12]) ensures that
for cofibrant X the functor X ⊗ − preserves weak equivalences between cofibrant
objects. So Axiom MC is a strengthening of Axiom M. That it implies Axiom
C is clear: if the right adjoint [X, −] preserves fibrations it preserves fibrant
objects.
For (C, ⊗, [−, −], I, ϕ) satisfying Axiom MC it follows that we can form the
left derived skew monoidal and right derived skew closed structures on Ho(C).
The following result establishes, as expected, that these form part of a monoidal
skew closed structure on Ho(C).
Theorem 5.11. Let (C, ⊗, [−, −], I, ϕ) be a monoidal skew closed category satisfying
Axiom MC. Then the left derived skew monoidal structure (Ho(C), ⊗l, I)
and the right derived skew closed structure (C, [−, −]r, I) together with the iso-
morphisms
ϕd
: Ho(C)(QA ⊗ QB, C) ∼= Ho(C)(A, [QB, RC])
form a monoidal skew closed structure on Ho(C).
The straightforward but long proof is deferred until the appendix. The following
result is the homotopical version of Eilenberg and Kelly’s theorem. Note that by
Proposition 5.7 conditions (1) and (2) amount to (C, [−, −], I) being homotopy
closed.
Theorem 5.12. Let (C, ⊗, [−, −], I) be a monoidal skew closed category satisfying
Axiom MC. Then (C, ⊗, I) is homotopy monoidal if and only if the following
three conditions are satisfied.
(1) For all cofibrant A and fibrant B the function v : C(A, B) → C(I, [A, B]) is
a bijection on homotopy classes of maps,
(2) For all fibrant A the map i : [I, A] → A is a weak equivalence,
(3) The transformation t : [A ⊗ B, C] → [A, [B, C]] is a weak equivalence whenever
A and B are cofibrant and C is fibrant.
Proof. Combining Theorems’ 5.11 and 2.6 we have that (Ho(C), ⊗l, I) is monoidal
just when (Ho(C), [−, −]r, I) is closed and the induced transformation td : [QA ⊗
QB, RC] → [QA, R[QB, RC]] is an isomorphism in Ho(C). By Proposition 5.7
28 JOHN BOURKE
closedness amounts to (1) and (2) above. From the proof of Theorem 5.11 the
transformation td is given by the composite
[Q(QA⊗QB),RC]
[p,1]−1
GG [QA⊗QB,RC]
t GG [QA,[QB,RC]]
[1,q]
GG [QA,R[QB,RC]]
and therefore is invertible just when the central component
tQA,QB,RC : [QA ⊗ QB, RC] → [QA, [QB, RC]]
is so for all A, B and C. For A, B cofibrant and C fibrant tQA,QB,RC is isomorphic
to tA,B,C : [A ⊗ B, C] → [A, [B, C]]. Since QA and QB are cofibrant and RC is
fibrant it follows that tQA,QB,RC is invertible for all A, B, C just when tA,B,C is
invertible for all cofibrant A,B and fibrant C.
Again we have a symmetric variant.
Theorem 5.13. Let (C, ⊗, [−, −], I) be a monoidal skew closed category satisfying
Axiom MC.
(1) If (C, [−, −], I) is homotopy closed and admits a natural symmetry isomorphism
s : [A, [B, C]] ∼= [B, [A, C]] satisfying S3 then (C, ⊗, I) is homotopy
monoidal.
(2) If, in addition to (1), (C, [−, −], I, s) is symmetric skew closed then (C, ⊗, I)
is homotopy symmetric monoidal.
Proof. By Proposition 5.8 if s satisfies S3 then so does sr with respect to
(Ho(C), [−, −]r, I). From Theorem 2.10 it follows that (Ho(C), ⊗l, I) is monoidal.
The second part follows again by application of Proposition 5.8 and 2.10.
6. Pseudo-commutative 2-monads and monoidal bicategories
In the category CMon of commutative monoids the set CMon(A, B) forms a
commutative monoid [A, B] with respect to the pointwise structure of B. This
is the internal hom of a symmetric monoidal closed structure on CMon whose
tensor product represents functions A × B → C that are homomorphisms in each
variable. From the monad-theoretic viewpoint the enabling property is that the
commutative monoid monad on Set is a commutative monad.
Extending this to dimension 2, Hyland and Power [13] introduced the notion of
a pseudo-commutative 2-monad T on Cat. Examples include the 2-monads for
categories with a class of limits, permutative categories, symmetric monoidal
categories and so on. For such T they showed that the 2-category of strict
algebras and pseudomorphisms admits the structure of a pseudo-closed 2-category
– a slight weakening of the notion of a closed category with a 2-categorical element.
Theorem 2 of ibid. described a bicategorical version of Eilenberg and Kelly’s
theorem, designed to produce a monoidal bicategory structure on T-Alg. However
they did not give the details of the proof, which involved lengthy calculations of
a bicategorical nature, and expressed their dissatisfaction with the argument.4
4 From [13]:“Naturally, we are unhappy with the proof we have just outlined. Since the data
we start from is in no way symmetric we expect some messy difficulties: but the calculations we
do not give are very tiresome, and it would be only too easy to have made a slip. Hence we
would like a more conceptual proof.”
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 29
In this section we take a slightly different route to the monoidal bicategory
structure on T-Alg. We begin by making minor modifications to Hyland and
Power’s construction to produce a skew closed structure on the 2-category T-Algs
of algebras and strict morphisms. This is simply the restriction of the pseudoclosed
2-category structure on T-Alg. We then obtain a monoidal skew closed
structure on T-Algs and, using Theorem 5.12, establish that it is homotopy
monoidal. The monoidal bicategory structure on T-Alg is obtained by transport
of structure from the full sub 2-category of T-Algs containing the cofibrant
objects.
6.1. Background on commutative monads. If V is a symmetric monoidal closed
category and T an endofunctor of V then enrichments of T to a V -functor
correspond to giving a strength: that is, a natural transformation tA,B : A⊗TB →
T(A⊗B) subject to associativity and identity conditions. One obtains a costrength
t∗
A,B : TA ⊗ B → T(A ⊗ B) related to the strength by means of the symmetry
isomorphism cA,B : A ⊗ B → B ⊗ A.
If (T, η, µ) is a V -enriched monad then we can consider the following diagram
(6.1) TA ⊗ TB T(TA ⊗ B) T2(A ⊗ B)
T(A ⊗ TB) T2(A ⊗ B) T(A ⊗ B)
t GG Tt GG
µ

t

Tt GG µ
GG
and if this commutes for all A and B then T is said to be a commutative monad
[17].
Now if T is commutative and V sufficiently complete and cocomplete then the
category of algebras V T is itself symmetric monoidal closed [18, 15]. Both tensor
product and internal hom represent T-bilinear maps – this perspective was
explored in [19] and more recently in [32]. More generally, a T-multilinear map
consists of a morphism f : A1 ⊗ . . . ⊗ An → B which is a T-algebra map in each
variable. This means that the diagram
A1 ⊗ . . . ⊗ TAi ⊗ . . . ⊗ An T(A1 ⊗ . . . ⊗ An) TB
A1 ⊗ . . . ⊗ Ai ⊗ . . . ⊗ An B
t GG Tf
GG
b

1⊗...⊗ai⊗...⊗1
 f
GG
is commutative for each i where the top row t : A1 ⊗ . . . ⊗ TAi ⊗ . . . ⊗ An →
T(A1 ⊗ . . . ⊗ An) is the unique map constructible from the strengths and
costrengths. T-multilinear maps form the morphisms of a multicategory of
T-algebras. Surprisingly, the multicategory perspective appears to have first
been explored in the more general 2-categorical setting of [13].
6.2. Background on 2-monads. The category of small categories Cat is cartesian
closed and hence provides a basis suitable for enriched category theory. In
particular one has the notions of Cat-enriched category – hence 2-category – and
30 JOHN BOURKE
of Cat-enriched monads – hence 2-monads. The appendage “2-” will always refer
to strict Cat-enriched concepts.
Given a 2-monad T = (T, η, µ) on a 2-category C one has the Eilenberg-Moore 2category
T-Algs of algebras. In T-Algs everything is completely strict. There are
the usual strict algebras A = (A, a) satisfying a◦Ta = a◦µA and a◦ηA = 1. The
strict morphisms f : A → B of T-Algs satisfy the usual equation b ◦ Tf = f ◦ a
on the nose, whilst the 2-cells α : f ⇒ g ∈ T-Algs(A, B) satisfy b ◦ Tα = α ◦ a.
T-Algs is just as well behaved as its Set-enriched counterpart. Important facts
for us are the following ones.
• The usual (free, forgetful)-adjunction lifts to a 2-adjunction F U where
U : T-Algs → C is the evident forgetful 2-functor.
• Suppose that C is a locally presentable 2-category: one, like Cat, that is
cocomplete in the sense of enriched category theory [16] and whose underlying
category is locally presentable [1]. If T is accessible – preserves λ-filtered
colimits for some regular cardinal λ – then T-Algs is also locally presentable.
There are accessible 2-monads T on Cat whose strict algebras are categories with
D-limits, permutative categories, symmetric monoidal categories and so on. In
particular the examples of skew closed 2-categories from Section 4.1 reside on
2-categories of the form T-Algs for T an accessible 2-monad on Cat.
So far we have discussed strict aspects of two-dimensional monad theory. Though
there are several possibilities, the only weak structures of interest here are
pseudomorphisms of strict T-algebras. A pseudomorphism f : A B consists
of a morphism f : A → B and invertible 2-cell f : b ◦ Tf ∼= f ◦ a satisfying two
coherence conditions [3]. These are the morphisms of the 2-category T-Alg into
which T-Algs includes via an identity on objects 2-functor ι : T-Algs → T-Alg.
The inclusion commutes with the forgetful 2-functors
(6.2)
T-Algs
U
44
ι GG T-Alg
U
||
C
to the base. Pseudomorphisms of T-algebras capture functors preserving categorical
structure up to isomorphism. For example, in the case that T is the
2-monad for categories with D-limits or permutative categories we obtain the
2-categories D-Lim and Perm as T-Alg.
An important tool in the study of pseudomorphisms are pseudomorphism classifiers.
If T is a reasonable 2-monad – for instance, an accessible 2-monad on Cat –
then by Theorem 3.3 of [3] the inclusion ι : T-Algs → T-Alg has a left 2-adjoint
Q. We call QA the pseudomorphism classifier of A since each pseudomorphism
f : A B factors uniquely through the unit qA : A QA as a strict morphism
QA → B. The counit pA : QA → A is a strict map with homotopy theoretic
content – see Section 6.4.1 below.
6.3. From pseudo-commutative 2-monads to monoidal skew closed 2-categories.
Given a 2-monad T on Cat we have, in particular, the corresponding strengths
t : T(A × B) → A × TB and costrengths T(A × B) → TA × B and can enquire
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 31
as to whether T is commutative. For those structures – such as categories with
finite products or symmetric monoidal categories – that involve an aspect of
weakness in their definitions the relevant diagram (6.1) rarely commutes on the
nose, but often commutes up to natural isomorphism. This leads to the notion of
a pseudo-commutative 2-monad T which is a 2-monad T equipped with invertible
2-cells
TA × TB T(TA × B) T2(A × B)
T(A × TB) T2(A × B) T(A × B)
t GG Tt GG
µ

t

Tt
GG
µ
GG
αA,B

subject to axioms (see Definition 5 of [13]) asserting the equality of composite
2-cells built from the above ones. If α commutes with the symmetry isomorphism
– in the sense that αB,A = TcA,B ◦αA,B ◦cTB,TA – then T is said to be a symmetric
pseudo-commutative 2-monad.
The 2-monad for categories with D-limits is symmetric pseudo-commutative [28]
as are the 2-monads for permutative and symmetric monoidal categories [13].
An example of a pseudo-commutative 2-monad which is not symmetric is the
2-monad for braided strict monoidal categories [5].
6.3.1. The 2-multicategory of algebras. For T pseudo-commutative one can define
T-multilinear maps. A T-multilinear map f : (A1, . . . , An) → B consists of a
functor f : A1 × . . . An → B together with a family of invertible 2-cells fi:
A1 × . . . × TAi × . . . × An T(A1 × . . . × An) TB
A1 × . . . × Ai × . . . × An B
t GG Tf
GG
c

1×...×ai×...×1
 f
GG
fi

satisfying indexed versions of the pseudomorphism equations, and a compatibility
condition involving the pseudo-commutativity. A nullary map (−) → B is defined
to be an object of the category B.
There are transformations of multilinear maps and these are the morphisms of a
category T-Alg(A1, A2 . . . An; B). Proposition 18 of ibid. shows that these are
the hom-categories of a 2-multicategory of T-algebras T-Alg and that, moreover,
if T is symmetric pseudo-commutative then T-Alg is a symmetric 2-multicategory.
T-Alg is itself recovered as the 2-category of unary maps.
Of course we can speak of multimaps (A1, . . . , An) → B which are strict in Ai:
those for which the natural transformation fi depicted above is an identity. Note
that this agrees with the formulation given in Definition 3.4.
Theorem 6.1 (Hyland-Power [13]). The 2-multicategory T-Alg is closed. Moreover
a multimap (A1, A2 . . . An, B) → C is strict in Ai just when the corresponding
map (A1, A2 . . . An) → [B, C] is so.
32 JOHN BOURKE
6.3.2. The skew closed structure. By definition T-Alg(−; A) = A. Since we have
a natural isomorphism T-Algs(F1, A) ∼= Cat(1, A) ∼= A and a suitable closed 2multicategory
T-Alg we can apply Theorem 3.6 to obtain a skew closed structure
on T-Algs.
Theorem 6.2. Let T be a pseudo-commutative 2-monad on Cat.
(1) Then (T-Alg, [−, −], L) is a semi-closed 2-category. Moreover [−, −] and L
restrict to T-Algs where they form part of a skew closed 2-category
(T-Algs, [−, −], F1, L, i, j).
(2) If T is symmetric then (T-Alg, [−, −], L) is symmetric semi-closed and
(T-Algs, [−, −], F1, L, i, j) is symmetric skew closed.
The skew closed 2-category (T-Algs, [−, −], F1, L, i, j) has components as
constructed in Section 3. Let us record, for later use, some further information
about these components.
(1) The underlying category of [A, B] is just T-Alg(A, B). More generally
U ◦ [−, −] = T-Alg(−, −) : T-Algop
× T-Alg → Cat.
(2) The underlying functor of L : [A, B] → [[C, A], [C, B]] is given by [C, −]A,B :
T-Alg(A, B) → T-Alg([C, A], [C, B]).
(3) The underlying functor of i : [F1, A] → A is the composite
T-Alg(F1, A)
UF1,A
GG Cat(T1, A)
Cat(η1,A)
GG Cat(1, A)
ev•
GG A
whose last component is the evaluation isomorphism.
(4) j : F1 → [A, A] is the transpose of the functor ˆ1 : 1 → T-Alg(A, A) selecting
the identity on A.
(1) follows from the construction of the hom algebra [A, B] in [13] as a 2-categorical
limit in T-Alg created by U : T-Alg → Cat.5
Theorem 11 of ibid. gives a full description
of the isomorphisms T-Alg(A1, . . . , An, B; C) ∼= T-Alg(A1 . . . An; [B, C]).
From this, it follows that the evaluation multimap ev : ([A, B], A) → B has
underlying functor T-Alg(A, B) × A → B acting by application, which is what is
required for (3). (2) follows from the analysis, given in Proposition 21 of ibid, of
how the same adjointness isomorphisms behave with respect to underlying maps.
(4) is by definition.
6.3.3. The monoidal skew closed structure on T-Algs. We now describe left 2adjoints
to the 2-functors [A, −] : T-Algs → T-Algs. For this let us further
suppose that T is an accessible 2-monad. We must show that each B admits a
reflection B ⊗ A along [A, −]. Since T-Algs is cocomplete the class of algebras
admitting such a reflection is closed under colimits; because each algebra is a
coequaliser of frees it therefore suffices to show that each free algebra admits a
5 This construction is accomplished in three stages by firstly forming an iso-inserter and
then a pair of equifiers and amounts to the construction of a descent object.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 33
reflection. With this is mind observe that the triangle
(6.3)
T-Algs
T-Alg(A,ι−) 66
[A,−]
GG T-Algs
Uzz
Cat
commutes. Because T is accessible we have the 2-adjunction Q ι and corresponding
isomorphism T-Algs(QA, −) ∼= T-Alg(A, ι−). Now the representable
T-Algs(QA, −) has a left adjoint −.QA given by taking copowers. It follows that
at C ∈ Cat the reflection FC ⊗ A is given by C.QA. We conclude:
Proposition 6.3. If T is an accessible pseudo-commutative 2-monad on Cat
then each [A, −] : T-Algs → T-Algs has a left 2-adjoint − ⊗ A. In particular
(T-Algs, ⊗, [−, −], F1) is a monoidal skew closed 2-category.
6.4. From T-Algs as a monoidal skew closed 2-category to T-Alg as a monoidal
bicategory. Our next goal is to show that (T-Algs, ⊗, [−, −], F1) is homotopy
monoidal. In order to do so requires understanding the Quillen model structure
on T-Algs and its relationship with pseudomorphisms. We summarise the key
points below and refer to the original source [22] for further details.
6.4.1. Homotopy theoretic aspects of 2-monads. Thought of as a mere category,
Cat admits a Quillen model structure in which the weak equivalences are the
equivalences of categories. The cofibrations are the injective on objects functors
and the fibrations are the isofibrations: functors with the isomorphism lifting
property. It follows that all objects are cofibrant and fibrant.
Equipped with the cartesian closed structure, Cat is a monoidal model category
[12]. Therefore one can speak of model 2-categories, of which Cat is the leading
example. It was shown in Theorem 4.5 of [22] that for an accessible 2-monad T
on Cat the model structure lifts along U : T-Algs → Cat to a model 2-category
structure on T-Algs: a morphism of T-Algs is a weak equivalence or fibration
just when its image under U is one. It follows immediately that the adjunction
F U : T-Algs Cat is a Quillen adjunction.
Since F preserves cofibrations each free algebra is cofibrant. In fact, the cofibrant
objects are the flexible algebras of [3] and were studied long before the connection
with model categories was made in [22]. Another source of cofibrant algebras
comes from pseudomorphism classifiers: each QA is cofibrant. In fact the counit
pA : QA → A of the adjunction Q ι : T-Algs T-Alg is a trivial fibration in
T-Algs; thus QA is a cofibrant replacement of A.
Theorem 4.7 of [3] ensures that if A is flexible then, for all B, the fully faithful
inclusion
ιA,B : T-Algs(A, B) → T-Alg(A, B)
is essentially surjective on objects: that is, an equivalence of categories. This
important fact can also be deduced from the model 2-category structure: the
inclusion ιA,B is isomorphic to T-Algs(pA, B) : T-Algs(A, B) → T-Algs(QA, B)
which is an equivalence since pA : QA → A is a weak equivalence of cofibrant
objects.
34 JOHN BOURKE
Finally we note that a parallel pair of algebra morphisms f, g : A B are right
homotopic just when they are isomorphic in T-Algs(A, B). This follows from
the fact that for each algebra B the power algebra [I, B] is a path object where
I is the walking isomorphism. In particular, if A is cofibrant then f and g are
homotopic just when they are isomorphic.
6.4.2. Homotopical behaviour of the skew structure.
Theorem 6.4. Let T be an accessible pseudo-commutative 2-monad on Cat.
(1) Then the monoidal skew closed 2-category (T-Algs, ⊗, [−, −], F1) satisfies
Axiom MC and (T-Algs, ⊗, F1) is homotopy monoidal.
(2) If T is symmetric then (T-Algs, ⊗, F1) is homotopy symmetric monoidal.
Proof. We observed above that each free algebra is cofibrant. Therefore the unit
F1 is cofibrant. We now verify Axiom MC in its closed form: in fact we establish
the stronger result that for all A the 2-functor [A, −] is right Quillen and that
[−, A] preserves all weak equivalences. For the first part consider the equality
U ◦ [A, −] = T-Algs(A, ι−) of (6.3). Since U reflects weak equivalences and
fibrations it suffices to show that T-Alg(A, ι−) : T-Algs → Cat is right Quillen.
Now T-Alg(A, ι−) ∼= T-Algs(QA, −) : T-Algs → Cat and this last 2-functor is
right Quillen since QA is cofibrant and T-Algs a model 2-category.
For the second part we use the commutativity U ◦[−, A] ∼= T-Alg(ι−, A). Arguing
as before it suffices to show that T-Alg(ι−, A) : T-Algs → Cat preserves all weak
equivalences or, equally, that the isomorphic T-Algs(Qι−, A) does so. Now if
f : B → C is a weak equivalence then Qιf is a weak equivalence of cofibrant
objects. As A, like all objects, is fibrant and T-Algs a model 2-category the
functor T-Algs(Qιf, A) is an equivalence.
We now apply Theorem 5.12 to establish that T-Algs is homotopy monoidal.
To verify the three conditions requires only the information on the underlying
functors of [−, −], L and i given in Section 6.3.2. Firstly we must show that the
underlying function of
vA,B : T-Algs(A, B) → T-Algs(F1, [A, B])
induces a bijection on homotopy classes of maps for cofibrant A. Since morphisms
with cofibrant domain are homotopic just when isomorphic it will suffice to show
that vA,B is an equivalence of categories. To this end consider the composite:
T-Algs(A, B) T-Algs(F1, [A, B]) Cat(1, T-Alg(A, B)) T-Alg(A, B)
vA,B
GG ϕ
GG ev•
GG
in which ϕ is the adjointness isomorphism – recall that U ◦ [−, −] = T-Alg(−, −)
– and in which ev• is the evaluation isomorphism. It suffices to show that the
composite is an equivalence. vA,B sends f : A → B to [A, f] ◦ j : F1 → [A, A] →
[A, B], whose image under ϕ is the functor T-Alg(A, f)◦ˆ1A : 1 → T-Alg(A, A) →
T-Alg(A, B). Evaluating at • thus returns f viewed as a pseudomap. The
action on 2-cells is similar and we conclude that the composite is the inclusion
ιA,B : T-Algs(A, B) → T-Alg(A, B). As per Section 6.4.1 this is an equivalence
since A is cofibrant.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 35
Secondly we show that
iA : [F1, A] → A
is a weak equivalence for all A: that its underlying functor iA : T-Alg(F1, A) → A
is an equivalence of categories. Since F1 is cofibrant this is equally to show that
the composite
iA ◦ ιF1,A : T-Algs(F1, A) → T-Alg(F1, A) → A
is an equivalence. An easy calculation shows that this is equally the composite
ev• ◦ ϕ : T-Algs(F1, A) → Cat(1, A) → A of the canonical adjunction and
evaluation isomorphisms. Hence iA is an equivalence for all A.
Let u : A → [B, A ⊗ B] denote the unit of the adjunction − ⊗ B [B, −]. We are
to show that the morphism tA,B,C given by the composite
[u, 1] ◦ L : [A ⊗ B, C] → [[B, A ⊗ B], [B, C]] → [A, [B, C]]
is a weak equivalence for cofibrant A and B. Now the underlying functor of this
composite is just the top row below.
T-Alg(A ⊗ B, C)
[B,−]
GG T-Alg([B, A ⊗ B], [B, C])
T-Alg(u,1)
GG T-Alg(A, [B, C])
T-Algs(A ⊗ B, C)
ι
yy
[B,−]
GG T-Algs([B, A ⊗ B], [B, C])
ι
yy
T-Algs(u,1)
GG T-Algs(A, [B, C])
ι
yy
In this diagram the left square commutes since [B, −] restricts from T-Alg to
T-Algs and the right square since u is a strict algebra map. The outer vertical
arrows are equivalences since both A ⊗ B and A are cofibrant: the former using
Axiom MC and the latter by assumption. The bottom row is the adjointness
isomorphism so that the top row is an equivalence by two from three.
Finally if T is symmetric then, by Theorem 6.2, the skew closed 2-category
(T-Algs, [−, −], F1) is symmetric skew closed. It now follows from Theorem 5.13
that (T-Algs, ⊗, F1) is homotopy symmetric monoidal.
6.4.3. The monoidal bicategory T-Alg. A monoidal bicategory is a bicategory C
equipped with a tensor product C × C C and unit I together with equivalences
α : (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C), l : I ⊗ A → A and r : A → A ⊗ I pseudonatural
in each variable, and satisfying higher dimensional variants of the axioms for a
monoidal category [10]. Note that here we mean equivalences in the 2-categorical
or bicategorical sense, as opposed to weak equivalences.
In particular, each skew monoidal 2-category in which the components α, l
and r are equivalences provides an example of a monoidal bicategory. The
skew monoidal 2-category (T-Algs, ⊗, F1) is not itself a monoidal bicategory.6
However (T-Algs, ⊗, F1) satisfies Axiom MC and hence, by Proposition 5.10, it
satisfies Axiom M. Therefore the skew monoidal structure restricts to the full
sub 2-category (T-Algs)c of cofibrant objects. Since (T-Algs, ⊗, F1) is homotopy
monoidal each component α, l and r is a weak equivalence of cofibrant objects.
6 In fact l : F1 ⊗ A → A is an equivalence just when A is equivalent to a flexible algebra.
Such algebras are called semiflexible [3].
36 JOHN BOURKE
Since all objects are fibrant such weak equivalences are homotopy equivalences:
thus equivalences in the 2-categorical sense. We conclude:
Proposition 6.5. The skew monoidal structure (T-Algs, ⊗, F1) restricts to a skew
monoidal 2-category ((T-Algs)c, ⊗, F1) which is a monoidal bicategory.
In fact (T-Algs)c is biequivalent to the 2-category T-Alg of algebras and
pseudomorphisms.
Lemma 6.6. The 2-adjunction Q ι : T-Algs T-Alg restricts to a 2-adjunction
Q ι : (T-Algs)c T-Alg whose unit and counit are pointwise equivalences. In
particular, the composite inclusion ι : (T-Algs)c → T-Alg is a biequivalence.
Proof. Because each QA is cofibrant/flexible the adjunction restricts. The unit
q : A QA is an equivalence by Theorem 4.2 of [3]. Since A is flexible the
counit pA : QA → A is an equivalence in T-Algs by Theorem 4.4 of ibid.
Just as monoidal structure can be transported along an adjoint equivalence of
categories, so the structure of a monoidal bicategory may be transported along
an adjoint biequivalence. And we obtain the following result: see Theorem 14 of
[13]. The present argument has the advantage of dealing solely with the strict
concepts of Cat-enriched category theory until the last possible moment.
Theorem 6.7. For T an accessible pseudo-commutative 2-monad on Cat the
2-category T-Alg admits the structure of a monoidal bicategory.
7. Bicategories
We now return to the skew closed category (Bicats, Hom, F1) of Section 4.2
and show that it forms part of a monoidal skew closed category that is homotopy
symmetric monoidal. A similar, but simpler, analysis yields the corresponding
result for the skew structure on 2-Cats discussed in Section 4.2 – this is omitted.
7.1. Preliminaries on Bicats. To begin with, it will be helpful to discuss some
generalities concerning homomorphism classifiers and the algebraic nature of
Bicats.
To this end, let us recall that the category Cat-Gph of Cat-enriched graphs
is naturally a 2-category – called CG in [23]. CG is locally presentable as a
2-category: that is, cocomplete as a 2-category and its underlying category
Cat-Gph is locally presentable. Section 4 of ibid. describes a filtered colimit
preserving 2-monad T on CG whose strict algebras are the bicategories, and
whose strict morphisms and pseudomorphisms are the strict homomorphisms
and homomorphisms respectively. The algebra 2-cells are called icons [24].
We write Icons and Iconp for the corresponding extensions of Bicats and Bicat
to 2-categories with icons as 2-cells. It follows from [3] that the inclusion
ι : Icons → Iconp has a left 2-adjoint Q: this assigns to a bicategory A its
homomorphism classifier QA.
As mentioned Cat-Gph is locally presentable. Since T preserves filtered colimits
it follows that the category of algebras Bicats is locally presentable too, and that
the forgetful right adjoint U : Bicats → Cat-Gph preserve limits and filtered
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 37
colimits. Now the three functors from Cat-Gph to Set sending a Cat-graph to its
set of (0/1/2)-cells respectively are represented by finitely presentable Cat-graphs.
It follows that the composite of each of these with U – the functors
(−)0, (−)1, (−)2 : Bicats → Set
sending a bicategory to its set of (0/1/2)-cells – preserves limits and filtered
colimits. Now a functor between locally presentable categories has a left adjoint
just when it preserves limits and is accessible: preserves λ-filtered colimits for
some regular cardinal λ. See, for instance, Theorem 1.66 of [1]. It follows that
each of the above three functors has a left adjoint – we used the adjoint F to
(−)0 to construct the unit F1 in Section 4.2.
7.2. The monoidal skew closed structure on Bicats. Our goal now is to show that
Hom(A, −) : Bicats → Bicats has a left adjoint for each A. We will establish this
by showing that Hom(A, −) preserves limits and is accessible. As pointed out
above, the functors (−)0, (−)1, (−)2 : Bicats → Set preserve limits and filtered
colimits. Since they jointly reflect isomorphisms they also jointly reflect limits and
filtered colimits. Accordingly it will be enough to show that the three functors
Hom(A, −)0, Hom(A, −)1, Hom(A, −)2 : Bicats → Set
preserve limits and are accessible. We argue case by case.
(1) Hom(A, B)0 is the set of homomorphisms from A to B. Hence Hom(A, −)0
is naturally isomorphic to Bicats(QA, −) where Q is the homomorphism
classifier. Like any representable functor Hom(A, −)0 preserves limits and
is accessible.
(2) Hom(A, B)1 is the set of pseudonatural transformations between homomorphisms.
Let Cyl(B) denote the following bicategory – first constructed, in
the lax case, in [2] . The objects of Cyl(B) are the morphisms of B whilst
morphisms (r, s, θ) : f → g are diagrams as below left
(7.1) a b
c d
r GG
f

s
GG
g

θ CQ
a b
c d
r GG
f
((
f
ÒÒ
s
GG
g
ÒÒ
α CQ θ CQ =
a b
c d
r GG
f
((
s
GG
g
((
g
ÒÒ
θ CQ β
CQ
in which θ is invertible. 2-cells of Cyl(B) consist of pairs of 2-cells (α, β)
satisfying the equality displayed above right. Note that here are strict projection
homomorphisms d, c : Cyl(B) B which, on objects, respectively
select the domain and codomain of an arrow.
It is straightforward to see that we have a natural isomorphism of functors
Hom(A, −)1
∼= Bicat(A, Cyl(−)). Combining this with Bicat(A, −) ∼=
Bicats(QA, −) gives an isomorphism Hom(A, −)1
∼= Bicats(QA, Cyl(−)).
Since this is the composite Bicats(QA, −) ◦ Cyl(−) : Bicats → Bicats → Set
whose second component is representable, it will suffice to show that Cyl(−)
preserves limits and is accessible.
For this, arguing as before, it is enough to show that each of Cyl(−)0,
Cyl(−)1 and Cyl(−)2 preserves limits and is accessible. Certainly we have
38 JOHN BOURKE
Cyl(B)0
∼= B1 naturally in B and (−)1 preserves limits and filtered colimits.
We construct Cyl(−)1 as a finite limit in four stages. These stages
correspond to the sets constructed below:
Opp(B2) = {(x, y) : x, y ∈ B2, sx = ty, tx = sy}
Comp(B1) = {(a, b) : a, b ∈ B1, ta = sb}
Iso(B2) = {(x, y) ∈ Opp(B2) : y ◦ x = ity, x ◦ y = itx}
Cyl(B)1 = {(a, b, c, d, x, y) : (x, y) ∈ Iso(B2),
(a, b), (c, d) ∈ Comp(B1), sx = b ◦ a, tx = d ◦ c}
the first three of which, in turn, define the sets of pairs of 2-cells pointing
in the opposite direction, of composable pairs of 1-cells, and of invertible
2-cells. Each stage corresponds to the finite limit in CAT(Bicats, Set) below.
Opp(B2) GG (B2)2
(sx,tx)
DD
(ty,sy)
PP (B1)2
Comp(B1) GG

B1
s

Iso(B2) GG

Opp(B2)
(y◦x,x◦y)

Cyl(B)1
GG

Iso(B2)
(sx,tx)

B1
t GG B0 (B1)2 i2
GG (B2)2 Comp(B1)2
(b◦a,d◦c)
GG (B1)2
Now each of the functors (−)0,(−)1 and (−)2 preserve finite limits and
filtered colimits. Since finite limits commute with limits and filtered colimits
in Set it follows that each constructed functor, and in particular, Cyl(−)1
preserves limits and filtered colimits. Another pullback followed by an
equaliser constructs Cyl(−)2 and shows it to have the same preservation
properties: we leave this case to the reader.
(3) Finally observe that we can express Hom(A, B)2 as the equaliser of the two
functions
Hom(A, Cyl(B))1 Bicat(A, B)2
sending an element α : f ⇒ g of Hom(A, Cyl(B))1 to the pair (df, cf) and
(dg, cg) respectively. This is natural in B. Therefore Hom(A, −)2 is a finite
limit of functors, each of which has already been shown to preserve limits
and be accessible. Since finite limits in Set commute with limits and with
λ-filtered colimits for each regular cardinal λ it follows that Hom(A, −)2
preserves limits and is itself accessible.
We conclude:
Proposition 7.1. For each bicategory A the functor Hom(A, −) : Bicats → Bicats
has a left adjoint − ⊗ A. In particular we obtain a monoidal skew closed category
(Bicats, ⊗, Hom, F1).
7.3. Homotopical behaviour of the skew structure. We turn to the homotopical
aspects of the skew structure.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 39
7.3.1. The model structure on Bicats. A 1-cell f : X → Y in a bicategory A is
said to be an equivalence if there exists g : Y → X and isomorphisms 1X
∼= gf
and 1Y
∼= fg. Now a homomorphism of bicategories F : A B is said to be a
biequivalence if it is essentially surjective up to equivalence (given Y ∈ B there
exists X ∈ A and an equivalence Y → FX) and locally an equivalence: each
functor FX,Y : A(X, Y ) → B(FX, FY ) is an equivalence of categories.
The relevant model structure on Bicats was constructed in [21]. The weak
equivalences are those strict homomorphisms that are biequivalences. A strict
homomorphism F : A → B is said to be a fibration if it has the following two
properties (1) if f : Y → FX is an equivalence then there exists an equivalence
f : Y → X with Ff = f and (2) each FX,Y : A(X, Y ) → B(FX, FY ) is an
isofibration of categories. We note that all objects are fibrant.
The only knowledge that we require of the cofibrant objects is that each homomorphism
classifier QA is cofibrant. To see this observe that if f : A → B is a trivial
fibration then there exists a homomorphism g : B A with f ◦ g = 1. Since the
inclusion ι : Bicats → Bicat sends each trivial fibration to a split epimorphism,
and since split epis can be lifted through any object, an adjointness argument
applied to Q ι shows that each QA is cofibrant. By Theorem 4.2 of [3] the
counit pA : QA → A is a surjective equivalence – equivalence plus split epi – in
the 2-category Iconp. Therefore pA is a trivial fibration and so exhibits QA as a
cofibrant replacement of A.
The right homotopy relation on Bicats(A, B) is equivalence in the bicategory
Hom(A, B). Where needed, we will use the term pseudonatural equivalence for
clarity. We note that a morphism η : F → G ∈ Hom(A, B) is an equivalence just
when each component ηX : FX → GX is an equivalence in B. That pseudonatural
equivalence coincides with right homotopy follows from the fact, used in ibid.,
that the full sub-bicategory PB of Cyl(B), with objects the equivalences, is a
path object for B. In particular, if A is a cofibrant bicategory then F, G : A B
are homotopic just when they are equivalent in Hom(A, B).
7.3.2. Homotopy monoidal structure. Finally, we are in a position to prove the
main theorem of this section.
Theorem 7.2. The monoidal skew closed structure (Bicats, ⊗, Hom, F1) satisfies
Axiom MC and is homotopy symmetric monoidal.
Proof. Firstly we show that the unit F1 is cofibrant. Recall that F is left adjoint
to (−)0 : Bicats → Set. Since (−)0 sends trivial fibrations to surjective functions,
and since surjective functions can be lifted through 1, it follows by adjointness
that F1 is cofibrant.
In order to verify the remainder of Axiom MC we use the well known fact, see
for example [34], that a homomorphism F : A B is a biequivalence if and
only if there exists G : B A and equivalences 1A → GF and 1B → FG. A
consequence is that if F : A → B is a biequivalence then so is Hom(C, F) and
Hom(F, D) for all C and D.
To verify Axiom MC, it remains to show that if C is cofibrant and F a fibration,
then Hom(C, F) is a fibration: in fact, we will show that this is true for all C.
To see that Hom(C, F) : Hom(C, A) → Hom(C, B) is locally an isofibration,
40 JOHN BOURKE
consider α : G → H ∈ Hom(C, A) and θ : β ∼= Fα. Then each component θX is
invertible in B and so lifts along F as depicted below.
GX HX
βX
99
αX
UU
θX

1 F GG FGX FHX
βX
99
FαX
UUθX

The components βX : GX → HX admit a unique extension to a pseudonatural
transformation β such that θ : β → α is a modification: at f : X → Y the
2-cell β f is given by:
Hf ◦ βX Hf ◦ αX αY ◦ Gf β∗
Y ◦ Gf
Hf◦θX
CQ
αf
CQ
(θY )−1◦Gf
CQ
Then Fβ = β and we conclude that Hom(C, F) is locally an isofibration.
It remains to show that Hom(C, F) has the equivalence lifting property. So
consider G : C → A and an equivalence α : H → FG ∈ Hom(C, B): a
pseudonatural transformation with each component αX : HX → FGX an
equivalence in B. Since F is a fibration there exists an equivalence βX : H X →
GX ∈ A with FβX = αX. Each such equivalence forms part of an adjoint
equivalence (ηx, βX ρx, x) and at f : X → Y we define H (f) : H X → H Y
as the conjugate
H X
βX
GG GX
Gf
GG GY
ρY
GG H Y
in which we take, as a matter of convention, this to mean (ρY ◦ Gf) ◦ βX. With
the evident extension to 2-cells H becomes a homomorphism. Moreover the
morphisms βX naturally extend to an equivalence β : H → G ∈ Hom(C, A).
Although FH X = HX for all X it is not necessarily the case that Hf = FH f.
Rather, we only have invertible 2-cells ϕf : Hf ∼= FH f corresponding to the
pasting diagram below.
HX
HY
FGX
FGY HY
Hf
SS αY
AA
FGf
SS
αX AA
FρY
GG
1Y
66
(αf )−1

FηY
Indeed ϕ : H ∼= FH is an invertible icon in the sense of [24]. Since F is locally
an isofibration these lift to invertible 2-cells ϕ (f) : H (f) ∼= H f. Moreover
H becomes a homomorphism, unique such that the above 2-cells yield an
invertible icon ϕ : H ∼= H . Composing ϕ : H ∼= H and β : H → G gives
the sought after lifted equivalence. This completes the verification of Axiom MC.
From Section 4.2 we know that (Bicats, Hom, F1) forms a symmetric skew closed
category. According to Theorem 5.13 the skew monoidal (Bicats, ⊗, F1) will form
part of a homotopy symmetric monoidal category so long as (Bicats, Hom, F1)
is homotopy closed.
Firstly we show that i = ev• : Hom(F1, A) → A is a biequivalence for each A. As
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 41
pointed out in Section 4.2 F1 has a single object and each parallel pair of 1-cells
is connected by a unique invertible 2-cell. Therefore the map ! : F1 → 1 is a
biequivalence. Hence Hom(!, A) : Hom(1, A) → Hom(F1, A) is a biequivalence
whereby it suffices to show that the composite ev• ◦ Hom(!, A) is a biequivalence.
This is just ev• : Hom(1, A) → A. It is straightforward, albeit tedious, to verify
that this last map is a biequivalence directly. For a quick proof we can use the
fact that for each bicategory A there is a strict 2-category st(A) and biequivalence
p : A st(A). Since evaluation is natural in all homomorphisms the square
below left commutes
Hom(1, A)
ev•

Hom(1,p)
GG Hom(1, st(A))
ev•

Ps(1, st(A))
ιoo
ev•
vv
A p
GG st(A)
and since both horizontal arrows are biequivalences it suffices to show that
ev• : Hom(1, st(A)) → st(A) is a biequivalence. Now let Ps(1, st(A)) →
Hom(1, st(A)) be the full sub 2-category containing the 2-functors. It is easy to
see that ι is essentially surjective up to equivalence – 1 is a cofibrant 2-category!
– and hence a biequivalence. Therefore we need only show that the composite
ev• : Ps(1, st(A)) → st(A) is a biequivalence. It is an isomorphism.
Finally we show that the function v : Bicats(A, B) → Bicats(F1, Hom(A, B))
given by the composite
Bicats(A, B) Bicats(Hom(A, A), Hom(A, B)) Bicats(F1, Hom(A, B))
Hom(A,−)
GG
Bicats(j,1)
GG
is a bijection on homotopy classes of maps for each cofibrant A. Firstly consider
the strict homomorphism
Hom(A, B)
L GG Hom(Hom(A, A), Hom(A, B))
Hom(j,1)
GG Hom(F1, Hom(A, B))
of bicategories. By (C2) it composes with i : Hom(F1, Hom(A, B)) → Hom(A, B)
to give the identity. Since this last map is a biequivalence so too is Hom(j, 1) ◦ L
by two from three. It follows that its underlying function Bicat(j, 1)◦Hom(A, −) :
Bicat(A, B) → Bicat(F1, Hom(A, B)) induces a bijection on equivalence classes
of objects: pseudonatural equivalence classes of homomorphisms.
Now we have a commutative diagram
Bicats(A, B)
ι

Bicats(j,1)◦Hom(A,−)
GG Bicats(F1, Hom(A, B))
ι

Bicat(A, B)
Bicat(j,1)◦Hom(A,−)
GG Bicat(F1, Hom(A, B))
in which the vertical functions are the inclusions. Each of the four functions is
well defined on pseudonatural equivalence classes: it follows, by two from three,
that the top function will determine a bijection on pseudonatural equivalence
classes if the two vertical inclusions do so. More generally, if X is a cofibrant
bicategory the inclusion ιX,Y : Bicats(X, Y ) → Bicat(X, Y ) induces a bijection
42 JOHN BOURKE
on pseudonatural equivalence classes. For we can identify this inclusion, up to
isomorphism, with the function
Bicats(pX, 1) : Bicats(X, Y ) → Bicats(QX, Y )
where pX : QX → X is the counit of the adjunction Q ι. Since pX : QX → X
exhibits QX as a cofibrant replacement of X, and so is a weak equivalence
between cofibrant objects, it follows – see, for instance, Proposition 1.2.5 of [12] –
that Bicats(pX, 1) induces a bijection on homotopy classes, that is, pseudonatural
equivalence classes, of morphisms.
From Section 4.2 we know that (Bicats, Hom, F1) forms a symmetric skew closed
category. Since it is homotopy closed we conclude from Theorem 5.13 that the
skew monoidal (Bicats, ⊗, F1) is homotopy symmetric monoidal.
8. Appendix
8.1. Proof of Theorem 5.11. The isomorphism
ϕd
: Ho(C)(A ⊗l B, C) ∼= Ho(C)(A, [B, C]r)
given by the composite
Ho(C)(QA⊗QB,C)
(1,q)
GG Ho(C)(QA⊗QB,RC)
ϕ
GG Ho(C)(QA,[QB,RC])
(p,1)−1
GG Ho(C)(A,[QB,RC])
is natural in each variable in Ho(C) since each component is natural in C.
Now the left and right derived structures have components (Ho(C), ⊗l, I, αl, ll, rl)
and (Ho(C), [−, −]r, I, Lr, ir, jr) respectively. We must prove that these components
are related by the equations (2.2), (2.3) and (2.5) of Section 2.3.
For (2.2) we must show that the diagram
(8.1)
Ho(C)(A, B)
(ll,1)
GG
vr
AA
Ho(C)(QI ⊗ QA, B)
ϕd

Ho(C)(I, [QA, RB])
commutes for all A and B. By naturality it suffices to verify commutativity in
the case that A is cofibrant and B is fibrant. By definition vr is the composite
Ho(C)(A,B)
Ho([QA,R−])
GG Ho(C)([QA,RA],[QA,RB])
([p,q],1)
GG Ho(C)([A,A],[QA,RB])
(j,1)
GG Ho(C)(I,[QA,RB])
Since A is cofibrant and B fibrant we can identify Ho(C)(A, B) with the set
of homotopy classes [f] : A → B of morphisms f : A → B; then vr([f]) is the
homotopy class of
I
j
GG [A, A]
[p,q]
GG [QA, RA]
[1,Rf]
GG [QA, RB]
which, by naturality of p and q, coincides with the homotopy class of
I
j
GG [A, A]
[1,f]
GG [A, B]
[p,q]
GG [QA, RB] .
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 43
Therefore the shorter path in the diagram below is vr. The longer path below is,
by definition, the longer path of the triangle (8.1). Accordingly we must show
that the following diagram commutes.
(8.2)
Ho(C)(A,B)
(p,1)
GG
(p,q) @@
v

Ho(C)(QA,B)
(l,1)
GG
(1,q)

Ho(C)(I⊗QA,B)
(1,q)

(p⊗1,1)
GG Ho(C)(QI⊗QA,B)
(1,q)

Ho(C)(QA,RB)
(l,1)
GG
v
AA
Ho(C)(I⊗QA,RB)
(p⊗1,1)
GG
ϕ

Ho(C)(QI⊗QA,RB)
ϕ

Ho(C)(I,[A,B])
(1,[p,q])
GG Ho(C)(I,[QA,RB])
(p,1)
GG
1
AA
Ho(C)(QI,[QA,RB])
(p−1,1)

Ho(C)(I,[QA,RB]
Each object above is of the form Ho(C)(X, Y ) for X cofibrant and Y fibrant and
we view each Ho(C)(X, Y ) as the set of homotopy classes of maps from X to Y .
The morphisms are of two kinds. Firstly there are those of the form Ho(C)(f, 1)
or Ho(C)(1, f) for f a morphism of C. Such morphisms respect the homotopy
relation and we view them as acting on homotopy classes. The other morphisms
are of the form v or ϕ and, because A is cofibrant and B fibrant, each occurence is
well defined on homotopy classes. Accordingly, to verify that the above diagram
commutes it suffices to verify that each sub-diagram commutes. Now apart from
the commutative triangle on the bottom right, each sub-diagram of (8.2) consists
of a diagram involving the hom-sets of C, but with components viewed as acting
on homotopy classes. Since in C itself these sub-diagrams commute, by naturality
or (2.2), they certainly commute when viewed as acting on homotopy classes.
Therefore (8.2) commutes.
According to (2.3) we must show that
(8.3)
Ho(C)(QA ⊗ QI, B)
ϕd

(rl,1)
AA
Ho(C)(A, [QI, RB])
(1,ir)
GG Ho(C)(A, B)
commutes for each A and B. Note that in Ho(C) the morphism 1⊗e : QA⊗I →
QA ⊗ QI is inverse to 1 ⊗ p : QA ⊗ QI → QA ⊗ I. Accordingly we can rewrite
rl as
A
p−1
GG QA
r GG QA ⊗ I
1⊗e
GG QA ⊗ QI
by substituting 1 ⊗ e for (1 ⊗ p)−1. The following diagram then establishes the
commutativity of (8.3).
Ho(C)(QA⊗QI,B)
(1⊗e,1)
GG
(1,q)

Ho(C)(QA⊗I,B)
(1,q)

(r,1)
@@
Ho(C)(QA⊗QI,RB)
ϕ

(1⊗e,1)
GG Ho(C)(QA⊗I,RB)
(r,1)
@@
ϕ

Ho(C)(QA,B)
(p−1,1)
@@
(1,q)

Ho(C)(QA,[QI,RB])
(p−1,1)

(1,[e,1])
GG Ho(C)(QA,[I,RB])
(p−1,1)

(1,i)
GGGG Ho(C)(QA,RB)
(p−1,1)
@@
Ho(C)(A,B)
(1,q)

1
99
Ho(C)(A,[QI,RB])
(1,[e,1])
GG Ho(C)(A,[I,RB])
(1,i)
GG Ho(C)(A,RB)
(1,q−1)
GG Ho(C)(A,B)
Next we calculate that td : [Q(QA⊗QB), RC] → [QA, R[QB, RC]] as constructed
in (2.4) has value:
[Q(QA⊗QB),RC]
[p,1]−1
GG [QA⊗QB,RC]
t GG [QA,[QB,RC]]
[1,q]
GG [QA,R[QB,RC]].
This calculation is given overleaf by the commutative diagram (8.4). All subdiagrams
of (8.4) commute in a routine manner. Apart from basic naturalities we
use the defining equation t = [u, 1] ◦ L of (2.4), the equation [1, p] ◦ k = p of (5.4)
and naturality of k as in (5.9). Furthermore, on the bottom right corner, we use
that the morphisms [QpA, 1], [pQA, 1] : [QA, R[QB, RC]] [[QQA, R[QB, RC]]
coincide in Ho(C). To see that this is so we argue as in the proof of Lemma 5.4.
Namely, pQA and QpA are left homotopic because they are coequalised by the
trivial fibration pA in C, and, since R[QB, RC] is fibrant, the desired equality
follows.
Finally, we use the calculation of td to prove that the diagram
Ho(C)(QA ⊗ Q(QB ⊗ QC), D)
ϕd

Ho(C)(αl,1)
GG Ho(C)(Q(QA ⊗ QB) ⊗ QC, D)
ϕd

Ho(C)(QA ⊗ QB, [QC, RD])
ϕd

Ho(C)(A, [QB ⊗ QC, RD])
Ho(C)(1,td)
GG Ho(C)(A, [QB, R[QC, RD]])
instantiating (2.5) commutes for all A, B, C and D. This is established overleaf in
the large, but straightforward, commutative diagram (8.5) whose only non-trivial
step is an application of (2.5) in C itself.
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 45
(8.4)
[Q(QA⊗QB),RC]
1
55
[p,1]−1

[Qq,1]−1
GG[QR(QA⊗QB),RC]
L
GG
[Qq,1]

[[QB,QR(QA⊗QB)],[QB,RC]]
[[1,Qq],1]

[k,1]
GG[Q[QB,R(QA⊗QB)],[QB,RC]]
[Q[1,q],1]

[1,q]
GG[Q[QB,R(QA⊗QB)],R[QB,RC]]
[Q[1,q],1]

[Q(QA⊗QB),RC]
L
GG[[QB,Q(QA⊗QB)],[QB,RC]]
[k,1]
GG[Q[QB,QA⊗QB],[QB,RC]]
[1,q]
GG
[Qu,1]

[Q[QB,QA⊗QB],R[QB,RC]]
[Qu,1]

[[QB,QA⊗QB],[QB,RC]]
[u,1]

[[1,p],1]
yy
[p,1]
UU
[QQA,[QB,RC]]
[1,q]
GG[QQA,R[QB,RC]]
[QA⊗QB,RC]
[p,1]
hh
L
RR
t
GG[QA,[QB,RC]]
[p,1]
UU
[1,q]
GG[QA,R[QB,RC]]
[pQA,1]=[QpA,1]
yy
(8.5)
Ho(C)(QA⊗Q(QB⊗QC),D)
(1,q)

((1⊗p)−1,1)
GGHo(C)(QA⊗(QB⊗QC),D)
(1,q)

(α,1)
GGHo(C)((QA⊗QB)⊗QC,D)
(1,q)

(p⊗1,1)
GGHo(C)(Q(QA⊗QB)⊗QC,D)
(1,q)

Ho(C)((QA⊗QB)⊗QC,RD)
ϕ

(p⊗1,1)
GGHo(C)(Q(QA⊗QB)⊗QC,RD)
ϕ

Ho(C)(QA⊗Q(QB⊗QC),RD)
ϕ

((1⊗p)−1,1)
GGHo(C)(QA⊗(QB⊗QC),RD)
ϕ

(α,1)
RR
Ho(C)(Q(QA⊗QB),[QC,RD])
(p−1,1)

Ho(C)(QA⊗QB,[QC,RD])
ϕ

(p,1)
QQ
1GG
(1,q)CC
Ho(C)(QA⊗QB,[QC,RD])
(1,q)

Ho(C)(QA,[Q(QB⊗QC),RD])
(1,[p,1]−1)
GG
(p−1,1)

Ho(C)(QA,[QB⊗QC,RD])
(p−1,1)

(1,t)BB
Ho(C)(QA⊗QB,R[QC,RD])
ϕ

Ho(C)(QA,[QB,[QC,RD]])
(1,[1,q])
GG
(p−1,1)

Ho(C)(QA,[QB,R[QC,RD]])
(p−1,1)

Ho(C)(A,[Q(QB⊗QC),RD])
(1,[p,1]−1)
GGHo(C)(A,[QB⊗QC,RD])
(1,t)
GGHo(C)(A,[QB,[QC,RD]])
(1,[1,q])
GGHo(C)(A,[QB,R[QC,RD]])
SKEW STRUCTURES IN 2-CATEGORY THEORY AND HOMOTOPY THEORY 47
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Categories, No. 20 (2011), 1–266.
Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 60000, Czech Republic
E-mail address: bourkej@math.muni.cz
Chapter 5
Monads and theories
This chapter contains the article Monads and theories by John Bourke and
Richard Garner, published in Advances in Mathematics 351 (2019) 1024–
1071. Both authors contributed an equal share to the results of this arti-
cle.
166
MONADS AND THEORIES
JOHN BOURKE AND RICHARD GARNER
Abstract. Given a locally presentable enriched category E together with a
small dense full subcategory A of arities, we study the relationship between
monads on E and identity-on-objects functors out of A, which we call Apretheories.
We show that the natural constructions relating these two
kinds of structure form an adjoint pair. The fixpoints of the adjunction are
characterised on the one side as the A-nervous monads—those for which
the conclusions of Weber’s nerve theorem hold—and on the other, as the
A-theories which we introduce here.
The resulting equivalence between A-nervous monads and A-theories is
best possible in a precise sense, and extends almost all previously known
monad–theory correspondences. It also establishes some completely new
correspondences, including one which captures the globular theories defining
Grothendieck weak ω-groupoids.
Besides establishing our general correspondence and illustrating its reach,
we study good properties of A-nervous monads and A-theories that allow
us to recognise and construct them with ease. We also compare them with
the monads with arities and theories with arities introduced and studied by
Berger, Melli`es and Weber.
1. Introduction
Category theory provides two approaches to classical universal algebra. On
the one hand, we have finitary monads on Set and on the other hand, we have
Lawvere theories. Relating the two approaches we have Linton’s result [26], which
shows that the category of finitary monads on Set is equivalent to the category
of Lawvere theories. An essential feature of this equivalence is that it respects
semantics, in the sense that the algebras for a finitary monad coincide up to
equivalence over Set with the models of the associated theory, and vice versa.
There have been a host of generalisations of the above story, each dealing
with algebraic structure borne by objects more general than sets. In many of
these [32, 31, 22, 23], one starts on one side with the monads on a given category
that preserve a specified class of colimits. This class specifies, albeit indirectly,
the arities of operations that may arise in the algebraic structures encoded by
such monads, and from this one may define, on the other side, corresponding
notions of theory and model. These are subtler than in the classical setting, but
2000 Mathematics Subject Classification. Primary:18C10, 18C20.
Both authors gratefully acknowledge the support of Australian Research Council Discovery
Project DP160101519; the second author also acknowledges Australian Research Council Future
Fellowship FT160100393.
1
2 JOHN BOURKE AND RICHARD GARNER
once the correct definitions have been found, the equivalence with the given class
of monads, and the compatibility with semantics, follows much as before.
The most general framework for a monad–theory correspondence to date
involves the notions of monad with arities and theory with arities. In this setting,
the permissible arities of operations are part of the basic data, given as a small,
dense, full subcategory of the base category. The monads with arities were
introduced first, in [35], as a setting for an abstract nerve theorem. Particular
cases of this theorem include the classical nerve theorem, identifying categories
with simplicial sets satisfying the Segal condition of [33], and also Berger’s nerve
theorem [8] for the globular higher categories of [7]. More saliently, when Weber’s
nerve theorem is specialised to the settings appropriate to the monad–theory
correspondences listed above, it becomes exactly the fact that the functor sending
the algebras for a monad to the models of the associated theory is an equivalence.
This observation led [29] and [9] to introduce theories with arities, and to prove,
by using Weber’s nerve theorem, their equivalence with the monads with arities.
The monad–theory correspondence obtained in this way is general enough to
encompass all of the instances from [32, 31, 22, 23].
Our own work in this paper has two motivations: one abstract and one concrete.
Our abstract motivation is a desire to explain the apparently ad hoc design choices
involved in the monad–theory correspondences outlined above. For indeed, while
these choices must be carefully balanced in order to obtain an equivalence, there
is no reason to believe that different careful choices might not yield more general
or more expressive results.
Our concrete motivation comes from the study of the Grothendieck weak
ω-groupoids introduced by Maltsiniotis [27], which, by definition, are models of
a globular theory in the sense of Berger [8]. Globular theories describe algebraic
structure on globular sets with arities drawn from the dense subcategory of
globular cardinals; see Example 8(v) below. However, globular theories are
not necessarily theories with arities, and in particular, those capturing higher
groupoidal structures are not. As such, they do not appear to one side of any of
the monad–theory correspondences described above.
The first goal of this paper is to describe a new schema for monad–theory
correspondences which addresses the gaps in our understanding noted above. In
this schema, once we have fixed the process by which a theory is associated to a
monad, everything else is forced. This addresses our first, abstract motivation.
The correspondence obtained in this way is in fact best possible, in the sense
that any other monad–theory correspondence for the same kind of algebraic
structure must be a restriction of this particular one. In many cases, this best
possible correspondence coincides with one in the literature, but in others, our
correspondence goes beyond what already exists. In particular, an instance of our
schema will identify the globular theories of [8] with a suitable class of monads
on the category of globular sets. This addresses our second, concrete motivation.
The further goal of this paper is to study the classes of monads and theories that
arise from our correspondence-schema. We do so both at a general level, where
we will see that both the monads and the theories are closed under essentially
all the constructions one could hope for; and also at a practical level, where we
MONADS AND THEORIES 3
will see how these general constructions allow us to give expressive and intuitive
presentations for the structure captured by a monad or theory.
To give a fuller account of our results, we must first describe how a typical
monad–theory correspondence arises. As in [35], the basic setting for such a
correspondence can be encapsulated by a pair consisting of a category E and
a small, full, dense subcategory K : A → E. For example, the Lawvere theory–
finitary monad correspondence for finitary algebraic structure on sets is associated
to the choice of E = Set and A = F the full subcategory of finite cardinals.
Given E and K : A → E, the goal is to establish an equivalence between
a suitable category of A-monads and a suitable category of A-theories. The
A-monads will be a certain class of monads on E; while the A-theories will be a
certain class of identity-on-objects functors out of A. We are being deliberately
vague about the conditions on each side, as they are among the seemingly ad
hoc design choices we spoke of earlier. But regardless of this, the monad–theory
correspondence itself always arises through application of the following two
constructions.
Construction A. For an A-monad T on E, the associated A-theory Φ(T) is the
identity-on-objects functor JT : A → AT arising from the (identity-on-objects,
fully faithful) factorisation
(1.1) A
JT
−−→ AT
VT
−−→ ET
of the composite FTK : A → E → ET. Here FT is the free functor into the Kleisli
category ET, so AT is equally the full subcategory of ET with objects those of A.
Construction B. For an A-theory J : A → T , the associated A-monad Ψ(T )
is obtained from the category of concrete T -models, which is by definition the
pullback
(1.2)
Modc(T ) GG
UT

[T op, Set]
[Jop,1]

E
NK =E(K−,1)
GG [Aop, Set] .
Since UT is a pullback of the strictly monadic [Jop, 1], it will be strictly monadic
so long as it has a left adjoint. The assumption that E is locally presentable
ensures that this is the case, and so we can take Ψ(T ) to be the monad whose
algebras are the concrete T -models.
There remains the problem of choosing the appropriate conditions on a monad
or theory for it to be an A-monad or A-theory. Of course, these must be carefully
balanced so as to obtain an equivalence, but this still seems to leave too many
degrees of freedom; one might hope that everything could be determined from
E and A alone. The main result of this paper shows that this is so: there are
notions of A-monad and A-theory which require no further choices to be made,
and which rather than being plucked from the air, may be derived in a principled
manner.
The key observation is that Constructions A and B make sense when given
as input any monad on E, or any “A-pretheory”—by which we mean simply an
4 JOHN BOURKE AND RICHARD GARNER
identity-on-objects functor out of A. When viewed in this greater generality,
these constructions yield an adjunction
(1.3) Mnd(E)
Φ
GG⊥ PrethA(E)
Ψoo
between the category of monads on E and the category of A-pretheories. Like
any adjunction, this restricts to an equivalence between the objects at which the
counit is invertible, and the objects at which the unit is invertible. Thus, if we
define the A-monads and A-theories to be the objects so arising, then we obtain
a monad–theory equivalence. By construction, it will be the largest possible
equivalence whose two directions are given by Constructions A and B.
Having defined the A-monads and A-theories abstractly, it behooves us to
give tractable concrete characterisations. In fact, we give a number of these,
allowing us to relate our correspondence to existing ones in the literature. We
also investigate further aspects of the general theory, and provide a wide range
of examples illustrating the practical utility of our results.
Before getting started, we conclude this introduction with a more detailed
outline of the paper’s contents. In Section 2, we begin by introducing our basic
setting and notions. We then construct, in Theorem 6, the adjunction (1.3)
between monads and pretheories. In Section 3, with this abstract result in
place, we introduce a host of running examples of our basic setting. To convince
the reader of the expressive power of our notions, we construct, via colimit
presentations, specific pretheories for a variety of mathematical structures.
In Section 4 we obtain our main result by characterising the fixpoints of the
monad–theory adjunction: the A-monads and A-theories described above. The
A-monads are characterised as what we term the A-nervous monads, since they
are precisely those monads for which Weber’s nerve theorem holds. The Atheories
turn out to be precisely those A-pretheories for which each representable
is a model; in the motivating case where E = Set and A = F, they are exactly
the Lawvere theories. With these characterisations in place, we obtain our main
Theorem 19, which describes the “best possible” equivalence between A-theories
and A-nervous monads.
Section 5 develops some of the general results associated to our correspondenceschema.
We begin by showing that our monad–theory correspondence commutes,
to within isomorphism, with the taking of semantics on each side. We also prove
that the functors taking semantics are valued in monadic right adjoint functors
between locally presentable categories. The final important result of this section
states that colimits of A-nervous monads and A-theories are algebraic, meaning
that the semantics functors send them to limits.
Section 6 is devoted to exploring what the A-nervous monads and A-theories
amount to in our running examples. In order to understand the A-nervous monads,
we prove the important result that they are equally the colimits, amongst all
monads, of free monads on A-signatures. We also introduce the notion of a
saturated class of arities as a setting in which, like in [32, 31, 22, 23], the A-nervous
monads can be characterised in terms of a colimit-preservation property. With
MONADS AND THEORIES 5
these results in place, we are able to exhibit many of these existing monad–theory
correspondences as instances of our general framework.
In Section 7, we examine the relationship between the monads and theories
of our correspondence, and the monads with arities and theories with arities
of [35, 29, 9]. In particular, we see that every monad with arities A is an Anervous
monad but that the converse implication need not be true: so A-nervous
monads are strictly more general. Of course, the same is also true on the theory
side. We also exhibit a further important point of difference: colimits of monads
with arities, unlike those of nervous monads, need not be algebraic. This means
that there is no good notion of presentation for monads or theories with arities.
Finally, in Section 8, we give a number of proofs deferred from Section 6.
2. Monads and pretheories
2.1. The setting. In this section we construct the monad–pretheory adjunction
(2.1) Mnd(E)
Φ
GG⊥ PrethA(E) .
Ψoo
The setting for this, and the rest of the paper, involves two basic pieces of data:
(i) A locally presentable V-category E with respect to which we will describe
the monad– pretheory adjunction; and
(ii) A notion of arities given by a small, full, dense sub-V-category K : A → E.
We will discuss examples in Section 2.1 below, but for now let us clarify some
of the terms appearing above. While in the introduction, we focused on the
unenriched context, we now work in the context of category theory enriched over
a symmetric monoidal closed category V which is locally presentable as in [13].
In this context, a locally presentable V-category [18] is one which is cocomplete
as a V-category, and whose underlying ordinary category is locally presentable.
We recall also some notions pertaining to density. Given a V-functor K : A → E
with small domain, the nerve functor NK : E → [Aop, V] is defined by NK(X) =
E(K–, X). We call a presheaf in the essential image of NK a K-nerve, and we
write K-Ner(V) for the full sub-V-category of [Aop, V] determined by these.
We say that K is dense if NK is fully faithful; whereupon NK induces an
equivalence of categories E K-Ner(V). Finally, we call a small sub-V-category
A of a V-category E dense if its inclusion functor K : A → E is so.
2.2. Monads. We write Mnd(E) for the (ordinary) category whose objects are Vmonads
on E, and whose maps S → T are V-transformations α: S ⇒ T compatible
with unit and multiplication. For each T ∈ Mnd(E) we have the V-category
of algebras UT : ET → E over E, but also the Kleisli V-category FT : E → ET
under E, arising from an (identity-on-objects, fully faithful) factorisation
(2.2)
E
FT

FT
11
ET
WT
GG ET
6 JOHN BOURKE AND RICHARD GARNER
of the free V-functor FT : E → ET; concretely, we may take ET to have objects
those of E, hom-objects ET(A, B) = E(A, TB), and composition and identities
derived using the monad structure of T. Each monad morphism α: S → T
induces, functorially in α, V-functors α∗ and α! fitting into diagrams
(2.3)
ET α∗
GG
UT 44
ES
US}}
E
FS
}}
FT
33
E ES
α!
GG ET ;
here α∗ sends an algebra a: TA → A to a ◦ αA : SA → A and is the identity on
homs, while α! is the identity on objects and has action on homs given by the
postcomposition maps αB ◦ (–): ES(A, B) → ET(A, B). In fact, every V-functor
ET → ES over E or V-functor ES → ET under E is of the form α∗ or α! for a
unique map of monads α—see, for example, [30]—and in this way, we obtain
fully faithful functors
(2.4) Mnd(E)op Alg
−−→ V-CAT/E and Mnd(E)
Kl
−−→ E/V-CAT .
2.3. Pretheories. An A-pretheory is an identity-on-objects V-functor J : A → T
with domain A. We write PrethA(E) for the ordinary category whose objects are
A-pretheories and whose morphisms are V-functors commuting with the maps
from A. While the A-pretheory is only fully specified by both pieces of data T
and J, we will often, by abuse of notation, leave J implicit and refer to such a
pretheory simply as T .
Just as any V-monad has a V-category of algebras, so any A-pretheory has a
V-category of models. Generalising (1.2), we define the V-category of concrete
T -models Modc(T ) by a pullback of V-categories as below left; so a concrete T model
is an object X ∈ E together with a chosen extension of E(K–, X): Aop → V
along Jop : Aop → T op. The reason for the qualifier “concrete” will be made
clear in Section 5.2 below, where we will identify a more general notion of model.
(2.5)
Modc(T )
PT
GG
UT

[T op, V]
[Jop,1]

Modc(S)
PS
GG
US

[Sop, V]
[Hop,1]
GG [T op, V]
[Jop,1]

E
NK
GG [Aop, V] E
NK
GG [Aop, V]
Remark 1. Avery considers a notion very similar to our A-pretheories under
the name prototheories [4, Definition 4.1.1]. The differences are that Avery’s
prototheories A → T are not enriched, and the hom-sets of T need not be small.
He also defines a category of (concrete) models for a prototheory, relative to a
given functor E → [Aop, C] called an aritation. When this functor is the nerve
NK : E → [Aop, Set], his category of models agrees with our Modc(T ).
Any A-pretheory map H : T → S gives a functor H∗ : Modc(S) → Modc(T )
over E by applying the universal property of the pullback left above to the
commuting square on the right. In this way, we obtain a semantics functor:
(2.6) PrethA(E)op Modc
−−−−→ V-CAT/E .
MONADS AND THEORIES 7
However, unlike (2.4), this is not always fully faithful. Indeed, in Example 10
below, we will see that non-isomorphic pretheories can have isomorphic categories
of concrete models over E.
2.4. Monads to pretheories. We now define the functor Φ: Mnd(E) → PrethA(E)
in (2.1). As in Construction A of the introduction, this will take the V-monad T
to the A-pretheory JT : A → AT arising as the first part of an (identity-on-objects,
fully faithful) factorisation of FTK : A → ET, as to the left in:
(2.7)
A
K

JT
GG AT
VT

A
K

JT
GG AT
KT

E
FT
GG ET E
FT
GG ET .
Since the comparison WT : ET → ET is fully faithful, we can also view JT as
arising from an (identity-on-objects, fully faithful) factorisation as above right;
the relationship between the two is that KT = WT ◦VT. Both perspectives will be
used in what follows, with the functor KT : AT → ET of particular importance.
To define Φ on morphisms, we make use of the orthogonality of identity-onobjects
V-functors to fully faithful ones; this asserts that any commuting square
of V-functors as below, with F identity-on-objects and G fully faithful, admits a
unique diagonal filler J making both triangles commute.
A
H GG
F

C
G

B
K GG
J
bb
D
Explicitly, J is given on objects by Ja = Ha, and on homs by
B(a, b)
Ka,b
−−−→ D(Ka, Kb) = D(GHa, GHb)
(GHa,Hb)−1
−−−−−−−−→ C(Ha, Hb) .
In particular, given a map α: S → T of Mnd(E), this orthogonality guarantees
the existence of a diagonal filler in the diagram below, whose upper triangle we
take to be the map Φ(α): Φ(S) → Φ(T) in PrethA(E):
A
JS

JT
GG AT
VT

AS
VS
GG
TT
ES α!
GG ET .
2.5. Pretheories to monads. Thus far we have not exploited the local presentability
of E. It will be used in the next step, that of constructing the left adjoint
to Φ: Mnd(E) → PrethA(E). We first state a general result which, independent
of local presentability, gives a sufficient condition for an individual pretheory to
have a reflection along Φ. Here, by a reflection of an object c ∈ C along a functor
U : B → C, we mean a representation for the functor C(c, U–): B → Set.
8 JOHN BOURKE AND RICHARD GARNER
Theorem 2. A pretheory J : A → T admits a reflection along Φ whenever the
forgetful functor UT : Modc(T ) → E from the category of concrete models has a
left adjoint FT . In this case, the reflection ΨT is characterised by an isomorphism
EΨT ∼= Modc(T ) over E, or equally, by a pullback square
(2.8)
EΨT GG
UΨT

[T op, V]
[Jop,1]

E
NK
GG [Aop, V] .
To prove this result, we will need a preparatory lemma, relating to the notion of
discrete isofibration: this is a V-functor U : D → C such that, for each f : c ∼= Ud
in C, there is a unique f : c ∼= d in D with U(f ) = f.
Example 3. For any V-monad T on C, the forgetful V-functor UT : CT → C is
a discrete isofibration. Indeed, if x: Ta → a is a T-algebra and f : b ∼= a in C,
then y = f−1 ◦ x ◦ Tf : Tb → b is the unique algebra structure on b for which
f : (b, y) → (a, x) belongs to CT. In particular, for any identity-on-objects Vfunctor
F : A → B between small V-categories, the functor [F, 1]: [B, V] → [A, V]
has a left adjoint and strictly creates colimits, whence is strictly monadic. It is
therefore a discrete isofibration by the above argument.
Lemma 4. Let U : A → B be a discrete isofibration and α: F ⇒ G: X → B an
invertible V-transformation. The displayed projections give isomorphisms between
liftings of F through U, liftings of α through U, and liftings of G through U:
A
U

X
F GG
¯F
cc
B
dom
←−−−
A
U

X
G
TTα
¯α
'5
F
@@
¯F
UU
¯G
qq
B
cod
−−→
A
U

X
G GG
¯G
cc
B .
Proof. Given ¯G: X → A as to the right, there is for each x ∈ X a unique lifting
of the isomorphism αx : Fx ∼= U ¯Gx to one ¯αx : ¯Fx ∼= ¯Gx. There is now a unique
way of extending x → ¯Fx to a V-functor ¯F : X → A so that ¯α: ¯F ∼= ¯G; namely, by
taking the action on homs to be ¯Fx,y = A(¯αx, ¯αy
−1)◦ ¯Gx,y : X(x, y) → A( ¯Fx, ¯Fy).
In this way, we have found a unique lifting of α through U whose codomain is
the given lifting of G through U. So the right-hand projection is invertible; the
argument for the left-hand one is the same on replacing α by α−1.
We can now give:
Proof of Theorem 2. UT has a left adjoint by assumption, and—as a pullback of
the strictly monadic [Jop, 1]: [T op, V] → [Aop, V]—strictly creates coequalisers
for UT -absolute pairs. It is therefore strictly monadic. Taking ΨT = UT FT to
be the induced monad, we thus have an isomorphism EΨT ∼= Modc(T ) over E.
It remains to exhibit isomorphisms Mnd(E)(ΨT , S) ∼= PrethA(E)(T , ΦS) natural
in S. We do so by chaining together the following sequence of natural
MONADS AND THEORIES 9
bijections. Firstly, by full fidelity in (2.4), monad maps α0 : ΨT → S correspond
naturally to functors α1 : ES → EΨT rendering commutative the left triangle in
(2.9)
ES
US
((
α1
GG EΨT
UΨT
ÒÒ
ES α2
GG
US

[T op, V]
[Jop,1]

E E
NK
GG [Aop, V] .
Since EΨT is defined by the pullback (2.14), such functors α1 correspond naturally
to functors α2 rendering commutative the square above right. Next, we observe
that there is a natural isomorphism in the triangle below left
(2.10)
ES US
GG
NF SK 22
∼=
E
NK
~~
ES α3
GG
NF SK 22
[T op, V]
[Jop,1]~~
[Aop, V] [Aop, V]
with components the adjointness isomorphisms E(Ka, USb) ∼= ES(FSKa, b). Since
Jop is identity-on-objects, [Jop, 1] is a discrete isofibration by Example 3, whence
by Lemma 4 there is a natural bijection between functors α2 as in (2.9) and ones
α3 as in (2.10). We should now like to transpose this last triangle through the
following natural isomorphisms (taking X = A, T ):
(2.11) V-CAT(ES
, [Xop
, V]) ∼= V-CAT(Xop
, [ES
, V]) .
However, since ES is large, the functor category [ES, V] will not always exist
as a V-category, and so (2.11) is ill-defined. To resolve this, note that NFSK
is, by its definition, pointwise representable; whence so too is α3, since J is
identity-on-objects. We may thus transpose the right triangle of (2.10) through
the legitimate isomorphisms
(2.12) V-CAT(ES
, [Xop
, V])pwr
∼= V-CAT(Xop
, [ES
, V]rep)
where on the left we have the category of pointwise representable V-functors, and
on the right, the legitimate V-category of representable V-functors ES → V. In
this way, we establish a natural bijection between functors α3 and functors α4
rendering commutative the left square in:
Aop
(FSK)op

Jop
GG T op
α4

(α5)op
zz
A
JS

J GG T
α5

α6
zz
(ES)op
Y
GG [ES, V]rep AS
KS
GG ES .
Now orthogonality of the identity-on-objects Jop and the fully faithful Y draws
the correspondence between functors α4 and functors α5 satisfying α5 ◦J = FSK
as left above. Finally, since AS fits in to an (identity-on-objects, fully faithful)
factorisation of FSK, orthogonality also gives the correspondence, as right above,
between functors α5 and functors α6 satisfying α6 ◦ J = JS, as required.
10 JOHN BOURKE AND RICHARD GARNER
We now show that the assumed local presentability of E ensures that every
pretheory has a reflection along Φ: Mnd(E) → PrethA(E), which consequently
has a left adjoint. The key result about locally presentable categories enabling
this is the following lemma.
Lemma 5. Consider a pullback square of V-categories
(2.13)
A
F GG
U

B
V

C
G GG D
in which G and V are right adjoints between locally presentable V-categories
and V is strictly monadic. Then U and F are right adjoints between locally
presentable V-categories and U is strictly monadic.
Proof. Since V is strictly monadic, it is a discrete isofibration, and so its pullback
against G is, by [14, Corollary 1], also a bipullback. By [10, Theorem 6.11] the
2-category of locally presentable V-categories and right adjoint functors is closed
under bilimits in V-CAT, so that both U and F are right adjoints between locally
presentable categories. Finally, since U is a pullback of the strictly monadic V ,
it strictly creates coequalisers for U-absolute pairs. Since it is already known to
be a right adjoint, it is therefore also strictly monadic.
With this in place, we can now prove:
Theorem 6. Let E be locally presentable. Then Φ: Mnd(E) → PrethA(E) has a
left adjoint Ψ: PrethA(E) → Mnd(E), whose value at the pretheory J : A → T
is characterised by an isomorphism EΨ(T ) ∼= Modc(T ) over E, or equally, by a
pullback square
(2.14)
EΨ(T ) GG
UΨ(T )

[T op, V]
[Jop,1]

E
NK
GG [Aop, V] .
Proof. Let J : A → T be a pretheory. The pullback square (2.5) defining Modc(T )
is a pullback of a right adjoint functor between locally presentable categories
along a strictly monadic one: so it follows from Lemma 5 that UT : Modc(T ) → E
is a right adjoint, whence the result follows from Theorem 2.
Remark 7. In Avery’s study of prototheories, he establishes a structure–semantics
adjunction [4, Theorem 4.4.8] of the form ProtoA(E)op CAT/E, where here
CAT is the category of large categories. By restricting to the locally small
prototheories to the left and to the strictly monadic functors to the right of this
adjunction, one can recover, via (2.4), the unenriched case of our adjunction (2.1).
3. Pretheories as presentations
In the next section, we will describe how the monad–pretheory adjunction (2.1)
restricts to an equivalence between suitable subcategories of A-theories and of
MONADS AND THEORIES 11
A-nervous monads. However, the results we have so far are already practically
useful. The notion of A-pretheory provides a tool for presenting certain kinds
of algebraic structure, by exhibiting them as categories of concrete T -models
for a suitable pretheory in a manner reminiscent of the theory of sketches [6].
Equivalently, via the functor Ψ, we can see A-pretheories as a way of presenting
certain monads on E.
3.1. Examples of the basic setting. Before giving examples of algebraic structures
presented by pretheories, we first describe a range of examples of the basic setting
of Section 2.1 above.
Examples 8. We begin by considering the unenriched case where V = Set.
(i) Taking E = Set and A = F the full subcategory of finite cardinals captures
the classical case of finitary algebraic structure borne by sets; so examples
like groups, rings, lattices, Lie algebras, and so on.
(ii) Taking E a locally finitely presentable category and A = Ef a skeleton
of the full subcategory of finitely presentable objects, we capture finitary
algebraic structure borne by E-objects. Examples when E = Cat include
finite product, finite colimit, and monoidal closed structure; for E = CRng,
we have commutative k-algebra, differential ring and reduced ring structure.
(iii) We can replace “finitary” above by “λ-ary” for any regular cardinal λ. For
example, when λ = ℵ1, this allows for the structure of poset with joins
of ω-chains [28] when E = Set, and for countable product structure when
E = Cat. When E = [O(X)op, Set] for some space X, and λ is suitably
chosen, it also permits sheaf or sheaf of rings structure.
(iv) Let G1 be the category freely generated by the graph 0 1, so that
E = [Gop
1 , Set] is the category of directed multigraphs, and let A = ∆0 be
the full subcategory of [Gop
1 , Set] on graphs of the form
[n] := 0 GG 1 GG · · · GG n for n > 0.
∆0 is dense in [Gop
1 , Set] because it contains the representables [0] and [1].
This example captures structure borne by graphs in which the operations
build vertices and arrows from paths of arrows: for example, the structures
of categories, involutive categories, and groupoids.
(v) The globe category G is freely generated by the graph
0
τ
GG
σ GG
1
τ
GG
σ GG
2
τ
GG
σ GG
· · ·
subject to the coglobular relations σσ = στ and τσ = ττ. This means that
for each m > n, there are precisely two maps σm−n, τm−n : n m, which
by abuse of notation we will write simply as σ and τ.
The category E = [Gop, Set] is the category of globular sets; it has a dense
subcategory A = Θ0, first described by Berger [8], whose objects have been
termed globular cardinals by Street [34]. The globular cardinals include the
representables—the n-globes Y n for each n—but also shapes such as the
12 JOHN BOURKE AND RICHARD GARNER
globular set with distinct cells as depicted below.
(3.1) • • •GG
00
dd
GG


The globular cardinals can be parametrised in various ways, for instance
using trees [7, 8]; following [27], we will use tables of dimensions—sequences
n = (n1, . . . , nk) of natural numbers of odd length with n2i−1 > n2i < n2i+1.
Given such a table n and a functor D: G → C, we obtain a diagram
Dn2 Dn4
. . .
Dnk−1
Dn1 Dn3 Dn5 Dnk−2 Dnk
Dτ
||
Dσ
44
Dτ
||
Dσ
44
Dτ
||
Dσ
44
whose colimit in C, when it exists, will be written as D(n), and called the
D-globular sum indexed by n. Taking D = Y : G → [Gop, Set], the category
Θ0 of globular cardinals is now defined as the full subcategory of [Gop, Set]
spanned by the Y -globular sums. For example, the globular cardinal in (3.1)
corresponds to the Y -globular sum Y (1, 0, 2, 1, 2).
This example captures algebraic structures on globular sets in which
the operations build globes out of diagrams with shapes like (3.1); these
include strict ω-categories and strict ω-groupoids, but also the (globular)
weak ω-categories and weak ω-groupoids studied in [7, 25, 3].
We now turn to examples over enriched bases.
(vi) Let V be a locally finitely presentable symmetric monoidal category whose
finitely presentable objects are closed under the tensor product (cf. [18]).
By taking E = V and A = Vf a skeleton of the full sub-V-category of
finitely presentable objects, we capture V-enriched finitary algebraic structure
on V-objects as studied in [32]. When V = Cat this means structure
on categories C built from functors and natural transformations CI → C
for finitely presentable I: which includes symmetric monoidal or finite
limit structure, but not symmetric monoidal closed or factorization system
structure. Similarly, when V = Ab, it includes A-module structure but not
commutative ring structure.
(vii) Taking V as before, taking E to be any locally finitely presentable Vcategory
[18] and taking A = Ef a skeleton of the full subcategory of
finitely presentable objects in E, we capture V-enriched finitary algebraic
structure on E-objects as studied in [31]. As before, there is the obvious
generalization from finitary to λ-ary structure.
(viii) This example builds on [23]. Let V be a locally presentable symmetric
monoidal closed category, and consider a class of V-enriched limit-types
Φ with the property that the free Φ-completion of a small V-category is
again small. A V-functor F : C → V with small domain is called Φ-flat if its
cocontinuous extension LanyF : [Cop, V] → V preserves Φ-limits, and a ∈ V
is Φ-presentable if V(a, –): V → V preserves colimits by Φ-flat weights.
MONADS AND THEORIES 13
Suppose that if C is small and Φ-complete, then every Φ-continuous
F : C → V is Φ-flat; this is Axiom A of [23]. Then by Proposition 3.4 and
§7.1 of ibid., we obtain an instance of our setting on taking E = V and
A = VΦ a skeleton of the full sub-V-category of Φ-presentable objects.
A key example takes V = E = Cat and Φ the class of finite products;
whereupon VΦ is the subcategory F of finite cardinals, seen as discrete categories.
This example captures strongly finitary [19] structure on categories
involving functors and transformations Cn → C; this includes monoidal or
finite product structure, but not finite limit structure.
(ix) More generally, we can take E = Φ-Cts(C, V), the V-category of Φ-continuous
functors C → V for some small Φ-complete C, and take A to be the full image
of the Yoneda embedding Y : Cop → Φ-Cts(C, V). This example is appropriate
to the study of “Φ-ary algebraic structure on E-objects”—subsuming most
of the preceding examples.
3.2. Pretheories as presentations. We will now describe examples of pretheories
and their models in various contexts; in doing so, it will be useful to avail ourselves
of the following constructions. Given a pretheory A → T and objects a, b ∈ T ,
to adjoin a morphism f : a → b is to form the V-category T [f] in the pushout
square to the left of:
(3.2)
2
a,b
GG
ι

T
¯ι

2 +2 2
f,g
GG
id,id

T
¯ι

2
f
GG T [f] 2
f=g
GG T [f =g] .
Here, ι: 2 → 2 is the inclusion of the free V-category on the set {0, 1} into the
free V-category 2 = {0 → 1} on an arrow. Since ι is identity-on-objects, its
pushout ¯ι may also be chosen thus, so that we may speak of adjoining an arrow
to a pretheory J : A → T to obtain the pretheory J[f] = ¯ι ◦ J : A → T [f].
Recall from (2.5) that a concrete T -model comprises X ∈ E and F ∈ [T op, V]
for which F ◦ Jop = E(K–, X): A → V. Thus, by the universal property of the
pushout (3.2), a concrete T [f]-model is the same as a concrete T -model (X, F)
together with a map [f]: E(Kb, X) → E(Ka, X) in V.
Similarly given parallel morphisms f, g: a b in the underlying category
of T we can form the pushout above right. In this way we may speak of
adjoining an equation f = g to a pretheory J : A → T to obtain the pretheory
J[f =g] = ¯ι◦J : A → T [f =g]. In this case, we see that a concrete T [f =g]-model
is a concrete T -model (X, F) such that Ff = Fg: E(Kb, X) → E(Ka, X).
Example 9. In the context of Examples 8(i) appropriate to classical finitary
algebraic theories— so V = E = Set and A = F—we will construct a pretheory
J : F → M whose category of concrete models is the category of monoids.
We start from the initial pretheory id: F → F whose concrete models are simply
sets, and construct from it a pretheory J1 : F → M1 by adjoining morphisms
(3.3) m: 1 → 2 and i: 1 → 0
14 JOHN BOURKE AND RICHARD GARNER
representing the monoid multiplication and unit operations, and also morphisms
(3.4) m1, 1m: 2 3 and i1, 1i: 2 1
which will be necessary later to express the monoid equations. Note that our
directional conventions mean that the input arity of these operations is in the
codomain rather than the domain. It follows from the preceding remarks that a
concrete M1-model is a set X equipped with functions
[m]: X2
→ X , [i]: 1 → X , [m1], [1m]: X3
X2
, [i1], [1i]: 1 X
interpreting the morphisms adjoined above. We now adjoin to M1 the eight
equations necessary to render commutative the following squares in M1:
(3.5)
1
m GG
ι1

2
ι1

1
id GG
ι1

1
ι1

1
i GG
ι1

0
!

1
id GG
ι1

1
ι1

1 + 1
m1 GG 2 + 1 1 + 1
1m GG 1 + 2 1 + 1
i1 GG 1 1 + 1
1i GG 1
1
ι2
yy
id GG 1
ι2
yy
1
ι2
yy
m GG 2
ι2
yy
1
ι2
yy
id GG 1
id
yy
1
ι2
yy
i GG 0
!
yy
where ι1, ι2 and ! are the images under J1 of the relevant coproduct injections or
maps from 0 in F; together with three equations which render commutative:
(3.6)
1
m

m GG 2
1m

1
1 00
m GG 2
i1

1
1 00
m GG 2
1i

2
m1 GG 3 1 1 .
A concrete model for the resulting theory J : F → M is a concrete M1-model
(X, F) for which Fop : M1 → Setop
sends each diagram in (3.5) and (3.6) to a
commuting one. Commutativity in (3.5) forces [m1] = [m] × id: X3 → X2 and
so on; whereupon commutativity of (3.6) expresses precisely the monoid axioms,
so that concrete M-models are monoids, as desired. Extending this analysis to
morphisms we see that Modc(M) is isomorphic to the category of monoids and
monoid homomorphisms.
Example 10. In the same way we can describe F-pretheories modelling any of the
categories of classical universal algebra—groups, rings and so on. Note that the
same structure can be presented by distinct pretheories: for instance, we could
extend the pretheory M of the preceding example by adjoining a further morphism
m11: 3 → 4 and two equations forcing it to become [m] × 1 × 1: X4 → X3 in
any model; on doing so, we would not change the category of concrete models.
However, in M, all of the maps 3 → 4 belong to F while in the new pretheory,
m11 does not. This non-canonicity will be rectified by the theories introduced in
Section 4 below; in particular, Corollary 24 implies that, to within isomorphism,
there is at most one F-theory which captures a given type of structure.
Example 11. In the situation of Examples 8(iv), where E = [Gop
0 , Set] is the
category of directed graphs and A = ∆0, we will describe a pretheory ∆0 → C
whose concrete models are categories. The construction is largely identical to
the example of monoids above. Starting from the initial ∆0-pretheory, we adjoin
MONADS AND THEORIES 15
composition and unit maps m: [1] → [2] and i: [1] → [0] as well as the morphisms
1m, m1: [2] [3] and i1, 1i: [2] [1] required to describe the category axioms.
We now adjoin the necessary equations. First, we have four equations ensuring
that composition and identities interact appropriately with source and target:
[0]
σ GG
σ

[1]
m

[0]
τ GG
τ

[1]
m

[0]
σ GG
id 11
[1]
i

[0]
τ GG
id 11
[1]
i

[1]
ι1
GG [2] [1]
ι2
GG [2] [0] [0]
where here we write σ, τ : [0] [1] for the two endpoint inclusions, and ι1, ι2
for the two colimit injections into [1] τ +σ [1] = [2]. We also require analogues
of the eight equations of (3.5) and three equations of (3.6). The modifications
are minor: replace n by [n], the coproduct inclusions ι1 : n → n + m ← m: ι2
by the pushout inclusions ι1 : [n] → [n] τ +σ [m] ← [m]: ι2, the first appearance
of !: 0 → 1 by σ: [0] → [1] and its second appearance by τ : [0] → [1]. After
adjoining these six morphisms and fifteen equations, we find that the concrete
models of the resulting pretheory ∆0 → C are precisely small categories.
We can extend this pretheory to one for groupoids. To do so, we adjoin a
morphism c: [1] → [1] modelling the inversion plus the further maps 1c: [2] → [2]
and c1: [2] → [2] required for the axioms. Now four equations must be adjoined
to force the correct interpretation of 1c and c1, plus the two equations for left and
right inverses. On doing so, the resulting pretheory ∆0 → G has as its concrete
models the small groupoids.
Example 12. In the situation of Examples 8(v), where E is the category of
globular sets and A = Θ0 is the full subcategory of globular cardinals, one can
similarly construct pretheories whose concrete models are strict ω-categories or
strict ω-groupoids. For instance, one encodes binary composition of n-cells along
a k-cell boundary (for k < n) by adjoining morphisms mn,k : Y (n) → Y (n, k, n)
to Θ0. In fact, all of the standard flavours of globular weak ω-category and weak
ω-groupoid can also be encoded using Θ0-pretheories; see Examples 44(v) below.
Example 13. Consider the case of Examples 8(viii) where V = E = Cat and
A = F, the full subcategory of finite cardinals (seen as discrete categories). We
will describe an F-pretheory capturing the structure of a monoidal category. In
doing so, we exploit the fact that our pretheories are no longer mere categories,
but 2-categories; so we may speak not only of adjoining morphisms and equations
between such, but also of adjoining an (invertible) 2-cell—by taking a pushout of
the inclusion 2 +2 2 → D2 of the parallel pair 2-category into the free 2-category
on an (invertible) 2-cell—and similarly of adjoining an equation between 2-cells.
To construct a pretheory for monoidal categories, we start essentially as
for monoids: freely adjoining the usual maps m, i, m1, 1m, i1, 1i to the initial
pretheory, but now also morphisms m11, 1m1, 11m: 3 → 4 and 1i1: 3 → 2 needed
for the monoidal category coherence axioms; thus, ten morphisms in all.
16 JOHN BOURKE AND RICHARD GARNER
We now add the 8 × 2 = 16 equations asserting that each of the morphisms
beyond m and i has the expected interpretation in a model, plus1
the equation
1m ◦ m11 = m1 ◦ 11m: 2 → 4. This being done, we next adjoin invertible 2-cells
1
m GG
m

α
2
m1

2
1m
GG 3
λ
2
i1

1
1
GG
m
PP
1
ρ
uƒ
2
1i

1
1
GG
m
PP
1
expressing the associativity and unit coherences, as well as the invertible 2-cells
2
m1 GG
m1

α1
3
m11

3
1m1
GG 4
2
1m GG
1m

1α
3
1m1

3
11m
GG 4
1λ
3
1i1

2
1
GG
1m
PP
2
ρ1
uƒ
3
1i1

2
1
GG
m1
PP
2
which will be needed to express the coherence axioms. Finally, we must adjoin
equations between 2-cells: the 2 × 4 = 8 equations ensuring that α1, 1α, 1λ and
ρ1 have the intended interpretation in any model, plus two equations expressing
the coherence axioms:
(3.7)
2
m1 GG
α
3
m11
22
1
m
bb
m GG
m
22
2
1m
bb
m1
22
α
4
2
1m
GG 3
11m
bb =
2
m1 GG
m1
22
α
3
m11
22
α1
1
m
bb
m
22
3 1m1 GG
1α
4
2
1m
GG
1m
bb
3
11m
bb
(3.8)
2 id
''
α
m1
00
ρ1
1
m
dd
m
00
3 1i1 GG
1λ
2
2 id
hh
1m
dd =
2 id
%%
1
m
dd
m
00
id 2 .
2 id
ii
All told, we have adjoined ten morphisms, seventeen equations between morphisms,
seven invertible 2-cells, and nine equations between 2-cells to obtain a
pretheory J : F → MC whose concrete models are precisely monoidal categories.
4. The monad–theory correspondence
In this section, we return to the general theory and establish our “best possible”
monad–theory correspondence. This will be obtained by restricting the
adjunction (2.1) to its fixpoints: the objects on the left and right at which the
counit and the unit are invertible. The categories of fixpoints are the largest
1It may be prima facie unclear why this is necessary; after all, if 1m, m11, m1 and 11m have
the intended interpretations in a model, then it is certainly the case that they will verify this
equality. Yet this equality is not forced to hold in the pretheory, and we need it to do so in
order for (3.7) to type-check.
MONADS AND THEORIES 17
subcategories on which the adjunction becomes an adjoint equivalence, and it is
in this sense that our monad–theory correspondence is the best possible.
4.1. A pullback lemma. The following lemma will be crucial in characterising
the fixpoints of (2.1) on each side. Note that the force of (2) below is in the “if”
direction; the “only if” is always true.
Lemma 14. A commuting square in V-CAT
A
F GG
H

B
K

C
G GG D
with G fully faithful and H, K discrete isofibrations is a pullback just when:
(1) F is fully faithful; and
(2) An object b ∈ B is in the essential image of F if and only if Kb is in the
essential image of G.
Proof. If the square is a pullback, then F is fully faithful as a pullback of G. As
for (2), if Kb ∼= Gc in D then since K is an isofibration we can find b ∼= b in B
with Kb = Gc; now by the pullback property we induce a ∈ A with Fa = b so
that b ∼= Fa as required. Suppose conversely that (1) and (2) hold. We form the
pullback P of K along G and the induced map L as below.
(4.1)
A F
11
H
33
L
22
P
P GG
Q

B
K

C
G GG D
P is fully faithful as a pullback of G, and F is so by assumption; whence by
standard cancellativity properties of fully faithful functors, L is also fully faithful.
In fact, discrete isofibrations are also stable under pullback, and also have
the same cancellativity property; this follows from the fact that they are the
exactly the maps with the unique right lifting property against the inclusion of
the free V-category on an object into the free V-category on an isomorphism.
Consequently, in (4.1), Q is a discrete isofibration as a pullback of K, and H is
so by assumption; whence by cancellativity, L is also a discrete isofibration.
If we can now show L is also essentially surjective, we will be done: for then L
is a discrete isofibration and an equivalence, whence invertible. So let (b, c) ∈ P.
Since Kb = Gc, by (2) we have that b is in the essential image of F. So there
is a ∈ A and an isomorphism β : b ∼= Fa. Now Kβ : Gc = Kb ∼= KFa = GHa
so by full fidelity of G there is γ : c ∼= Ha with Gγ = Kβ; and so we have
(β, γ): (b, c) ∼= La exhibiting (b, c) as in the essential image of L, as required.
4.2. A-theories. We first use the pullback lemma to describe the fixpoints of (2.1)
on the pretheory side.
18 JOHN BOURKE AND RICHARD GARNER
Definition 15. An A-pretheory J : A → T is said to be an A-theory if each
T (J–, a) ∈ [Aop, V] is a K-nerve. We write ThA(E) for the full subcategory of
PrethA(E) on the A-theories.
In the language of Section 5.2 below, a pretheory T is an A-theory just
when each representable T (–, a): T op → V is a (non-concrete) T -model. When
V = E = Set and A = F, an A-pretheory is an A-theory precisely when it is a
Lawvere theory; see Examples 44(i) below.
Theorem 16. An A-pretheory J : A → T is an A-theory if and only if the unit
component ηT : T → ΦΨT of (2.1) is invertible.
Proof. The unit ηT : T → ΦΨT is obtained by starting with α0 = 1: ΨT → ΨT
and chasing through the bijections of Theorem 6 to obtain α6 = ηT . Doing this,
we quickly arrive at α2 equal to P, the projection in the depicted pullback square
(4.2)
EΨT P GG
UΨT

[T op, V]
[Jop,1]

Aop
(JΨT )op

Jop
GG T op
α4
α5
op
xx
αop
6
ss
E
NK
GG [Aop, V] (AΨT )op
(KΨT )op
GG (EΨT )op
Y
GG [EΨT , V]rep
defining EΨT . Now α3 : EΨT → [T op, V] is obtained by lifting an isomorphism
through [Jop, 1] and so we have α3
∼= P. We obtain α4 by transposing α3 through
the isomorphism (–)t : V-CAT(EΨT , [T op, V])pwr
∼= V-CAT(T op, [EΨT , V]rep) displayed
in (2.12). The relationships between α4, α5 and the unit component
ηT = α6 are depicted in the commutative diagram above right.
The identity-on-objects unit ηT = α6 will be invertible just when it is fully
faithful which, since KΨT is fully faithful, will be so just when α5 is fully faithful.
Now, since P ∼= α3 = (α4)t = (Y ◦ αop
5 )t = Nα5 , and P is fully faithful, as the
pullback of the fully faithful NK, it follows that Nα5 : EΨT → [T op, V] is also
fully faithful. As a consequence, α5 is fully faithful just when there exists a
factorisation to within isomorphism:
(4.3) Y ∼= Nα5 ◦ G: T → EΨT
→ [T op
, V] .
Indeed, in one direction, if α5 is fully faithful then the canonical natural transformation
Y ⇒ Nα5 ◦ α5 is invertible. In the other, given a factorisation as displayed,
G is fully faithful since Nα5 and Y are. Moreover we have isomorphisms
EΨT
(α5b, –) ∼= [T op
, V](Y b, Nα5 –) ∼= [T op
, V](Nα5 Gb, Nα5 –) ∼= EΨT
(Gb, –)
natural in b. So by Yoneda, α5
∼= G and so α5 is fully faithful since G is so.
This shows that ηT is invertible just when there is a factorisation (4.3). Since
Nα5 is fully faithful this in turn is equivalent to asking that each Y b = T (–, b) lies
in the essential image of α5, or equally in the essential image of the isomorphic P.
As the left square of (4.2) is a pullback, Lemma 14 asserts that this is, in turn,
equivalent to each [Jop, 1](Y b) = T (J–, b) being in the essential image of NK;
which is precisely the condition that J is an A-theory.
MONADS AND THEORIES 19
4.3. A-nervous monads. We now characterise the fixpoints on the monad side.
In the following definition, AT, JT and KT are as in (2.7).
Definition 17. A V-monad T on E is called A-nervous if
(i) The fully faithful KT : AT → ET is dense;
(ii) A presheaf X ∈ [AT
op
, V] is a KT-nerve if and only if X ◦ JT
op
is a K-nerve.
We write MndA(E) for the full subcategory of Mnd(E) on the A-nervous monads.
Note that the adjointness isomorphisms ET(KTJTX, Y ) = ET(FTKX, Y ) ∼=
E(KX, UTY ) for the adjunction FT UT give a pseudo-commutative square
(4.4)
ET
NKT
GG
UT

∼=
[AT
op
, V]
[JT
op,1]

E
NK
GG [Aop, V] ;
as a result of which, [Jop
T , 1] maps KT-nerves to K-nerves. Thus the force of
clause (ii) of the preceding definition lies in the if direction.
Theorem 18. The counit component εT : ΨΦT → T of (2.1) at a monad T on E
is invertible if and only if T is A-nervous.
Proof. εT is obtained by taking α6 = 1: JT → JT and proceeding in reverse
order through the series of six natural isomorphisms in the proof of Theorem 6.
Doing this, we quickly reach α3 = NKT
. Then α2 : ET → [(AT)op, V] is obtained
by lifting the natural isomorphism ϕ of (4.4) through the discrete isofibration
[Jop
T , 1], yielding a commutative square as left below.
(4.5)
ET α2
GG
UT

[(AT)op, V]
[Jop
T ,1]

E
NK
GG [Aop, V]
ET
UT
99
α1
GG EΨΦT
UΨΦT

E.
The map α1 : ET → EΨΦT is the unique map to the pullback, and α0 = εT the
corresponding morphism of monads. It follows that εT is invertible if and only
the square to the left of (4.5) is a pullback. Both vertical legs are discrete
isofibrations and NK is fully faithful, so by Lemma 14 this happens just when,
firstly, α2 is fully faithful, and, secondly, X ∈ [Aop
T , V] is in the essential image of
α2 if and only if XJT is a K-nerve. But as α2
∼= NKT
, and natural isomorphism
does not change either full fidelity or essential images, this happens just when T
is A-nervous.
4.4. The monad–theory equivalence. Putting together the preceding results now
yields the main result of this paper.
Theorem 19. The adjunction (2.1) restricts to an adjoint equivalence
(4.6) MndA(E)
Φ
GG⊥ ThA(E)
Ψoo
between the category of A-nervous monads and the category of A-theories.
20 JOHN BOURKE AND RICHARD GARNER
Proof. Any adjunction restricts to an adjoint equivalence between the objects
with invertible unit and counit components respectively, and Theorems 16 and 18
identify these objects as the A-theories and the A-nervous monads.
Note that there is an asymmetry between the conditions found on each side.
On the one hand, the condition characterising the A-theories among the Apretheories
is intrinsic, and easy to check in practice. On the other hand, the
condition defining an A-nervous monad refers to the associated pretheory, and is
non-trivial to check in practice. Indeed, one of the main points of [35, 9] is to
provide a general set of sufficient conditions under which a monad can be shown
to be A-nervous.
In the sections which follow, we will provide a number of more tractable
characterisations of the A-theories and A-nervous monads; the crucial fact which
drives all of these is that the adjunction (2.1) is in fact idempotent. Recall that
an adjunction L R: D → C is idempotent if the monad RL on C is idempotent,
and that this is equivalent to asking that the comonad LR is idempotent, or that
any one of the natural transformations Rε, εL, ηR and Lη is invertible.
Theorem 20. The adjunction (2.1) is idempotent.
Proof. We show for each T ∈ Mnd(E) that the unit ηΦT : ΦT → ΦΨΦT is
invertible. By Theorem 16, this is equally to show that JT : A → AT is an Atheory,
i.e., that each AT(JT–, JTa) ∈ [Aop, V] is a K-nerve. But AT(JT–, JTa) ∼=
ET(FTK–, FTKa) ∼= E(K–, UTFTKa) = E(K–, TKa) as required.
Exploiting the alternative characterisations of idempotent adjunctions listed
above, we immediately obtain the following result, which tells us in particular that
a monad T is A-nervous if and only if it can be presented by some A-pretheory.
Corollary 21. A monad T on E is A-nervous if and only if T ∼= ΨT for some
A-pretheory J : A → T ; while an A-pretheory J : A → T is an A-theory if and
only if T ∼= ΦT for some monad T on E.
The next result also follows directly from the definition of idempotent adjunc-
tion.
Corollary 22. The full subcategory MndA(E) ⊆ Mnd(E) is coreflective via ΨΦ,
while the full subcategory ThA(E) ⊆ PrethA(E) is reflective via ΦΨ.
5. Semantics
In the next section, we will explicitly identify the A-nervous monads and
A-theories for the examples listed in Section 2.1. Before doing this, we study
further aspects of the general theory, namely those related to the taking of
semantics.
5.1. Interaction with the semantics functors. We begin by examining the interaction
of our monad–theory correspondence with the semantics functors of
Section 2. In fact, we begin at the level of the monad–pretheory adjunction (2.1).
MONADS AND THEORIES 21
Proposition 23. There is a natural isomorphism θ as on the left in:
PrethA(E)op Ψop
GG
Modc 44
θks
Mnd(E)op
Alg||
Mnd(E)op Φop
GG
Alg 44
¯θ CQ
PrethA(E)op
Modc||
V-CAT/E V-CAT/E .
Let ¯θ be its mate under the adjunction Φop Ψop, as right above. The component
of ¯θ at T ∈ Mnd(E) is invertible if and only if T is A-nervous.
Proof. For the first claim, Theorem 6 provides the necessary natural isomorphisms
θT : EΨT → Modc(T ) over E. For the second, if we write as before εT : ΨΦT → T
for the counit component of (2.1) at T ∈ Mnd(E), then the T-component of ¯θ
is the composite θΨT ◦ (εT)∗ : ET → EΨΦT → Modc(ΦT ) over E. Since θΨT is
invertible and since Alg is fully faithful, ¯θT will be invertible just when εT is so;
that is, by Theorem 18, just when T is A-nervous.
From this and the fact that each monad ΨT is A-nervous, it follows that an Apretheory
T and its associated theory ΦΨT have isomorphic categories of concrete
models. By contrast, the passage from a monad T to its A-nervous coreflection
ΨΦT may well change the category of algebras. For example, the power-set monad
on Set, whose algebras are complete lattices, has its F-nervous coreflection given
by the finite-power-set monad, whose algebras are ∨-semilattices. However, if we
restrict to A-nervous monads and A-theories, then the situation is much better
behaved.
Theorem 24. The monad–theory equivalence (4.6) commutes with the semantics
functors; that is, we have natural isomorphisms:
(5.1)
ThA(E)op Ψop
GG
Modc 33
θks
MndA(E)op
Alg}}
MndA(E)op Φop
GG
Alg 33
¯θ CQ
ThA(E)op
Modc}}
V-CAT/E V-CAT/E .
Moreover, both Modc : ThA(E)op → V-CAT/E and Alg: MndA(E)op → V-CAT/E
are fully faithful functors.
Proof. The first statement follows from Proposition 23. For the second, note
that Alg: MndA(E)op
→ V-CAT/E is obtained by restricting the fully faithful
Alg: Mnd(E)op
→ V-CAT/E along a full embedding, and so is itself fully faithful.
It follows that Modc
∼= Alg◦Ψop : ThA(E)op → V-CAT/E is also fully faithful.
Full fidelity of Modc : ThA(E)op → V-CAT/E means that an A-theory is
determined to within isomorphism by its category of concrete models over E.
This rectifies the non-uniqueness of pretheories noted in Example 10 above.
5.2. Non-concrete models. In Section 2.3 we defined a concrete model of an Apretheory
T to be an object X ∈ E endowed with an extension of E(K–, X): Aop →
V to a functor T op → V. In the literature, one often encounters a looser notion
of model for a theory, in which an underlying object in E is not provided. In our
22 JOHN BOURKE AND RICHARD GARNER
setting, this notion becomes the following one: by an (unqualified) T -model, we
mean a functor F : T op → V whose restriction FJop : Aop → V is a K-nerve.
The T -models span a full sub-V-category Mod(T ) of [T op, V]. Recalling from
Section 2.1 that K-Ner(V) denotes the full sub-V-category of [Aop, V] on the
K-nerves, we may also express Mod(T ) as a pullback as to the right in:
(5.2)
Modc(T )
PT
CC
GG
UT

Mod(T ) GG
WT

[T op, V]
[Jop,1]

E
NK
GG K-Ner(V) GG [Aop, V] .
On the other hand, Modc(T ) is the pullback around the outside, and so there
is a canonical induced map Modc(T ) → Mod(T ) as displayed. By the usual
cancellativity properties, the left square above is now also a pullback. Moreover,
WT is an isofibration, as a pullback of the discrete isofibration [Jop, 1], and
NK : E → K-Ner(V) is an equivalence. Since equivalences are stable under
pullback along isofibrations, we conclude that:
Proposition 25. The comparison Modc(T ) → Mod(T ) in (5.2) is an equivalence.
Taking non-concrete models gives rise to a semantics functor landing in
V-CAT/K-Ner(V) which, like before, is not fully faithful on A-pretheories, but
is so on the subcategory of A-theories. Note that the “underlying K-nerve” of a
T -model is more natural than it might seem, being the special case of the functor
Mod(T ) → Mod(S) induced by a morphism of A-pretheories for which S is the
initial pretheory. However, in the following result, for simplicity, we view the
semantics functors for T -models as landing simply in V-CAT.
Theorem 26. The monad–theory equivalence (4.6) commutes with the non-concrete
semantics functors in the sense that we have natural transformations
ThA(E)op Ψop
GG
Mod 33
θks
MndA(E)op
Alg}}
MndA(E)op Φop
GG
Alg 33
¯θ CQ
ThA(E)op
Mod}}
V-CAT V-CAT
whose components are equivalences in V-CAT.
Proof. First postcompose the natural isomorphisms (5.1) with the forgetful
functor V-CAT/E → V-CAT. Then paste with the natural transformation
Modc ⇒ Mod: ThA(E)op → V-CAT coming from the previous proposition.
5.3. Local presentability and algebraic left adjoints. Next in this section, we
consider the categorical properties of the V-categories and V-functors in the
image of the semantics functors. We begin with the case of pretheories.
Proposition 27. (i) If J : A → T is an A-pretheory then Modc(T ) is locally
presentable and UT : Modc(T ) → E is a strictly monadic right adjoint.
(ii) If H : T → S is a map of A-pretheories, then H∗ : Modc(S) → Modc(T ) is
a strictly monadic right adjoint.
MONADS AND THEORIES 23
Proof. (i) follows from Lemma 5 and the description in (2.5) of Modc(T ) → E
as a pullback. For (ii), applying the standard cancellativity properties to the
pullbacks defining Modc(S) and Modc(T ) yields a pullback square
Modc(S)
PS
GG
H∗

[Sop, V]
[Hop,1]

Modc(T )
PT
GG [T op, V] .
Since [Hop, 1] is strictly monadic and PT is a right adjoint between locally
presentable categories, the result follows again from Lemma 5.
Composing with the equivalence Modc(T ) Mod(T ) of Proposition 25, this
result immediately implies the local presentability of the category Mod(T ) of
non-concrete models. Likewise, in the non-concrete setting, the analogue of
Proposition 27 remains true on replacing “strict monadicity” by “monadicity”
throughout. On the other hand, taken together with Proposition 23, it immediately
implies the corresponding result for nervous monads. We state this here as:
Proposition 28. (i) If T is an A-nervous monad then ET is locally presentable,
and UT : ET → E is a strictly monadic right adjoint.
(ii) If α: T → S is a map of A-nervous monads, then α∗ : ES → ET is a strictly
monadic right adjoint.
5.4. Algebraic colimits of monads and theories. To conclude this section, we
examine the interaction of the semantics functors with colimits. We begin with the
more-or-less classical case of the semantics functor for monads Alg: Mnd(E)op →
V-CAT/E.
In general, Mnd(E) need not be cocomplete. Indeed, when V = E = Set, it
does not even have all binary coproducts; see [5, Proposition 6.10]. However
many colimits of monads do exist, and an important point about these is that,
in the terminology of [16], they are algebraic. That is, they are sent to limits by
the semantics functor Alg: Mnd(E)op → V-CAT/E.
To prove this, we use the following lemma, which is a mild variant of the
standard result that right adjoints preserve limits.
Lemma 29. Let C be a complete (ordinary) category with a strongly generating
class of objects X and consider a functor U : A → C. If each x ∈ X admits a
reflection along U then U preserves any limits that exist in A.
Proof. As X is a strong generator, the functors C(x, –) with x ∈ X jointly reflect
isomorphisms, and so jointly reflect limits. Accordingly U preserves any limits
that are preserved by C(x, U–) for each x ∈ X. But each C(x, U–) is representable
and so preserves all limits; whence U preserves any limits that exist.
In the setting of Set-enriched categories the following result, expressing the
algebraicity of colimits of monads, is a special case of Proposition 26.3 of [16].
Proposition 30. Alg: Mnd(E)op → V-CAT/E preserves limits.
24 JOHN BOURKE AND RICHARD GARNER
Proof. The V-functors F : X → E with small domain form a strong generator
for V-CAT/E. Moreover, it is shown in [12, Theorem II.1.1] that each such F
has a reflection along Alg: Mnd(E)op → V-CAT/E given by its codensity monad
RanF (F): E → E. The result thus follows from Lemma 29.
We now adapt the above results concerning Mnd(E) to the cases of PrethA(E),
MndA(E) and ThA(E). In Theorem 38 below, we will see that these categories
are locally presentable; in particular, and by contrast with Mnd(E), they are
cocomplete. It is also not difficult to prove the cocompleteness directly.
Proposition 31. Each of the semantics functors Alg: MndA(E)op → V-CAT/E,
Modc : PrethA(E)op → V-CAT/E and Modc : ThA(E)op → V-CAT/E preserves
limits.
Proof. These three functors are isomorphic to the respective composites:
MndA(E)op inclop
−−−−→Mnd(E)op Alg
−−→ V-CAT/E(5.3)
PrethA(E)op Ψop
−−−→Mnd(E)op Alg
−−→ V-CAT/E(5.4)
ThA(E)op Ψop
−−−→Mnd(E)op Alg
−−→ V-CAT/E ;(5.5)
for (5.3) this is clear, while for (5.4) and (5.5) it follows from Proposition 23. The
common second functor in each composite is limit-preserving by Proposition 30,
while the first functor is limit-preserving in each case since it is the opposite
of a left adjoint functor—by Corollary 22, Theorem 6 and Theorem 19 (taken
together with Corollary 22) respectively.
We leave it to the reader to formulate this result also for non-concrete models.
6. The monad–theory correspondence in practice
In this section, we return to the examples of our general setting described
in Section 2.1, with the goal of describing as explicitly as possible what the
A-nervous monads, the A-theories, and the corresponding models look like in
each case. By way of these descriptions, we will re-find many of the monad–theory
correspondences existing in the literature as instances of our main Theorem 19.
To obtain our explicit descriptions, we will require some further results which
characterise A-theories and A-nervous monads in particular situations. We begin
this section by describing these results.
6.1. Theories in the presheaf context. A number of the examples of our basic
setting described in Section 3.1 arise in the following manner. We take E = [Cop, V]
a presheaf category, and take A to be any full subcategory of E containing the
representables. In this situation, we then have a factorisation
(6.1) C
I GG A
K GG [Cop, V] = E
of the Yoneda embedding. The Yoneda lemma implies that Y : C → [Cop, V] is
dense, whereupon by Theorem 5.13 of [17], both I and K are too. In particular,
K provides an instance of our basic setting; we will call this the presheaf context.
Each of Examples 8(i), (iv), (v), (vi), and (viii) arise in this way.
MONADS AND THEORIES 25
Lemma 32. In the presheaf context, we have NI
∼= K and NK
∼= RanIop .
Moreover, a functor F : Aop → V is a K-nerve just when it is the right Kan
extension of its restriction along Iop : Cop → Aop.
Proof. For the first isomorphism we calculate that
(6.2) NI(x) = A(I–, x) ∼= [Cop
, V](KI–, Kx) = [Cop
, V](Y –, Kx) ∼= Kx
by full fidelity of K and the Yoneda lemma. For the second, since LanY K NK
and [Iop, 1] RanIop it suffices to show LanY K ∼= [Iop, 1]: [Aop, V] → [Cop, V].
Since both are cocontinuous, it suffices to show (LanY K)Y ∼= [Iop, 1]Y , which
follows since (LanY K)Y ∼= K ∼= NI = [Iop, 1]Y using full fidelity of Y and (6.2).
Finally, since Iop is fully faithful, F : Aop → V is a right Kan extension along
Iop just when it is the right Kan extension of its own restriction. Thus the final
claim follows using the isomorphism NK
∼= RanIop .
In this setting, we have practically useful characterisations of the A-theories
and their (non-concrete) models.
Proposition 33. Let J : A → T be an A-pretheory in the presheaf context (6.1).
(i) A functor F : T op → V is a T -model just when FJop : Aop → V is the right
Kan extension of its restriction along Iop : Cop → Aop;
(ii) J : A → T is itself an A-theory just when it is the pointwise left Kan
extension of its restriction along I : C → A.
Proof. (i) follows immediately from Lemma 32 since, by definition, F is a T model
just when FJop is a K-nerve. For (ii), note that by Proposition 4.46
of [17], J : A → T is the pointwise left Kan extension of its restriction along
I just when, for each x ∈ T , the functor T (J–, x): Aop → V is the right Kan
extension of its restriction along Iop. By Lemma 32, this happens just when each
T (J–, x) is a K-nerve—that is, just when J is a A-theory.
We can sharpen these results using Day’s notion of density presentation [11].
The density of an ordinary functor K : C → D is often introduced as the assertion
that each object of D is the colimit of a certain diagram in the image of K. It is
this perspective that the notion of density presentation generalises.
A family of colimits Φ in the ordinary category D is a class of diagrams
(Di : Ji → D)i∈I each of which has a colimit in D. In the enriched case, a family
of colimits Φ in the V-category D is a class of pairs (Wi ∈ [J op
i , V], Di : Ji → D)i∈I
such that each weighted colimit Wi Di exists in D. In either case, a full replete
subcategory B of D is closed in D under Φ-colimits if it contains the (weighted)
colimit of any Di in Φ whenever it contains each vertex of Di. We say that D
is the closure of B under Φ-colimits if the only replete full subcategory of D
containing B and closed under Φ-colimits is D itself.
Now given a fully faithful K : C → D, we say that a colimit in D is K-absolute
if it is preserved by NK, or equivalently, by each representable D(Kx, –): D → V.
If D is the closure of C under a family Φ of K-absolute colimits then Φ is said to
be a density presentation for K. The nomenclature is justified by Theorem 5.19
of [17], which, among other things, says that the fully faithful K has a density
presentation just when it is dense.
26 JOHN BOURKE AND RICHARD GARNER
We will make use of density presentations in the presheaf context (6.1) with
respect not to the dense K, but to the dense I. By Lemma 32 we have NI
∼= K,
and so the I-absolute colimits are in this case those preserved by K : A → E.
We will see numerous instances of this situation in Section 6.3 below; we give a
couple of examples now to clarify the ideas.
Examples 34.
(i) Example 8(i) corresponds to the presheaf context
1
I GG F
K GG Set ,
and here I has a density presentation given by all finite copowers of 1 ∈ F;
these are I-absolute since K preserves them. In fact, F has all finite coproducts
and these are preserved by K, so that there is a larger density
presentation given by all finite coproducts in F.
(ii) Example 8(iv) yields the presheaf context below, wherein I has a density
presentation given by the wide pushouts [n] ∼= [1] +[0] [1] +[0] . . . +[0] [1]:
G1
I GG ∆0
K GG [G1
op
, Set] .
The reason we care about density presentations is the following result, which
comprises various parts of Theorem 5.29 of [17].
Proposition 35. Let K : C → D be fully faithful and dense. The following are
equivalent:
(i) F : D → E is the pointwise left Kan extension of its restriction along K;
(ii) F sends Φ-colimits to colimits for any density presentation Φ of K;
(iii) F sends K-absolute colimits to colimits.
Combined with Proposition 33, this yields the desired sharper characterisation
of the A-theories and their models.
Theorem 36. Let J : A → T be an A-pretheory in the presheaf context (6.1), and
let Φ be a density presentation for I.
(i) A functor F : T op → V is a T -model just when FJop : Aop → V sends
Φ-colimits in A to limits in V;
(ii) J : A → T is an A-theory just when it sends Φ-colimits to colimits.
6.2. Nervous monads, signatures and saturated classes. We now turn from characterisations
for A-theories to characterisations for A-nervous monads. We know
from Corollary 21 that a monad is A-nervous just when it is isomorphic to ΨT
for some A-pretheory J : A → T , and the examples in Section 3 make it an
intuitively reasonable idea that these are the monads which can be “presented
by operations and equations with arities from A”.
Our first characterisation result makes this idea precise by exhibiting the
category of A-nervous monads as monadic over a category of signatures. We
defer the proof of this result to Section 8.
Definition 37. The category SigA(E) of signatures is the category V-CAT(ob A, E).
We write V : Mnd(E) → SigA(E) for the functor sending T to (Ta)a∈A.
MONADS AND THEORIES 27
Theorem 38. V : Mnd(E) → SigA(E) has a left adjoint F : SigA(E) → Mnd(E)
taking values in A-nervous monads. Moreover:
(i) The restricted functor V : MndA(E) → SigA(E) is monadic;
(ii) A monad T ∈ Mnd(E) is A-nervous if and only if it is a colimit in Mnd(E)
of monads in the image of F;
(iii) Each of MndA(E), PrethA(E) and ThA(E) is locally presentable.
The idea behind this result originates in [20]. A signature Σ ∈ SigA(E) specifies
for each a ∈ A an E-object Σa of “operations of input arity a”. The free monad
FΣ on this signature has as its algebras the Σ-structures: objects X ∈ E endowed
with a function E(a, X) → E(Σa, X) for each a ∈ A. The above result implies
that a monad T ∈ Mnd(E) is A-nervous just when it admits a presentation as a
coequaliser FΓ FΣ T—that is, a presentation by a signature Σ of basic
operations together with a family Γ of equations between derived operations.
We now turn to our second characterisation result for A-nervous monads. This
is motivated by the fact, noted in the introduction, that in many monad–theory
correspondences the class of monads can be characterised by a colimit-preservation
property. To reproduce this result in our setting, we require a closure property of
the arities in the subcategory A which, roughly speaking, says that substituting
A-ary operations into A-ary operations again yields A-ary operations.
Definition 39. An endo-V-functor F : E → E is called A-induced if it is the
pointwise left Kan extension of its restriction along K. We call A a saturated
class of arities if A-induced endofunctors of E are closed under composition.
Example 40. In the case of K : F → Set, there is a density presentation for K
given by all filtered colimits in Set, so that by Proposition 35, an endofunctor
Set → Set is F-induced just when it preserves filtered colimits. Thus F → Set is
a saturated class of arities.
Example 41. More generally, if Φ is a class of enriched colimit-types and K : A → E
exhibits E as the free cocompletion of A under Φ-colimits, then there is a density
presentation of K given by all Φ-colimits, and an endofunctor of E is K-induced
just when it preserves Φ-colimits. Thus A is a saturated class of arities.
Example 42. Let K : A → Set be the inclusion of the one-object full subcategory
A on the two-element set 2 = {0, 1}. Since the dense generator 1 of Set is a
retract of 2, and taking retracts does not change categories of presheaves, A is
dense in Set. We claim it does not give a saturated class of arities.
To see this, note first that (–)2 : Set → Set is A-induced, being a left Kan
extension along K of the representable A(2, –): A → Set. We claim that (–)2◦(–)2
is not A-induced. For indeed, by the Yoneda lemma, any X ∈ [A, Set] has an
epimorphic cover by copies of the unique representable A(2, –). Since left Kan
extension preserves epimorphisms, each LanK(X) admits an epimorphic cover by
copies of (–)2. But (–)2 ◦ (–)2 ∼= (–)4 can admit no such cover, since the identity
map on 4 does not factor through 2, and so cannot be A-induced.
The proof of the following result will again be deferred to Section 8 below.
28 JOHN BOURKE AND RICHARD GARNER
Theorem 43. Let A be a saturated class of arities in E. The following are
equivalent properties of a monad T ∈ Mnd(E):
(i) T is A-nervous;
(ii) T : E → E is A-induced;
(iii) T : E → E preserves Φ-colimits for any density presentation Φ of K.
6.3. The monad–theory equivalence in practice. We now apply our characterisation
results to the examples of Section 2.1. In many cases, the explicit descriptions
we obtain of the A-nervous monads, the A-theories, and their models will allow
us to reconstruct a familiar monad–theory correspondence from the literature.
Examples 44. As before, we begin with the unenriched examples where V = Set.
(i) The case E = Set and A = F corresponds to the instance of the presheaf
context described in Examples 34(i). Applied to the density presentations
for I given there, Theorem 36 tells us that an F-pretheory J : F → T is an Ftheory
just when it preserves finite copowers of 1, or equally (using the larger
density presentation) all finite coproducts. It thus follows that the F-theories
are the Lawvere theories of [24]. Moreover a functor F : T op → Set is a T model
if and only if FJop : Fop → Set preserves finite products. Since, in this
case, J also reflects finite coproducts, this happens just when F : T op → Set
is itself finite-product-preserving, that is, just when F is a model of the
Lawvere theory T .
On the other hand, by Example 40, F is a saturated class of arities, and the
F-induced endofunctors are the finitary ones; so by Theorem 43, a monad on
Set is F-nervous just when it is finitary. Theorem 19 thus specialises to the
classical finitary monad–Lawvere theory correspondence, while Theorem 26
recaptures its compatibility with semantics.
(ii) When E is locally finitely presentable and A = Ef , the category of K-nerves
is, by [13, Kollar 7.9], the full subcategory of [Eop
f , Set] on the finite-limitpreserving
functors. So an Ef -pretheory J : Ef → T is an Ef -theory just when
each T (J–, a): Ef
op
→ Set preserves finite limits. By the Yoneda lemma,
this happens just when J preserves finite colimits, so that the Ef -theories
are precisely [31]’s Lawvere E-theories.
The concrete T -models in this setting are exactly the models of [31,
Definition 2.2]. The general T -models are those functors F : T op → Set
for which FJop : Eop
f → Set is a K-nerve, i.e., finite-limit-preserving; these
are the more general models of [22, Definition 12], and the correspondence
between the two notions in Proposition 25 recaptures Proposition 15 of ibid.
On the monad side, since K : Ef → E exhibits E as the free filtered-colimit
completion of Ef , Example 41 and Theorem 43 imply that Ef is a saturated
class, and that the Ef -nervous monads are the finitary ones. So in this case,
Theorem 19 and Corollary 24 reconstruct (the unenriched version of) the
monad–theory correspondence given in [31, Theorem 5.2].
(iii) More generally, when E is locally λ-presentable and A = Eλ is a skeleton
of the full subcategory of λ-presentable objects, the Eλ-theories are those
pretheories J : Eλ → T which preserve λ-small colimits; the T -models are
MONADS AND THEORIES 29
functors F : T op → Set for which FJop preserves λ-small limits; and the
Eλ-nervous monads are those whose endofunctor preserves λ-filtered colimits.
(iv) When E = [Gop
1 , Set] and A = ∆0, we are in the presheaf context of Examples
34(ii). For the density presentation for I given there, Theorem 36
tells us that a pretheory J : ∆0 → T is a ∆0-theory just when it preserves
the wide pushouts [n] ∼= [1] +[0] [1] +[0] . . . +[0] [1]. Moreover, a functor
X : T op → Set is a T -model just when it sends each of these wide pushouts
to a limit in Set. This is precisely the Segal condition of [33]; in elementary
terms, it requires the invertibility of each canonical map
(6.3) Xn −→ X1 ×X0 X1 ×X0 · · · ×X0 X1 .
In Corollary 49 below we will see that ∆0 is not a saturated class of arities,
and so we have no more direct characterisations of the ∆0-nervous monads
than is given by Corollary 21 or Theorem 38. However, Example 11 provides
us with natural examples of ∆0-nervous monads: namely, the monads T
and Tg for categories and for groupoids on [Gop
1 , Set]. As was already noted
in [35], the nervosity of T recaptures the classical nerve theorem relating
categories and simplicial sets. Indeed, the ∆0-theory associated to T is the
first part of the (bijective-on-objects, fully faithful) factorisation
∆0
JT
−−→ ∆
KT
−−→ Cat
of the composite FTK : ∆0 → Cat. The interposing object here is the
topologist’s simplex category ∆, with KT the standard inclusion into Cat.
Thus, to say that T is ∆0-nervous is to say that:
(a) The classical nerve functor NKT
: Cat → [∆op, Set] is fully faithful;
(b) The essential image of NKT
comprises those X ∈ [∆op, Set] for which
XJT is a K-nerve.
This much is already done in [35], but our use of density presentations allows
for a small improvement. To say that XJT is a K-nerve in (b) is equally
to say that X is a T -model, or equally that X satisfies the Segal condition
expressed by the invertibility of each (6.3). This is a mild sharpening of [35],
where the “Segal condition” is left in the abstract form given in (b) above.
In a similar way, the nervosity of the monad Tg for small groupoids
captures the “symmetric nerve theorem”. This states that the functor
Gpd → [Fop
+ , Set] sending a groupoid to its symmetric nerve—indexed by
the category of non-empty finite sets—is fully faithful, and characterises the
essential image once again as the functors satisfying the Segal condition (6.3).
(v) With E = [Gop, Set] and A = Θ0, we are now in the presheaf context
G
I GG Θ0
K GG [Gop, Set] .
I has a density presentation given by the I-globular sums (n1, . . . , nk) ∼=
(n1) +(n2) +(n3) + . . . +(nk−1) (nk) in Θ0; whence by Theorem 36, a pretheory
J : Θ0 → T is a Θ0-theory when it preserves these I-globular sums—that is,
30 JOHN BOURKE AND RICHARD GARNER
when it is a globular theory in the sense of [8]2
. A functor F : T op → Set is
a T -model when it sends I-globular sums to limits, thus when each map
Xn −→ Xn1 ×Xn2 Xn3 ×Xn4 . . . ×Xnk−1
Xnk
is invertible. Once again, Θ0 is not a saturated class of arities, and so
there is no direct characterisation of the Θ0-nervous monads; however,
their interaction with Θ0-theories is important in the literature on globular
approaches to higher category theory, as we now outline.
Globular theories can describe structures on globular sets such as strict
or weak ω-categories and ω-groupoids. For the strict variants, we pointed
out in Section 3.2 that these may be modelled by Θ0-pretheories; and since
reflecting a pretheory T into a theory ΦΨT does not change the models, it
is immediate that there are Θ0-theories modelling these structures too.
The original definition of globular weak ω-category was given by Batanin
in [7]; he defines them be globular sets equipped with algebraic structure
controlled by a globular operad. Globular operads can be understood as
certain cartesian monads on globular sets. Berger [8] introduced globular
theories and described the passage from a globular operad T to a globular
theory ΘT just as in Section 2.4 above. In our language, his Theorem 1.17
states exactly that each globular operad T is Θ0-nervous, so that algebras for
the globular operad are the same as models of the associated theory ΘT . In
particular, Batanin’s weak ω-categories are the models of a globular theory3
.
On the other hand, Grothendieck weak ω-groupoids [27] are, by definition,
models for certain globular theories called coherators.
We now proceed to our examples over a more general base for enrichment V.
(vi) With V = E a locally finitely presentable symmetric monoidal category and
with A = Vf , we are in the presheaf context
I
I GG Vf
K GG V ,
wherein I has a density presentation given by the class of all finite tensors—
tensors by finitely presentable objects of V. Thus by Theorem 36, the Vf theories
are the pretheories J : Vf → T which preserve finite tensors, which
are precisely the Lawvere V-theories of [32, Definition 3.1]. Furthermore,
like in (i), a functor F : T op → V is a T -model just when it preserves finite
cotensors, just as in Definition 3.2 of ibid. On the other hand, Vf → V
exhibits V as the free filtered-colimit completion of Vf ; whence by Example 41
it is a saturated class of arities, and by Theorem 43 the Vf -nervous monads
are again the finitary ones. So Theorems 19 and 26 specialise to Theorems 4.3,
3.4 and 4.2 of [32].
2The definition of globular theory in [8] has the extra condition, satisfied in most cases, that
J be a faithful functor.
3As an aside, we note that a complete understanding of those globular theories corresponding
to globular operads was obtained in Theorem 6.6.8 of [2]. See also Section 3.12 of [9].
MONADS AND THEORIES 31
(vii) Now taking E to be any locally finitely presentable presentable V-category
and A = Ef , we may argue as in (ii) to recapture the fully general enriched
monad–theory correspondence of [31], and its interaction with semantics.
(viii) Now suppose we are in the situation of Examples 8(viii), provided with a
class Φ of enriched colimit-types satisfying Axiom A of [23]. With E = V
and A = VΦ, we are now in the presheaf context
I
I GG VΦ
K GG V .
By [17, Theorem 5.35], I has a density presentation given by Φ-tensors (i.e.,
tensors by objects in Φ) while by [23, Theorem 3.1], K exhibits V as the
free Φ-flat cocompletion of VΦ. Arguing as in the preceding parts, we see
that VΦ-theories are pretheories J : Vop
Φ → T which preserve Φ-tensors, that
T -models are Φ-tensor-preserving functors F : T op → V, and that a monad
is VΦ-nervous if its underlying endofunctor preserves Φ-flat colimits. This
sharpens slightly the results obtained in [23] in the special case E = V.
(ix) Finally, in the situation of Examples 8(ix), we find that the A-theories
are the Φ-colimit preserving pretheories J : A → T ; that the T -models
are functors F : T op → V such that FJop preserves Φ-limits; and that a
monad is A-nervous just when it preserves Φ-flat colimits. In this way, our
Theorems 19 and 26 reconstruct Theorems 7.6 and 7.7 of [23].
7. Monads with arities and theories with arities
In the introduction, we mentioned the general framework for monad–theory
correspondences obtained in [35, 9]. Similar to this paper, the basic setting
involves a category E and a small, dense subcategory K : A → E; given these
data, one defines notions of monad with arities A and theory with arities A, and
proves an equivalence between the two that is compatible with semantics.
In this section, we compare this framework with ours by comparing the classes
of monads and of theories. We will see that our setting yields strictly larger
classes of monads and theories which are better-behaved in practically useful
ways. On the other hand, in the more restrictive setting of [35, 9], checking that
a monad or theory is in the required class may give greater combinatorial insight
into the structure which it describes.
7.1. Monads with arities versus nervous monads. In [35, 9] the authors work in
the unenriched setting; the introduction to [9] states that the results “should
be applicable” also in the enriched one. To ease the comparison to our results,
we take it for granted that this is true, and transcribe their framework into the
enriched context without further comment.
Another difference is that we assume local presentability of E while [35] assumes
only cocompleteness, and [9] not even that. Given a small dense subcategory,
there is no readily discernible difference between cocompleteness and local present-
ability4
; however, cocompleteness is substantively different from nothing, so that
4Indeed, if there were, then it would negate the large cardinal axiom known as Vopˇenka’s
principle [1, Chapter 6].
32 JOHN BOURKE AND RICHARD GARNER
in this respect [9]’s results are more general than ours. However all known
applications are in the context of a locally presentable E, and so we do not lose
much in restricting to this context. In conclusion, when we make our comparison
we will work in exactly the same general setting as in Section 2.1, and now have:
Definition 45. [35, Definition 4.1] An endofunctor T : E → E is said to have arities
A if the composite V-functor NKT : E → [Aop, V] is the left Kan extension of its
own restriction along K. A monad T ∈ Mnd(E) is a monad with arities A if its
underlying endofunctor has arities A.
We consider the following way of restating this to be illuminating.
Proposition 46. An endofunctor T : E → E has arities A if and only if it sends
K-absolute colimits to K-absolute colimits. In particular, each endofunctor with
arities A is A-induced.
Proof. By Proposition 35, T has arities A just when NKT sends K-absolute
colimits to colimits. Since NK is fully faithful, it reflects colimits, and so T has
arities A just when T sends K-absolute colimits to colimits which are preserved
by NK—that is, to K-absolute colimits.
For the second claim, recall from Definition 39 that an endofunctor T : E → E is
A-induced if it is the left Kan extension of its own restriction to A, or equivalently,
by Proposition 35, when it sends K-absolute colimits to colimits.
Recall also that we call a class of arities A saturated when A-induced endofunctors
are closed under composition. Example 42 shows that this condition
is not always satisfied. In light of the preceding result, the endofunctors with
arities A can be seen as a natural subclass of the A-induced endofunctors for
which composition-closure is always verified.
The reason that Weber introduced monads with arities was in order to prove
his nerve theorem [35, Theorem 4.10], which in our language may be restated as:
Theorem 47. Monads with arities A are A-nervous.
One may reasonably ask whether the classes of monads with arities and Anervous
monads in fact coincide. In many cases, this is true; in particular, in the
situation of Example 41, where K : A → E exhibits E as the free Φ-cocompletion
of A for some class of colimit-types Φ. Indeed, this condition implies that a
monad T is A-nervous precisely when T sends Φ-colimits to Φ-colimits; since
Φ-colimits are K-absolute, this in turn implies that NKT sends Φ-colimits to
colimits, and so is the left Kan extension of its own restriction along K. So in
this case, every A-nervous monad has arities A; so in particular, the two notions
coincide in each of Examples 8(i), (ii), (iii), (vi), (vii), (viii) and (ix).
However, they do not coincide in general. That is, in some instances of our
basic setting, there exist monads which are A-nervous but do not have arities A.
We give three examples of this. The first two arise in the setting of Example 8(iv),
and concern the monads for groupoids and involutive graphs respectively.
Proposition 48. The monad T on Grph := [Gop
1 , Set] whose algebras are groupoids
is ∆0-nervous but does not have ∆0-induced underlying endofunctor. It follows
that T does not have arities ∆0.
MONADS AND THEORIES 33
Proof. From Example 11 we know that T is ∆0-nervous. To see that T is not
∆0-induced, consider the graph X with vertices and arrows as to the left in:
(7.1) a
r
−→ b
s
←− c
[0]
τ GG
τ

[1]
s

[1]
r GG X .
This X is equally the K-absolute pushout right above; so if T were ∆0-induced
then it would preserve this pushout. But T[1] +T[0] T[1] is the graph
a1a WW r
GG b
r−1
oo
oo
s
1b
ÖÖ
c
GGs−1
1cdd
wherein, in particular, there is no edge a → c; while in TX we have s−1 ◦r: a → c.
So the pushout is not preserved. This shows that T is not ∆0-induced and so, by
Proposition 46, that T does not have arities ∆0.
Since the above result exhibits a ∆0-nervous monad whose underlying endofunctor
is not ∆0-induced, we can apply Theorem 43 to deduce:
Corollary 49. K : ∆0 → Grph is not a saturated class of arities.
Our second example, originally due to Melli`es [29, Appendix III], shows that
even monads with ∆0-induced endofunctor need not have arities ∆0. In this
example, we call a graph s, t: X1 X0 involutive if it comes endowed with an
order-2 automorphism i: X1 → X1 reversing source and target, i.e., with si = t
(and hence also ti = s).
Proposition 50. The monad T on Grph := [Gop
1 , Set] whose algebras are involutive
graphs is ∆0-nervous and has ∆0-induced underlying endofunctor, but does not
have arities ∆0.
Proof. The value of T at s, t: X1 X0 is given by s, t , t, s : X1 +X1 X0. It
follows that T is cocontinuous and so certainly ∆0-induced. To see it does not have
arities ∆0, consider again the graph (7.1) and its K-absolute pushout presentation.
If this were preserved by NKT : Grph → [∆op
0 , Set] then, on evaluating at [2],
the maps Grph([2], T[1]) Grph([2], TX) given by postcomposition with Tr
and Ts would be jointly surjective. To show this is not so, consider the map
f : [2] → TX picking out the composable pair (r: a → b, i(s): b → c). Since
neither Tr nor Ts are surjective on objects, the bijective-on-objects f cannot
factor through either of them. This shows that T does not have arities ∆0.
Our final example shows that not even free monads on A-signatures—which
are A-nervous by Theorem 38 above—need necessarily have arities A.
Proposition 51. Let V = E = Set and let A be the one-object full subcategory on
a two-element set. The free monad on the terminal A-signature does not have
A-induced underlying endofunctor and therefore does not have arities A.
34 JOHN BOURKE AND RICHARD GARNER
Proof. The algebras for the free monad T on the terminal signature are sets
equipped with a binary operation. Elements of the free T-algebra on X are
binary trees with leaves labelled by elements of X, yielding the formula
TX = n∈N Cn × Xn+1
where Cn is the nth Catalan number. In particular, T contains at least one
coproduct summand (–)4 and so, as in Example 42, is not A-induced; in particular,
by Proposition 46, it does not have arities A.
7.2. Theories with arities A versus A-theories. The paper [9] introduced theories
with arities A. These are A-pretheories J : A → T for which the composite
(7.2) [Aop
, V]
LanJ
−−−→ [T op
, V]
[Jop,1]
−−−−→ [Aop
, V]
takes K-nerves to K-nerves. This functor takes the representable A(–, x) to
T (J–, x), so that in this language, we may describe the A-theories as the pretheories
for which (7.2) takes each representable to a K-nerve. It follows that:
Proposition 52. Theories with arities A are A-theories.
Proof. It suffices to observe that each representable A(–, x) is a K-nerve since
A(–, x) ∼= E(K–, Kx) = NK(Kx).
Theorem 3.4 of [9] establishes an equivalence between the categories of monads
with arities A and of theories with arities A. The functor taking a monad with
arities to the corresponding theory with arities is defined in the same way as
the Φ of Section 2.4, and so it follows that:
Proposition 53. The equivalence of monads with arities A and theories with
arities A is a restriction of the equivalence between A-nervous monads and
A-theories.
In particular, there exist A-theories which are not theories with arities A; it is
this statement which was verified in in [29, Appendix III].
7.3. Colimits of monads with arities. In Theorem 38 we saw that the A-nervous
monads are the closure of the free monads on A-signatures under colimits in
Mnd(E). Since colimits of monads are algebraic, this allows us to give intuitive
presentations for A-nervous monads as suitable colimits of frees. The pretheory
presentations of Section 3 can be understood as particularly direct descriptions
of such colimits.
Since not every A-nervous monad has arities A, the monads with arities are
not the colimit-closure of the frees on signatures. We already saw one explanation
for this in Proposition 51: the free monads on signatures need not have arities.
However, this leaves open the possibility that the monads with arities A are the
colimit-closure of some smaller class of basic monads—which would allow for
the same kind of intuitive presentation as we have for A-nervous monads. The
following result shows that even this is not the case.
Theorem 54. Monads with arities A need not be closed in Mnd(E) under colimits.
MONADS AND THEORIES 35
Proof. We saw in Proposition 50 that, when E = Grph and A = ∆0, the monad
T for involutive graphs does not have arities ∆0. To prove the result it will
therefore suffice to exhibit T as a colimit in Mnd(Grph) of a diagram of monads
with arities ∆0. This diagram will be a coequaliser involving a pair of monads P
and Q, whose respective algebras are:
• For P: graphs X endowed with a function u: X1 → X0;
• For Q: graphs X endowed with an order-2 automorphism i: X1 → X1.
We construct this coequaliser of monads in terms of the categories of algebras.
The category GrphT
of involutive graphs is an equaliser in CAT as to the left in:
GrphT GG E GG GrphQ
F GG
G
GG GrphP
P
ϕ
GG
γ
GG Q
ε GG GG T
where the functors F and G send a Q-algebra (X, i) to the respective P-algebras
(X, si) and (X, t). Since each of these functors commutes with the the forgetful
functors to Grph, we have an equaliser of forgetful functors in CAT/Grph. Since
the functor Alg: Mnd(Grph)op → CAT/Grph is fully faithful, this equaliser must
be the image of a coequaliser diagram in Mnd(Grph) as right above.
It remains to show that in this coequaliser presentation both P and Q
have arities ∆0. By Proposition 35, this means showing that NKP and NKQ
send K-absolute colimits to colimits, or equally, that each Grph([n], P–) and
Grph([n], Q–) sends K-absolute colimits to colimits. To see this, we calculate P
and Q explicitly. On the one hand, the free P-algebra on a graph X is obtained
by freely adjoining an element u(f) to X0 for each f ∈ X1. On the other hand,
the free Q-algebra on X is obtained by freely adjoining an element i(f) ∈ X1 for
each f ∈ X1. Thus we have
PX = X + X1 · [0] and QX = X + X1 · [1]
where we use · to denote copower. Since each [n] ∈ Grph is connected, and since
each hom-set Gph([n], [m]) has cardinality max(0, m − n + 1), we conclude that
(7.3)
Grph([n], PX) =
Grph([0], X) + Grph([1], X) if n = 0;
Grph([n], X) if n > 0.
Grph([n], QX) =



Grph([0], X) + 2 · Grph([1], X) if n = 0;
Grph([1], X) + Grph([1], X) if n = 1;
Grph([n], X) if n > 1.
Now by definition, NK sends K-absolute colimits to colimits, whence also
each Grph([n], –): Grph → Set. The functors with this property are closed
under colimits in [Grph, Set], and so (7.3) ensures that each Grph([n], P–) and
Grph([n], Q–) sends K-absolute colimits to colimits as desired.
It is not even clear to us if the category of monads with arities A is always
cocomplete. The argument for local presentability of MndA(E) in Theorem 38
does not seem to adapt to the case of monads with arities, and no other obvious
argument presents itself. In any case, the preceding result shows that, even
if the category of monads with arities does have colimits, they do not always
36 JOHN BOURKE AND RICHARD GARNER
coincide with the usual colimits of monads, and, in particular, are not always
algebraic. This dashes any hope we might have had of giving a sensible notion of
presentation for monads with arities.
8. Deferred proofs
8.1. Identifying the monads. In this section, we complete the proofs of the results
deferred from Section 6 above, beginning with Theorem 38. Recall that the
category SigA(E) of signatures is the (ordinary) category V-CAT(ob A, E), and
that V : Mnd(E) → SigA(E) is the functor sending T to (Ta)a∈A.
Proposition 55. V : Mnd(E) → SigA(E) has a left adjoint F which takes values
in A-nervous monads.
Proof. We can decompose V as the composite
Mnd(E)
V1
−−→ V-CAT(E, E)
V2
−−→ SigA(E)
where V1 takes the underlying endofunctor, and V2 is given by evaluation at each
a ∈ ob A. Since V2 is equally given by restriction along ob A → A → E, it has a
left adjoint F2 given by pointwise left Kan extension, with the explicit formula:
F2(Σ) = a∈A E(Ka, –) · Σa: E → E ,
where · denotes V-enriched copower. So it suffices to show that the free monad on
each endofunctor F2(Σ) exists and is A-nervous. By [16, Theorem 23.2], such a
free monad T is characterised by the property that EF(Σ) ∼= ET over E, where on
the left we have the V-category of algebras for the mere endofunctor F2(Σ). Thus,
to complete the proof, it suffices by Theorem 6 to exhibit EF(Σ) as isomorphic
to the V-category of concrete models of some A-pretheory.
To this end, we let B be the collage of the V-functor NKΣ: ob A → [Aop, V].
Thus B is the V-category with object set ob A + ob A and the following homobjects,
where we write , r: ob A → ob B for the two injections:
B( a , a) = A(a , a) B(ra , ra) = (ob A)(a , a)
B( a , ra) = E(Ka , Σa) B(ra , a) = 0 .
Let : A → B and r: ob A → B be the two injections into the collage, and now
form the pushout J : A → T of , r : A + ob A → B along 1, ι : A + ob A → A.
Since , r is identity-on-objects, so is J : A → T , and so we have an A-pretheory.
To conclude the proof, it now suffices to show that EF2(Σ) ∼= Modc(T ) over E.
By the universal property of the collage and the pushout, to give a functor
H : T → X is equally to give a functor F = HJ : A → X together with V-natural
transformations αa : E(K–, Σa) ⇒ X(F–, Fa) for each a ∈ ob A. In particular,
taking X = Vop and F = E(K–, X), we see that a concrete T -model structure
on X ∈ E is given by an ob A-indexed family of V-natural transformations
αa : E(K–, Σa) ⇒ [E(Ka, X), E(K–, X)]
or equally under transpose, by a family of maps
E(Ka, X) → [Aop
, V](E(K–, Σa), E(K–, X)) .
MONADS AND THEORIES 37
By full fidelity of NK, the right-hand side above is isomorphic to E(Σa, X),
and so concrete T -model structure on X is equally given by a family of maps
E(Ka, X) → E(Σa, X). Finally, using the universal properties of copowers and
coproducts, this is equivalent to giving a single map
¯α: a∈A E(Ka, X) · Σa → X
exhibiting X as an F2(Σ)-algebra. We thus have a bijection over E between
objects of EF(Σ) and objects of Modc(T ).
A similar analysis shows that a morphism A → E(X, Y ) in V lifts through the
monomorphism Modc(T )((X, α), (Y, β)) → E(X, Y ) if and only if it lifts through
the monomorphism EF2(Σ)((X, ¯α), (Y, ¯β)) → E(X, Y ). It follows that we have an
isomorphism of V-categories EF(Σ) ∼= Modc(T ) over E as desired.
In proving the rest of Theorem 38, the following lemma will be useful.
Lemma 56. Let C1 ⊆ C2 be replete, full, colimit-closed sub-V-categories of C; for
example, they could be coreflective. If V : C → D has a left adjoint F taking
values in C1, and the restriction V |C2
: C2 → D is monadic, then C1 = C2.
Proof. Since F takes values in C1 ⊆ C2, the left adjoint to V |C2
: C1 → D is still
given by F. So monadicity of V |C2
means that each X ∈ C2 can be written as
a coequaliser in C2, and hence also in C, of objects in the image of F. Since
im F ⊆ C1 and since C1 is closed in C under colimits, it follows that X ∈ C1.
Theorem 38. V : Mnd(E) → SigA(E) has a left adjoint F : SigA(E) → Mnd(E)
taking values in A-nervous monads. Moreover:
(i) The restricted functor V : MndA(E) → SigA(E) is monadic;
(ii) A monad T ∈ Mnd(E) is A-nervous if and only if it is a colimit in Mnd(E)
of monads in the image of F;
(iii) Each of MndA(E), PrethA(E) and ThA(E) is locally presentable.
Proof. We begin with (i). Let H : PrethA(E) → V-CAT(ob A, [Aop, V]) be the
functor sending a pretheory J : A → T to the family of presheaves (T (J−, Ja))a∈A.
Since an A-pretheory is a theory just when each of these presheaves is a K-nerve,
we have a pullback square as to the right in:
(8.1)
MndA(E)
Φ GG
V

∼=
ThA(E)
P

GG PrethA(E)
H

V-CAT(ob A, E)
NK ◦(–)
GG V-CAT(ob A, K-Ner) GG V-CAT(ob A, [Aop, V]) .
Since K-Ner → [Aop, V] is replete, this square is a pullback along a discrete
isofibration, and so by [14, Corollary 1] also a bipullback. On the other hand, to
the left, we have a pseudocommuting square as witnessed by the isomorphisms:
(PJT)(A) = AT(JT–, JTA) = ET
(FT
K–, FT
KA) ∼= E(K–, TKA) = NK(TKA) .
Since both horizontal edges of this square are equivalences, it is also a bipullback.
To show the required monadicity, we must prove that V creates V -absolute
coequalisers. Since the large rectangle is a bipullback—as the pasting of two
38 JOHN BOURKE AND RICHARD GARNER
bipullbacks—it suffices to show that H creates H-absolute coequalizers. As the
definition of H depends only on A and not E, we lose no generality in proving
this if we assume that E = [Aop, V] and K = Y . In this case, every presheaf on
A is a K-nerve, and so the horizontal composites in (8.1) are equivalences; and
so, finally, it suffices to prove that V is monadic when E = [Aop, V] and K = Y .
Note that, in this case, A is a saturated class of arities: for indeed, by the universal
property of free cocompletion, a functor F : [Aop, V] → [Aop, V] is A-induced
if and only if it is cocontinuous. It thus follows from Proposition 58 below that the
restriction Vc : Mndc(E) → SigA(E) of V to cocontinuous monads is monadic; so
we will be done if Mndc(E) = MndA(E). In this case, Ψ: PrethA(E) → MndA(E)
sends J : A → T to a monad which is isomorphic to that induced by the adjunction
LanJ : [Aop, V] [T op, V]: [Jop, 1], and so MndA(E) ⊆ Mndc(E). To obtain
equality, we apply Lemma 56. We have that:
• MndA(E) and Mndc(E) are coreflective in Mnd(E) by Corollary 22 and
Lemma 57 respectively;
• V : Mnd(E) → SigA(E) has a left adjoint taking values in MndA(E);
• The restriction Vc : Mndc(E) → SigA(E) is monadic;
and so MndA(E) = Mndc(E). This proves monadicity of V in the special case
E = [Aop, V], whence also, by the preceding argument, in the general case.
In order to prove (ii), we let C1 be the colimit-closure in Mnd(E) of the
image of F. Since MndA(E) contains this image and is colimit-closed, we
have C1 ⊆ MndA(E) ⊆ Mnd(E). Thus, applying Lemma 56 to this triple and
V : MndA(E) → SigA(E) gives MndA(E) = C1 as desired.
Finally we prove (iii). The monadicity of V above implies that of P and
hence also of H (by taking E = [Aop, V]). Since filtered colimits of A-pretheories
can be computed at the level of underlying graphs, the forgetful H preserves
them; which is to say that PrethA(E) is finitarily monadic over the locally
presentable V-CAT(ob A, [Aop, V]), whence locally presentable by [13, Satz 10.3].
So in the right-hand and the large bipullback squares in (8.1), the bottom and
right sides are right adjoints between locally presentable categories. Since by [10,
Theorem 2.18], the 2-category of locally presentable categories and right adjoint
functors is closed under bilimits in CAT, we conclude that each ThA(E) and each
MndA(E) is also locally presentable.
8.2. Saturated classes. We now turn to the deferred proof of Theorem 43. Recall
the context: an endo-V-functor F : E → E is called A-induced when the pointwise
left Kan extension of its restriction along K, and A is a saturated class of arities
if A-induced endofunctors of E are composition-closed.
We begin by recording the basic properties of this situation. We write A-End(E)
and A-Mnd(E) for the full subcategories of End(E) = V-CAT(E, E) and Mnd(E)
on, respectively, the A-induced endofunctors, and the monads with A-induced
underlying endofunctor.
MONADS AND THEORIES 39
Lemma 57. A-End(E) is coreflective in End(E) = V-CAT(E, E) via the coreflector
R(F) = LanK(FK), as on the left in:
(8.2) A-End(E)
I
GG End(E)
Roo
A-Mnd(E)
I
GG Mnd(E) .
Roo
If A is a saturated class, then A-End(E) is right-closed monoidal, and the coreflection
left above lifts to the corresponding categories of monads as on the right.
Proof. Restriction and left Kan extension along the fully faithful K exhibits
A-End(E) as equivalent to V-CAT(A, E), whence locally presentable. Since
restriction along K is a coreflector of End(E) into V-CAT(A, E), it follows that
R(F) = LanK(FK) is a coreflector of End(E) into A-End(E).
If A is saturated then A-End(E) is monoidal under composition. Since each
endofunctor (–) ◦ F of End(E) is cocontinuous, and A-End(E) is closed in End(E)
under colimits, each endofunctor (–) ◦ F of A-End(E) is cocontinuous, and so has
a right adjoint by local presentability. Thus A-End(E) is right-closed monoidal.
Furthermore, the inclusion of A-End(E) into End(E) is strict monoidal, whence
by [15, Theorem 1.5] the coreflection to the left of (8.2) lifts to a coreflection in the
2-category MONCAT of monoidal categories, lax monoidal functors and monoidal
transformations. Applying the 2-functor MONCAT(1, –): MONCAT → CAT
yields the coreflection to the right of (8.2).
The key step towards establishing Theorem 43 above is now:
Proposition 58. The left adjoint F of V : Mnd(E) → SigA(E) takes values in
A-induced monads; furthermore, the restriction of V to A-Mnd(E) is monadic.
Proof. For any T ∈ Mnd(E), its A-induced coreflection εT : IR(T) → T has as
underlying map in End(E) the component LanK(TK) → T of the counit of
the adjunction given by restriction and left Kan extension along K. Since K
is fully faithful, the restriction of this map along K is invertible, whence in
particular, V ε: V IR ⇒ V : Mnd(E) → SigA(E) is invertible. So η: id ⇒ V F
factors through V εF : V IRF ⇒ V F whence, by adjointness, id: F ⇒ F factors
through εF . Therefore each F(Σ) is a retract of IRF(Σ); since A-Mnd(E) is
closed under colimits in Mnd(E), it is retract-closed and so each F(Σ) belongs
to A-Mnd(E).
It remains to prove that the restriction of V to A-Mnd(E) is monadic. To do
so, we decompose this restriction as
A-Mnd(E)
V1
−−→ A-End(E)
V2
−−→ SigA(E) ,
where V1 forgets the monad structure and V2 is given by precomposition with
ob A → A → E, and apply the following result, which is [21, Theorem 2]:
Theorem. Let M be a right-closed monoidal category, and V2 : M → N a monadic
functor for which there exists a functor : M×N → N with natural isomorphisms
X V Y ∼= V (X ⊗ Y ). If the forgetful functor V1 : Mon(M) → M has a left
adjoint, then the composite V2V1 : Mon(M) → N is monadic.
40 JOHN BOURKE AND RICHARD GARNER
Indeed, by Lemma 57, A-End(E) is a right-closed monoidal category, and
A-Mnd(E) the category of monoids therein. Under the equivalence A-End(E)
V-CAT(A, E), we may identify V2 with precomposition along ob A → A. It is
thus cocontinuous, and has a left adjoint given by left Kan extension; whence is
monadic. Now since V2V1 has a left adjoint and V2 is monadic, it follows that V1
also has a left adjoint. Finally, we have a functor
: A-End(E) × SigA(E) → SigA(E)
defined by (F, G) → FG, and this clearly has the property that M(FG) =
F M(G). So applying the above theorem yields the desired monadicity.
We are now ready to prove:
Theorem 43. Let A be a saturated class of arities in E. The following are
equivalent properties of a monad T ∈ Mnd(E):
(i) T is A-nervous;
(ii) T : E → E is A-induced;
(iii) T : E → E preserves Φ-colimits for any density presentation Φ of K.
Proof. For (i) ⇔ (ii), the monadicity of V : A-Mnd(E) → SigA(E) verified in the
previous proposition implies, as in the proof of Theorem 38(iii), that A-Mnd(E) is
the colimit-closure in Mnd(E) of the free monads on signatures. Since MndA(E)
is also this closure, we have MndA(E) = A-Mnd(E) as desired. For (ii) ⇔ (iii),
we apply Proposition 35.
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Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2,
Brno 61137, Czech Republic
E-mail address: bourkej@math.muni.cz
Department of Mathematics, Macquarie University, NSW 2109, Australia
E-mail address: richard.garner@mq.edu.au
Chapter 6
Adjoint functor theorems for
homotopically enriched
categories
This chapter contains the article Adjoint functor theorems for homotopically
enriched categories by John Bourke, Steve Lack and Luk´aˇs Vokˇr´ınek.
This 2020 article is available on the Arxiv at https://arxiv.org/abs/2006.
07843 and has been submitted for publication. All three authors contributed
an equal share to the results of this article.
208
ADJOINT FUNCTOR THEOREMS FOR
HOMOTOPICALLY ENRICHED CATEGORIES
JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Abstract. We prove a weak adjoint functor theorem in the setting
of categories enriched in a monoidal model category V admitting
certain limits. When V is equipped with the trivial model
structure this recaptures the enriched version of Freyd’s adjoint
functor theorem. For non-trivial model structures, we obtain new
weak adjoint functor theorems of a homotopical flavour — in particular,
when V is the category of simplicial sets we obtain a homotopical
adjoint functor theorem appropriate to the ∞-cosmoi of
Riehl and Verity. We also investigate accessibility in the enriched
setting, in particular obtaining homotopical cocompleteness results
for accessible ∞-cosmoi.
1. Introduction
The general adjoint functor theorem of Freyd describes conditions
under which a functor U : B → A has a left adjoint: namely,
• U satisfies the solution set condition and
• B is complete and U : B → A preserves limits.
The first applications of this result that one typically learns about
are the construction of left adjoints to forgetful functors between algebraic
categories, and the cocompleteness of algebraic categories. In the
present paper we shall describe a strict generalisation of Freyd’s result,
applicable to categories of a homotopical nature, and shall demonstrate
similar applications to homotopical algebra.
The starting point of our generalisation is to pass from ordinary Setenriched
categories to categories enriched in a monoidal model category
V. For instance, we could take B to be
• the 2-category of monoidal categories and strong monoidal functors
(here V = Cat);
The first and third named authors acknowledge the support of the Grant Agency
of the Czech Republic under the grant 19-00902S. The second-named author acknowledges
with gratitude the hospitality of Masaryk University, and the support
of an Australian Research Council Discovery Project DP190102432.
1
2 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
• the ∞-cosmos of quasicategories admitting a class of limits and
morphisms preserving those limits (here V = SSet);
• the category of fibrant objects in a model category (here V =
Set but with what we call the split model structure)
and U : B → A the appropriate forgetful functor to the base.
Now in none of these cases is the V-category B complete in the sense
of enriched category theory, but each does admit certain limits captured
by the model structure in which we are enriching — in particular,
cofibrantly-weighted limits.
If we keep the usual solution set condition, which holds in many
examples including those described above, the question turns to what
kind of left adjoint we can hope for? Our answer here will fit into
the framework of enriched weakness developed by the second-named
author and Rosick´y in [25]. Given a class E of morphisms in V, one
says that U : B → A admits an E-weak left adjoint if for each object
A ∈ A there exists an object A ∈ B and a morphism ηA : A → UA
for which the induced map
B(A , B)
U GG A(UA , UB)
A(ηA,UB)
GG A(A, UB) (1.1)
belongs to E for all B ∈ B. Here one recovers classical adjointness
on taking E to be the isomorphisms. In this paper, we take E to be
the class of morphisms in the model category V with the dual strong
deformation retract property (see Section 3 below). We follow [9] in
calling such morphisms shrinkable.1
Our first main result, Theorem 7.8, is our generalisation of Freyd’s
adjoint functor theorem, and gives a sufficient condition for a V-functor
to have such an E-weak left adjoint. It is a generalisation, since when
V = Set with the trivial model structure, all colimits are cofibrantlyweighted,
the shrinkable morphisms are the isomorphisms, and E-weak
left adjoints are just ordinary left adjoints.
Our second main result, Theorem 8.9, applies our adjoint functor
theorem to the construction of E-weak colimits. Namely, we prove that
if C is an accessible V-category admitting certain limits, then C admits
E-weak colimits.
The remainder of the paper is devoted to giving applications of our
two main results in various settings. In particular, we interpret them
for 2-categories and ∞-cosmoi, where we obtain new results concerning
adjoints of a homotopical flavour and homotopy colimits. We also
provide numerous concrete examples to which these results apply.
1And we thank Karol Szumilo for suggesting the name.
WEAK ADJOINT FUNCTOR THEOREMS 3
Let us now give a more detailed overview of the paper. In Section
2 we introduce our basic setting, including four running examples
to which we return throughout the paper. These four examples capture
classical category theory, weak category theory, 2-category theory
and ∞-cosmoi. Furthermore, we assign to each V-category C its “homotopy
ER-category” h∗C: a certain category enriched in equivalence
relations. Since this is a simple kind of 2-category, it admits notions of
equivalence, bi-initial objects, and so on — accordingly, we can transport
these notions to define equivalences and bi-initial objects in the
V-category C.
The next five sections walk through generalisations of the various
steps in the proof of the adjoint functor theorem given in [27], which
we summarise below.
• The technical core of the general adjoint functor theorem is the
result that a complete category C with a weakly initial set of objects
has an initial object. In Section 5 we generalise this result,
establishing conditions under which C has a bi-initial object.
• In the classical setting the next step is to apply the preceding
result to obtain an initial object in each comma category A/U.
Accordingly Section 4 is devoted to the subtle world of enriched
comma categories.
• The next step, trivial in the classical case, is the observation
that if η: A → UA is initial in A/U then (1.1) is invertible.
Another way of saying this is that (1.1) is terminal in the slice
Set/B(A, UB). In Section 3 we consider the class E of shrinkable
morphisms, which turn out to be bi-terminal objects in enriched
slice categories.
• Putting all this together in Section 6, we prove our generalised
adjoint functor theorem in the case where the unit I ∈ V is
terminal. In Section 7 we adapt this to cover the general case
by passing from V-enrichment to V/I-enrichment.
In Section 8 we consider accessible V-categories and prove that each accessible
V-category with powers and enough cofibrantly-weighted limits
admits E-weak colimits.
In Section 9 we interpret our two main theorems in each of our four
running examples. In the first two examples we recover, and indeed
extend, the classical theory. Our focus, however, is primarily on 2categories
and ∞-cosmoi, where we describe many examples to which
our two main results apply. In particular, we show that many natural
examples of ∞-cosmoi are accessible, which allows us to conclude that
they admit flexibly-weighted homotopy colimits in the sense of [33].
4 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
2. The setting
We start with a monoidal model category V, as in [12] for example, in
which the unit object I is cofibrant. We may sometimes allow ourselves
to write as if the monoidal structure were strict.
We shall now give four examples, to which we shall return throughout
the paper. Several further examples may be found in Section 3 below.
Example 2.1. Any complete and cocomplete symmetric monoidal
closed category (V, ⊗, I) may be equipped with the trivial model structure,
in which the weak equivalences are the isomorphisms, and all
maps are cofibrations and fibrations. The case of classical category
theory corresponds the cartesian closed category (Set, ×, 1).
Example 2.2. For our second example, involving weak category theory,
we take our motivation from the case of Set, in which case the
relevant model structure has cofibrations the injections, fibrations the
surjections, and all maps as weak equivalences.
For general V we refer to the corresponding model structure as the
split model structure, which we now define. By Proposition 2.6 of
[36], in any category with binary coproducts there is a weak factorisation
whose left class consists of the retracts of coproduct injections
inj : X → X + Y , while the right class consists of the split epimorphisms.
Here the canonical factorisation of an arrow f : X → Y is
given by the cograph factorisation
X
inj
GG X + Y
f,1
GG Y.
As with any weak factorisation system (on a category with, say, finite
limits and colimits), we can extend it to a model structure whose
cofibrations and fibrations are the two given classes, and for which all
maps are weak equivalences. Doing this in the present case produces
what we shall refer to as the split model structure. It is a routine
exercise to show that if (V, ⊗, I) is a complete and cocomplete symmetric
monoidal closed category, then the split model structure on V
is monoidal.
Example 2.3. The cartesian closed category (Cat, ×, 1) becomes a
monoidal model category when it is equipped with the canonical model
structure: here the weak equivalences are the equivalences of categories,
the cofibrations are the functors which are injective on objects, and
the fibrations are the isofibrations. All objects are cofibrant and fibrant.
The trivial fibrations are the equivalences which are surjective
on objects; these are often known as surjective equivalences or retract
equivalences.
WEAK ADJOINT FUNCTOR THEOREMS 5
Example 2.4. Our final main example is the cartesian closed category
(SSet, ×, 1) of simplicial sets equipped with the Joyal model structure.
(Using the Kan model structure would give another example, but it
turns out that this leads to a strictly weaker adjoint functor theorem.)
We follow the standard notation in enriched category theory of writing
A0 for the underlying ordinary category of a V-category A. This
has the same objects as A, and the hom-set A0(A, B) is given by the
set of morphisms I → A(A, B) in V. Similarly, we write F0 : A0 → B0
for the underlying ordinary functor of a V-functor F : A → B.
We shall begin by showing how to associate to any V-category a
2-category whose hom-categories are both groupoids and preorders.
2.1. Intervals. An interval in V will be an object J equipped with a
factorisation
I + I
(d c)
GG J
e GG I
of the codiagonal for which (d c) is a cofibration and e a weak equivalence.
We shall then say that “(J, d, c, e) is an interval”. Observe
that if (d c) is a cofibration, then since I is cofibrant, d and c are also
cofibrations, and the following are equivalent:
• e is a weak equivalence
• d is a trivial cofibration
• c is a trivial cofibration.
We may always obtain an interval by factorising the codiagonal I +
I → I as a cofibration followed by a trivial fibration; an interval of this
type will be called a standard interval. (J, d, c, e) is an interval if and
only if (J, c, d, e) is one. There are also two constructions on intervals
that will be needed.
Proposition 2.5. If (J1, d1, c1, e1) and (J2, d2, c2, e2) are intervals, so
is (J, d2d1, c1c2, e) where J is constructed as the pushout
J1 d2
11
e1
&&
I
c1 cc
d2
11
J
e GG I.
J2
c1
cc
e2
ii
Proof. Since d2 and c1 are pushouts of trivial cofibrations they are
trivial cofibrations. Thus both d2d1 and c1c2 are trivial cofibrations.
6 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
On the other hand we can construct J as the pushout
I + I + I + I
(d1 c1)+(d2 c2)
GG
1+ +1

J1 + J2
d2+c1

I + I i1,3
GG I + I + I GG J
and so the lower horizontal is also a cofibration.
Proposition 2.6. If (J1, d1, c1) and (J2, d2, c2) are intervals then J1 ⊗
J2 becomes an interval when equipped with the maps
I
∼= GG I ⊗ I
d1⊗d2
GG J1 ⊗ J2
I
∼= GG I ⊗ I
c1⊗c2
GG J1 ⊗ J2
J1 ⊗ J2
e1⊗e2
GG I ⊗ I
∼= GG I.
2.2. ER-categories. Let ER be the full subcategory of Cat consisting
of the equivalence relations: that is, the groupoids which are also
preorders. This is closed in Cat under both products and internal
homs; it follows that ER becomes a cartesian closed category in its
own right, and that we can consider categories enriched over ER.
An ER-category is just a 2-category for which each hom-category
is an equivalence relation. Equivalently, it is an ordinary category in
which each hom-set is equipped with an equivalence relation, and this
relation is respected by composition on either side (whiskering, if you
will).
Since an ER-category is a special sort of 2-category, we can consider
standard 2-categorical notions. Thus morphisms f, g: A → B
are isomorphic in the 2-category just when they are related under the
given equivalence relation. We may as well therefore use ∼= to denote
this relation. An equivalence in an ER-category consists of morphisms
f : A → B and g: B → A for which both gf ∼= 1A and fg ∼= 1B.
Definition 2.7. An object T of an ER-category C is bi-terminal if for
each object C, the induced map C(C, T) → 1 is an equivalence.
In elementary terms, this means that there exists a morphism C → T
and it is unique up to isomorphism. Dually there are bi-initial objects.
2.3. The homotopy ER-category of a V-category. Any monoidal
category has a monoidal functor to Set given by homming out of the
WEAK ADJOINT FUNCTOR THEOREMS 7
unit object; in our case, this has the form V(I, −): V → Set. The
monoidal structure is defined by
V(I, X) × V(I, Y ) GG V(I, X ⊗ Y )
(x, y) 1 GG x ⊗ y
(2.1)
with unit defined by the map 1 → V(I, I) picking out the identity.
Given X ∈ V and x, y: I → X, write x ∼ y if there is an interval
(J, d, c, e) and a morphism h: J → X with hd = x and hc = y.
This clearly defines a symmetric reflexive relation on V(I, X); and it is
transitive by Proposition 2.5. Given f : X → Y , the function V(I, f)
is a morphism of equivalence relations, since if x = hd ∼ hc = y
then also fx = fhd ∼ fhc = fy for any f. Thus we have a functor
V → ER. Thanks to Proposition 2.6, the functions (2.1) lift to
morphisms of equivalence relations. Since the unit map 1 → V(I, I)
is trivially a morphism of equivalence relations, the monoidal functor
V(I, −): V → Set lifts to a monoidal functor h: V → ER.
This h induces a 2-functor h∗ : V-Cat → ER-Cat. Explicitly, for
any V-category C, the induced ER-category h∗(C) is given by the underlying
ordinary category of C, equipped with the equivalence relation
∼= on each hom-set C0(A, B), where f ∼= g just when, seen as
maps I → C(A, B), they are ∼-related; in other words, if there is a
factorisation
I + I
(f g)
GG
(d c)

C(A, B)
J
h
WW
for some interval (J, d, c, e). We call this relation ∼= the V-homotopy
relation.
Remark 2.8. For f and g as above, let us refer to a given extension h as
above as a (V, J)-homotopy, and denote it as h: f ∼=J g. We say that
such an h: J → C(A, B) is trivial if it factorises through e: J → I, in
which case of course f = g.
Given a: X → A and b: B → Y we naturally obtain (V, J)-homotopies
h◦a: f ◦a ∼=J g ◦a and b◦h: b◦f ∼=J b◦g. It is only when considering
transitivity of the relation ∼= that we need to change the interval J.
Remark 2.9. A priori, we now have two “homotopy relations” for morphisms
x, y: I → X in V: on the one hand x ∼ y and on the other
x ∼= y. But these are clearly equivalent.
8 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
More generally, morphisms f, g: X → Y in V are V-homotopic if
any (and thus all) of the following diagrams has a filler:
X + X
(f g)
GG
(d·X c·X)

Y I + I
(f g)
GG
(d c)

[X, Y ] X GG
(f
g) 55
Y J
(Y d
Y c)

J · X
``
J
XX
Y × Y.
In the case that X is cofibrant then e · X : J · X → X is a weak
equivalence so that J ·X is a cylinder object for X; in this case the left
diagram above shows V-homotopy implies left homotopy in the model
categorical sense of [31]. Similarly if Y is fibrant the third diagram
shows that V-homotopy implies right homotopy, but in general there
need be no relation between V-homotopy and left or right homotopy.
Since a V-functor F : A → B induces an ER-functor h∗(F): h∗(A) →
h∗(B), if f ∼= g in A then Ff ∼= Fg in B. This is one useful feature of
V-homotopy not true of the usual left and right homotopy relations in
the model category V.
Definition 2.10. An object of a V-category A is bi-terminal or biinitial
if it is so in h∗(A).
In elementary terms, T is bi-terminal in A if for each A ∈ A there
is a morphism A → T, which is unique up to V-homotopy.
If A has an actual terminal object 1, then T is bi-terminal if and
only if the unique map T → 1 is an equivalence. Explicitly, this means
that there is a morphism t: 1 → T with t◦! ∼= 1T . The dual remarks
apply to bi-initial objects.
Proposition 2.11. Right adjoint V-functors preserve bi-terminal ob-
jects.
Proof. Let U : B → A be a V-functor with F U a left adjoint, and let
T ∈ B be bi-terminal. For any A ∈ A, there is a morphism FA → T
in B and so a morphism A → UT in A. If f, g: A → UT are two
such morphisms in A, then their adjoint transposes f , g : FA → T
are V-homotopic in B, hence f = Uf ◦ ηA
∼= Ug ◦ ηA = g in A.
3. Shrinkable morphisms and E-weak adjoints
Definition 3.1. A morphism f : A → B in a V-category C is said to
be shrinkable if there exists a morphism s: B → A with f ◦ s = 1B,
an interval J and a (V, J)-homotopy h: sf ∼=J 1A such that fh: f =
fsf ∼=J f is trivial.
WEAK ADJOINT FUNCTOR THEOREMS 9
In elementary terms, such an h is a morphism as below
I + I
(d c)

(1A s◦f)
GG C(A, A)
C(A,f)

J e
GG
h
TT
I
f
GG C(A, B)
(3.1)
rendering the diagram commutative.
Let us record a few easy properties.
Proposition 3.2. Shrinkable morphisms are closed under composition,
contain the isomorphisms, and are preserved by any V-functor.
Proof. Closure under composition follows from the fact — see Proposition
2.5 — that intervals can be composed. Verification of the remaining
facts is routine.
Proposition 3.3. Let C be a model V-category. Any trivial fibration
f : A → B in C with cofibrant domain and codomain is shrinkable.
Proof. Since B is cofibrant, f has a section s. Since A is cofibrant, the
induced A + A → J · A is a cofibration, and so we have an induced
h: J · A → A, corresponding to the desired J → C(A, A).
Proposition 3.4. Let C be a model V-category. If f : A → B is shrinkable
in C and either d · A: A → J · A or d A: J A → A is a weak
equivalence then so is f. In particular, any shrinkable morphism whose
domain is either cofibrant or fibrant is a weak equivalence.
Proof. Suppose first that d · A is a weak equivalence. We have commutative
diagrams
A
d·A GG
c·A
GG
1
66
sf
XXJ · A
h GG A A
d·A GG
c·A
GG
1
66
1
XXJ · A
e·A GG A
and since d · A is a weak equivalence it follows successively that e · A,
c · A, h, and finally sf are so. But since fs = 1 it follows that f (and
s) are also weak equivalences.
Since d is a trivial cofibration, if A is cofibrant then d · A will be a
trivial cofibration, and in particular a weak equivalence.
The cases of d A a weak equivalence, and of A fibrant, are similar.
Using the preceding two propositions, we have the following result.
10 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Corollary 3.5. If all objects of C are cofibrant, then each trivial fibration
is shrinkable and each shrinkable morphism is a weak equivalence
with a section.
Remark 3.6. If all objects of V are fibrant and I = 1, then in any
interval (J, d, c, e), the morphism e: J → I = 1 is not just a weak
equivalence but a trivial fibration. Thus in this case any two intervals
give the same notion of V-homotopy, and the shrinkable morphisms
can be described using any one fixed interval.
Definition 3.7. We write E for the collection of all shrinkable morphisms
in V itself.
Examples 3.8. In each of our four main examples, all objects are
cofibrant, and so by Corollary 3.5 every trivial fibration is shrinkable,
and every shrinkable morphism is a weak equivalence with a section.
In the first three of them, every weak equivalence with a section is
a trivial fibration, and so the shrinkable morphisms are precisely the
trivial fibrations.
(1) For V equipped with the trivial model structure, the trivial
fibrations and the shrinkable morphisms are both just the iso-
morphisms.
(2) For V with the split model structure, the trivial fibrations and
the shrinkable morphisms are both just the split epimorphisms.
(3) For Cat equipped with the canonical model structure, the trivial
fibrations and the shrinkable morphisms are both just the
surjective equivalences.
(4) Now consider SSet equipped with the Joyal model structure.
By Corollary 3.5 the shrinkable morphisms lie between the weak
equivalences and the trivial fibrations.
Not every shrinkable morphism is a trivial fibration. For example,
let J be the nerve of the free isomorphism, with its
unique interval structure, then form the pushout of the two
morphisms 1 → J to obtain a new interval J , as in Proposition
2.5. The induced map J → J is a shrinkable morphism
but is not a trivial fibration: in particular, J is fibrant while J
is not, so J → J cannot be a fibration.
In general, all that we can say is that the shrinkable morphisms
are weak equivalences with sections; however for a shrinkable
morphism f : X → Y between quasicategories, the fibrant
objects, we can say a little more. First recall that an equivalence
of quasicategories is a map f : X → Y for which there
exists a g: Y → X with fg ∼=J 1Y and gf ∼=J 1X where J is the
WEAK ADJOINT FUNCTOR THEOREMS 11
nerve of the free isomorphism as discussed above. Let us call f
a surjective/retract equivalence if in fact fg = 1Y . Now since
all objects are cofibrant in SSet each weak equivalence between
quasicategories is a homotopy equivalence relative to any choice
of interval; in particular, each shrinkable morphism f : X → Y
between quasicategories is a surjective equivalence. (Note, however,
that we do not assert that the (V, J)-homotopy gf ∼=J 1X
is f-trivial — indeed, while there is an f-trivial V-homotopy
relative to some interval, it does not seem that f-triviality is
independent of the choice of interval.)
Example 3.9. The category CGTop of compactly generated topological
spaces is cartesian closed, and the standard model structure for
topological spaces restricts to CGTop (see [12, Section 2.4] for example).
Here the notion of shrinkable morphism is dual to that of strong
deformation retract, and it is in this case that the name “shrinkable”
originated, as explained in the introduction.
Example 3.10. Another example is given by the monoidal model category
Gray: this is the category of (strict) 2-categories and 2-functors
with the model structure of [22], as corrected in [23]: the weak equivalences
are the biequivalences, the trivial fibrations are the 2-functors
which are surjective on objects, and retract equivalences on homs. The
monoidal structure is the Gray tensor product, as in [11]. We shall
see that in this case, the shrinkable morphisms are exactly the trivial
fibrations which have a section.
Not all objects are cofibrant, and not all trivial fibrations have sections,
so not all trivial fibrations are shrinkable. On the other hand,
all objects are fibrant, so we can use a standard interval for all Vhomotopies;
also all shrinkable morphisms are weak equivalences. Taking
as standard interval the free adjoint equivalence, a V-homotopy between
two 2-functors is then a pseudonatural equivalence. If p: E → B
is shrinkable, via s: B → E and a pseudonatural equivalence sp 1,
then clearly p is surjective on objects and an equivalence on homcategories.
Thus it will be a trivial fibration provided that it is full
on 1-cells. For this, given x, y ∈ E and a morphism β : px → py in
B, we may compose with suitable components of the pseudonatural
equivalence to get a morphism
x
σ GG spx
sβ
GG spy
σ GG y
and by p-triviality of σ, both pσ and pσ are identities, and so p is
indeed full on 1-cells.
12 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Suppose conversely that p: E → B is a trivial fibration, and that
s: B → E is a section of p. For each object x ∈ E, we have pspx = px
and so there is a morphism σx : spx → x with pσx = 1 and similarly a
morphism σx : x → spx, also over the identity. These form part of an
equivalence, also over the identity. For any α: x → y the square
spx
σx
GG
spα

x
α

spy σy
GG y
need not commute, but both paths lie over the same morphism pα in B,
and so there is a unique invertible 2-cell in the diagram, which is sent
by p to an identity. It now follows that σ determines an equivalence
sp 1 over B, and so that p is shrinkable.
Example 3.11. Let R be a finite-dimensional cocommutative Hopf algebra,
which is Frobenius as a ring. Then the category of modules over
R is a monoidal model category [12, Section 2.2 and Proposition 4.2.15].
All objects are cofibrant. The trivial fibrations are the surjections with
projective kernel. In particular, the codiagonal R ⊕ R → R is a trivial
fibration, and so R ⊕ R itself is an interval. Thus the shrinkable
morphisms are just the split epimorphisms. Every trivial fibration is
shrinkable, while the converse is true (if and) only if every module is
projective.
Example 3.12. Let R be a commutative ring, and Ch(R) the symmetric
monoidal closed category of unbounded chain complexes of Rmodules.
This has a model structure [12, Section 2.3] for which the
fibrations are the pointwise surjections and the weak equivalences the
quasi-isomorphisms. A morphism f : X → Y is shrinkable when it has
a section s and a chain homotopy between sf and 1 which is f-trivial.
This is of course rather stronger than being a trivial fibration.
The paper [25] introduced the notion of E-weak left adjoint for any
class of morphisms E. Here we consider the instance of this concept
obtained by taking the class E of shrinkable morphisms.
Definition 3.13. Let U : B → A be a V-functor. An E-weak reflection
of A ∈ A is a morphism ηA : A → UA with the property that for all
B ∈ B the induced morphism
B(A , B)
U GG A(UA , UB)
A(η,UB)
GG A(A, UB)
belongs to E. If each object A ∈ A admits an E-weak reflection then
we say that U admits an E-weak left adjoint.
WEAK ADJOINT FUNCTOR THEOREMS 13
Examples 3.14. (1) For V with the trivial model structure, a Vfunctor
U : B → A as above admits an E-weak left adjoint just
when it admits a genuine left adjoint.
(2) For V with the split model structure, U : B → A admits an Eweak
left adjoint if we have morphisms η: A → UA for each A
such that the induced B(A , B) → A(A, UB) is a split epimorphism.
In the case that V = Set, this recovers ordinary weakness
— that is, for each f : A → UB there exists f : A → B
with Uf ◦ηA = f. In this case, setting FA = A for each A ∈ B
and Ff = (ηB ◦ f) : FA → FB for f : A → B equips F with
the structure of a graph morphism, but there is no reason why
it should preserve composition.
(3) In the Cat case each B(A , B) → A(A, UB) is a retract equivalence.
In this setting we can form F as in the previous example,
and with a similar definition α → Fα on 2-cells. Then given
f : A1 → A2 and g: A2 → A3, since B(A1, A3) → A(A1, UA3)
is a retract equivalence, we obtain a unique invertible 2-cell
such that Ff,g : Fg ◦ Ff ∼= F(g ◦ f). Continuing in this way, F
becomes a pseudofunctor and we obtain a biadjunction in the
sense of [20].
(4) For SSet equipped with the Joyal model structure, if B and
A are locally fibrant — that is, if they are enriched in quasicategories
— then each B(A , B) → A(A, UB) is a retract
equivalence of quasicategories. As before, we can define F as a
graph morphism as in the preceding two examples. Then given
f : A1 → A2 and g: A2 → A3 let us write k: B(A1, A3) →
A(A1, UA3) for the retract equivalence, with section s, and
write a = F(g ◦ f) and b = Fg ◦ Ff. Then the (SSet, J)homotopies
1 ∼= sk and sk ∼= 1 give a composite homotopy
F(g ◦ f) = a ∼= ska = skb ∼= b = Fg ◦ Ff
in the hom-quasicategory B(A1, A3), suggesting that F could
be turned into a pseudomorphism of simplicially enriched categories
given a suitable formalisation of that concept.
4. Comma categories and slice categories
The proof of Freyd’s general adjoint functor theorem given in [27]
uses the fact that in order to construct a left adjoint to U : B → A, it
suffices to construct an initial object in each comma category A/U, for
A ∈ A. Indeed, the universal property of the initial object ηA : A →
14 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
UA implies that the induced map
B(A , B)
U GG A(UA , UB)
A(ηA,UB)
GG A(A, UB)
is a bijection, so that the maps ηA : A → UA give the components of
the unit for a left adjoint.
In order to frame this in a matter suitable for generalisation, observe
that to say that the above function is a bijection is equally to say that
it is a terminal object in the slice category Set/A(A, UB) — thus one
passes from an initial object in the comma category to a terminal object
in the slice.
In the present section we develop the necessary results about comma
categories in the enriched context. In our setting, we shall only be able
to construct bi-initial objects in the comma category A/U and so are
led to consider bi-terminal objects in enriched slice categories, which
turn out to be the shrinkable morphisms of the previous section.
4.1. Comma categories and slice categories. Given V-functors
G: C → A and U : B → A the enriched comma-category G/U is the
comma object, below left
G/U C
B A
P GG
Q

G

U
GG
θ
{Ó
G/U
(P
Q)

GG A2
(dom
cod )

C × B G×U
GG A × A
in the 2-category V-Cat of V-categories. This has the universal property
that, given V-functors R: D → C and S : D → B, and a V-natural
transformation λ: G ◦ R ⇒ U ◦ S, there exists a unique V-functor
K : D → G/U such that P ◦ K = R, Q ◦ K = S and θ ◦ K = λ, as well
as a 2-dimensional aspect characterising V-natural transformations out
of such a K.
For the reader unfamiliar with 2-dimensional limits, we point out
that it is equally the pullback on the right above in which A2
is the
(enriched) arrow category, and dom
cod
the projection which sends a morphism
to its domain and codomain.
The objects of G/U consist of triples (C, α: GC → UB, B) where
C ∈ C, B ∈ B, and α: GC → UB is in A0; and with hom-objects as
WEAK ADJOINT FUNCTOR THEOREMS 15
in the pullback below.
G/U((C, α, B), (C , α , B ))

GG C(C, C )
G

A(GC, GC )
A(1,α )

B(B, B ) U
GG A(UB, UB )
A(α,1)
GG A(GC, UB )
A standard result of importance to us is the following one.
Proposition 4.1. Suppose that B and C admit W-limits for some
weight W : D → V and that U : B → A preserves them. Then G/U
admits W-weighted limits for any G: C → A, and they are preserved
by the projections P : G/U → C and Q: G/U → B.
Proof. Consider a diagram T = (R, λ, S): D → G/U and let us write
W -cone(X, T) := [D, V](W, G/U(X, T−))
for X = (C, α, B) ∈ G/U. We must prove that W -cone(−, T): G/Uop
→
V is representable. First observe that the definition of the hom-objects
in G/U gives us the components of a pullback
G/U(X, T−) C(C, S−)
A(GC, GS−)
B(B, R−) A(UB, UR−) A(GC, UR−)
P GG
G

λ∗

Q

U
GG
α∗
GG
in [D, V]. Applying [D, V](W, −), we obtain a pullback
W -cone(X, T) W -cone(C, S)
W -cone(GC, GS)
W -cone(B, R) W -cone(UB, UR) W -cone(GC, UR)
P∗
GG
G∗

λ∗

Q∗

U∗
GG
α∗
GG
(4.1)
16 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
in V. Since the limits {W, S} and {W, R} exist and since U preserves
the latter, this pullback square is isomorphic to
W -cone(X, T) C(C, {W, S})
A(GC, G{W, S})
B(B, {W, R}) A(UB, U{W, R}) A(GC, U{W, R})
GG
G

ϕ∗

U
GG
α∗
GG
for ϕ: G{W, S} → U{W, R} the morphism induced by
W GG C({W, S}, S−)
G GG A(G{W, S}, GS−)
λ∗
GG A(G{W, S}, UR−)
and the universal property of U{W, R}. By definition, the pullback in
this square is G/U((C, α, B), ({W, S}, ϕ, {W, R})). Hence the object
({W, S}, ϕ, {W, R}) represents W -cone(−, T) and is therefore the limit
{W, T}.
Let I denote the unit V-category, which has a single object with
object of endomorphisms the monoidal unit I of V. An object A ∈ A
then corresponds to a V-functor A: I → A. We denote by A/U the
comma object depicted below left
A/U I
B A
P GG
Q

A

U
GG
θ
{Ó
A/A A
I A
P GG
Q

1

A
GG
θ
{Ó
in which we shall consider bi-initial objects, and the enriched slice
category A/A as above right in which we shall consider bi-terminal
ones. We record for later use that if (B, b: B → A) and (C, c: C → A)
are objects of A/A, the corresponding hom is given by the pullback
(A/A)((B, b), (C, c)) GG

A(B, C)
A(B,c)

I
b
GG A(B, A).
(4.2)
Remark 4.2. If the unit object of V is also terminal, then I is the
terminal V-category, and is complete and cocomplete. We may then
use Proposition 4.1 to deduce that, if B has W-limits and U : B → A
preserves them, then A/U has W-limits for any A ∈ A. But if I = 1,
then I need not be complete; indeed, almost by definition, I has a
terminal object if and only if I = 1.
WEAK ADJOINT FUNCTOR THEOREMS 17
Proposition 4.3. Let A be a V-category and A an object of A. The
identity morphism on A is a terminal object in the slice V-category
A/A if and only if I = 1.
Proof. For any f : B → A we have the pullback
(A/A)((B, f), (A, 1A)) GG

B(B, A)
B(B,1A)

I
f
GG B(B, A)
and since the pullback of an isomorphism is an isomorphism, we conclude
that (A/A)((B, f), (A, 1A)) ∼= I, which is terminal if and only if
I is so.
Example 4.4. By the preceding result, 1: A → A is never terminal in
the V-category A/A unless I = 1. For a concrete example, let V = Ab,
and consider Ab/A for a given abelian group A. An object consists of
a homomorphism f : B → A, and a morphism from (B, f) to (A, 1A)
consists of a pair (h, n) where h: B → A is a homomorphism, n ∈ Z,
and h = n.f. There is such a morphism (n.f, n) for any n ∈ Z, and
these are all distinct, thus (A, 1A) is not terminal.
Proposition 4.5. Suppose that I = 1, and let U : B → A and A ∈ A.
(1) Then A/U has underlying category A/U0. Furthermore, given
parallel morphisms in A/U
A
b
}}
c
33
UB
Uf
GG
Ug
GG UC
a V-homotopy f ∼= g in A/U is a V-homotopy h: f ∼= g in B
such that Uh◦b is trivial, which we will refer to as a V-homotopy
under A.
(2) Then A/A has underlying category A0/A. Furthermore, given
parallel morphisms in A/A
B
b 11
f
GG
g
GG C
c

A
18 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
a V-homotopy f ∼= g in A/A is a V-homotopy h: f ∼= g in B
such that c◦h is trivial, which we will refer to as a V-homotopy
over A.
Proof. A morphism in (A/U)0 is given by a map into the pullback as
in
I
99
f =1
AA
f
@@
(A/U)((B, b), (C, c)) GG

I
c

B(B, C) U
GG A(UB, UC)
b∗
GG A(A, UC)
where the map f : I → I is necessarily the identity by virtue of the
assumption that I = 1. Therefore it amounts to a morphism f : B →
C ∈ B0 such that Uf ◦ b = c: that is, a morphism of A/U0. Similarly,
for J an interval, a (V, J)-homotopy f ∼= g in A/U is specified by a
map from J into the pullback whose component in I is fixed as the
structure map for the interval e: J → I, again by the assumption that
I = 1. The component in B(B, C) then specifies a (V, J)-homotopy
h: f ∼= g for which Uh ◦ b is trivial.
This analysis applies to the special case A/A and so, by duality,
establishes the corresponding claim for A/A.
Proposition 4.6. If I = 1 then f : A → B is shrinkable if and only if
it is bi-terminal in C/B.
Proof. By Proposition 4.3 the identity 1B is terminal in C/B. Therefore
f : A → B is bi-terminal if and only if the unique map f : (A, f) →
(B, 1B) is an equivalence. Now a morphism s: (B, 1B) → (A, f) is specified
by a section s: B → A of f so that we have f ◦s = 1B : (B, 1B) →
(B, 1B). By Proposition 4.5 a V-homotopy s ◦ f ∼= 1A in C/B is specified
by a V-homotopy h: s ◦ f ∼= 1A in C such that f ◦ h is trivial, as
required.
In the case I is not terminal, the above proposition need not hold.
We deal with this, and other issues related to a non-terminal I, by passing
from V to the slice category V/I, which has an induced monoidal
structure for which the unit is terminal. Moreover, the comma Vcategories
A/U and A/A naturally give rise to V/I-categories. See
Section 7 below.
WEAK ADJOINT FUNCTOR THEOREMS 19
Theorem 4.7. Suppose that I = 1. Let B be a V-category with powers,
and U : B → A a V-functor which preserves powers; if η: A → UA is
bi-initial in A/U then
B(A , B)
U GG A(UA , UB)
A(η,UB)
GG A(A, UB) (4.3)
is bi-terminal in V/A(A, UB).
Proof. By Proposition 4.5 we need to find, for any object α of V/A(A, UB)
a morphism
X
α
66
f
GG B(A , B)
xx
A(A, UB)
and for any two such morphisms a V-homotopy between them over
A(A, UB).
Using the universal property of the power X B and the fact that
U(X B) ∼= X UB, the above problem is translated to A/U as
below.
A
η
}}
α
66
UA
Uf
GG U(X B)
A
f
GG X B
Since η is bi-initial, the required morphism f and so f exists. If f
and g are two such morphisms then we can, by bi-initiality of η, find a
V-homotopy between f and g under A; writing it as
h : A → J (X B) = X (J B),
we obtain h: X → B(A , J B) ∼= J B(A , B) and this is the
required V-homotopy between f and g over A(A, UB)
Remark 4.8. A more conceptual explanation for the preceding result is
as follows. Given U : B → A and objects B ∈ B and A ∈ A, we obtain
a V-functor K : (A/U)op
→ V/A(A, UB) sending η: A → UA to the
morphism (4.3). If B has powers and U preserves them, then K has a
left V-adjoint. The preceding result then follows immediately.
20 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
5. Enough cofibrantly-weighted limits
In this section we describe the completeness and continuity conditions
which will arise in our weak adjoint functor theorem. We start
by recalling the notion of cofibrantly-weighted limit.
5.1. Cofibrantly-weighted limits. Let D be a small V-category, and
consider the V-category [D, V] of V-functors from D to V. The morphisms
in (the underlying ordinary category of) [D, V] are the V-natural
transformations. We shall say that such a V-natural transformation
p: F → G is a trivial fibration if it is so in the pointwise/levelwise
sense: that is, if each component pD: FD → GD is a trivial fibration
in V.
We say that an object Q ∈ [D, V] is cofibrant, if for each trivial
fibration p: F → G in [D, V] the induced morphism
[D, V](Q, p): [D, V](Q, F) → [D, V](Q, G)
is a trivial fibration in V. In many cases, the projective model structure
on [D, V] will exist and make [D, V] into a model V-category, and then
these notions of cofibrant object and trivial fibration will be the ones
that apply there.
Example 5.1. Each representable D(D, −) is cofibrant: this follows
immediately from the Yoneda lemma, since [D, V](D(D, −), p) is just
pD: FD → GD.
Proposition 5.2. The cofibrant objects in [D, V] are closed under coproducts
and under copowers by cofibrant objects of V.
Proof. The case of coproducts is well-known. As for copowers, let
Q: D → V and X ∈ V be cofibrant. If p: F → G is a trivial fibration
in [D, V] then [D, V](X · Q, p) is (up to isomorphism) given by
[X, [D, V](Q, p)]. Since Q is cofibrant and p is a trivial fibration, it
follows that [D, V](Q, p) is a trivial fibration; and now since also X is
cofibrant it follows that [X, [D, V](Q, p)] is also a trivial fibration.
Many of the V-categories to which we shall apply our weak adjoint
functor theorem have all cofibrantly-weighted limits, but the hypotheses
of the theorem will be weaker than this. There are several reasons
for this, and in particular for not requiring idempotents to split. For
instance, the weak adjoint functor theorem of Kainen [16] which we
wish to generalise does not assume them. Furthermore, in 2-category
theory there exist many natural examples of 2-categories admitting
enough (but not all) cofibrantly-weighted limits. An example is the
WEAK ADJOINT FUNCTOR THEOREMS 21
2-category of strict monoidal categories and strong monoidal functors:
see [8, Section 6.2]. We start with the case where I = 1.
5.2. Enough cofibrantly-weighted limits (when I = 1). We now
define our key completeness and continuity conditions in the case I = 1;
these will be modified later to deal with the case where I = 1.
Definition 5.3. Suppose that the unit object I of V is terminal. We
say that a V-category B has enough cofibrantly-weighted limits if for any
small V-category D, there is a chosen cofibrant weight Q: D → V for
which the unique map Q → 1 is a trivial fibration, and for which B has
Q-weighted limits. Similarly, a V-functor U : B → A preserves enough
cofibrantly-weighted limits if it preserves these Q-weighted limits for
each small D.
In particular, B will have enough cofibrantly-weighted limits if it has
all cofibrantly-weighted limits.
Remark 5.4. In our definition of “enough cofibrantly-weighted limits”,
we have asked that there be a chosen fixed weight Q: D → V for each
small D. But in fact it is possible to allow Q to depend upon the
particular diagram D → B of which one wishes to form the (weighted)
limit. The cost of doing this is that it becomes more complicated to
express what it means to preserve enough cofibrantly-weighted limits,
but our main results can still be proved with essentially unchanged
proofs.
Example 5.5. If V has the trivial model structure, then the trivial
fibrations in [D, V] are just the isomorphisms, and all objects are cofibrant.
Thus in this case a V-category has enough cofibrantly-weighted
limits if and only if it has all (unweighted) limits of V-functors (with
small domain).
Example 5.6. If V has the split model structure, then a V-category
will have enough cofibrantly-weighted limits provided it has products.
To see this, let D be a small V-category and consider the terminal
weight 1: D → V. Then D∈D D(D, −) is a coproduct of representables,
and so is cofibrant. The unique map q: D∈D D(D, −) → 1
has component at C ∈ D given by D∈D D(D, C) → 1, which has
a section picking out the identity 1 → D(C, C). Thus q is a pointwise
split epimorphism, and so a trivial fibration in [D, V]. And the
D D(D, −)-weighted limit of S : D → B is just D∈D SD.
Example 5.7. In the case where V = Cat, the enriched projective
model structure on [D, Cat] exists and the cofibrant weights are precisely
the flexible ones in the sense of [4]: see Theorem 5.5 and Section 6
22 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
of [24]. Thus a 2-category will have all cofibrantly-weighted limits just
when it has all flexible limits. But it will have enough cofibrantlyweighted
limits provided that it has all weighted pseudolimits, and so
in particular if it has products, inserters, and equifiers; in other words,
if it has PIE limits in the sense of [30]. Once again, the 2-category of
strict monoidal categories and strong monoidal functors is an example
which has PIE limits but not flexible ones: see [8, Section 6.2].
Example 5.8. In the case of SSet, the enriched projective model
structure on [D, SSet] exists (see Proposition A.3.3.2 and Remark A.3.3.4
of [26]) and has generating cofibrations
I = {∂∆n
· D(X, −) → ∆n
· D(X, −): n ∈ N, X ∈ D}
obtained by copowering the boundary inclusions of simplices by representables.
The I-cellular weights are, in this context, what Riehl and
Verity call flexible weights — in particular, these are particular cofibrant
weights. Since Quillen’s small object argument applied to the
set I produces a cellular cofibrant replacement of each weight, a simplicially
enriched category will have enough cofibrantly-weighted limits
provided that it has all flexible limits — that is, those weighted by flexible
weights. In particular, each ∞-cosmos in the sense of [35] admits
enough cofibrantly-weighted limits.
Lemma 5.9. Let Q: D → V be a cofibrant weight for which the unique
map Q → 1 is a trivial fibration. Then Q is bi-terminal in the full
subcategory of [D, V] consisting of the cofibrant weights.
Proof. Let G be cofibrant. Then the unique map from [D, V](G, Q) to
[D, V](G, 1) is a trivial fibration. Since the left vertical in each of the
following diagrams is a cofibration in V
∅

GG [D, V](G, Q)

I
SS
GG 1 = [D, V](G, 1)
I + I

GG [D, V](G, Q)

J
SS
GG 1 = [D, V](G, 1)
it follows that each diagram has a filler. This implies the existence and
essential uniqueness of maps G → Q.
Consider a diagram J : D → B, and let L = {Q, J}. Now Q is
bi-terminal with respect to cofibrant objects by Lemma 5.9, and each
representable is cofibrant by Example 5.1, so that there exists a morphism
sD : D(D, −) → Q. Since weighted limits are (contravariantly)
functorial in their weights, we obtain a morphism
pD : L = {Q, J} → {D(D, −), J} = JD.
WEAK ADJOINT FUNCTOR THEOREMS 23
By bi-terminality of Q once again, for any f : C → D in D there exists
a V-homotopy in the triangle below left
D(C, −)
D(D, −) Q
D(f,−)
cc
sC
11
sD
GG
∼=
L JC
JD
Jf

pC
GG
pD
11
∼=
and so, on taking limits of J by the given weights, a V-homotopy in
the triangle on the right.
Lemma 5.10. If J : D → B is fully faithful, and L = {Q, J} as above,
then the projection pD : L → JD satisfies f ◦pD
∼= 1L for any morphism
f : JD → L.
Proof. Using fully faithfulness of J we define k: Q → D(D, −) as the
unique morphism rendering commutative the upper square below
Q
η
GG
k

B(L, J−)
B(f,J−)

D(D, −)
J
∼=
GG
sD

B(JD, J−)
B(pD,J−)

Q η
GG B(L, J−)
in which η is the unit of the limit L = {Q, J}, and the lower commutes
essentially by definition of pD. Bi-terminality of Q among cofibrant
objects in [D, V] implies that the composite sD ◦ k is V-homotopic to
the identity.
Consider the weighted limit V-functor {−, J}: [D, V]op
B → B, where
the domain is restricted to the full subcategory containing those weights
W for which B admits all W-weighted limits. Since [D, V]B contains Q
and D(D, −) we use the fact that V-functors preserve V-homotopies to
deduce that {k, J} ◦ {sD, J} ∼= 1; in other words that f ◦ pD
∼= 1.
Example 5.11. Let Eq be the free parallel pair, involving maps f, g: A →
B, made into a free V-category. Consider the constant functor ∆I : Eq →
V, which is equally the terminal weight. We first show how a (standard)
interval gives a cofibrant replacement of ∆I and interpret the
corresponding limit. Then we relate this special type of cofibrant replacement
to a general one.
24 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Given a standard interval — that is, an interval
I
d GG
c
GG J
e GG I
in which e is a trivial fibration, one can show that the pair d, c constitutes
a cofibrant weight Q: Eq → V and admits a pointwise trivial
fibration
I
c
GG
d GG
1

J
e

I
1
GG
1 GG
I
to the constant functor ∆I. A Q-cone from an object X to a diagram
f, g: A B amounts to a natural transformation
I
d

c

k GG C(X, A)
f∗

g∗

J
h
GG C(X, B)
or equivalently a morphism k: X → A equipped with a homotopy
h: fk ∼= gk. Thus, a limit {Q, F} possesses a universal such pair
and is thus what one might call a Q-isoinserter, or isoinserter if Q is
understood.
Next, suppose that Q : Eq → V is cofibrant and q : Q → ∆I is a
pointwise trivial fibration, as in
I
d GG
c
GG
a

J
b

I
1 GG
1
GG I
and let u be a section of the trivial fibration a. Factorise the induced
map (d u c u) as a cofibration followed by a trivial fibration as in
I + I
u+u
GG
(d c)

I + I
(d c )

J
v GG J
to give an interval
I
d GG
c
GG J
bv GG I.
WEAK ADJOINT FUNCTOR THEOREMS 25
together with a natural transformation
I
d GG
c
GG
u

J
v

I
d GG
c
GG J .
Thus, any Q -cone gives a Q-cone and, in particular, a morphism
k: X → A and a homotopy h: fk ∼= gk. We call the limit {Q , F}, the
Q -isoinserter of f and g, or just the isoinserter if Q is understood.
5.3. Enough cofibrantly-weighted limits (I = 1). For I the unit
V-category we write I : I → V for the V-functor selecting the object I.
Definition 5.12. We say that a V-category B has enough cofibrantlyweighted
limits if for each small V-category D and each V-functor
P : D → I there is a cofibrant V-weight Q: D → V and pointwise
trivial fibration Q → IP for which B has Q-weighted limits. Similarly,
U : B → A preserves enough cofibrantly-weighted limits if it preserves
such Q-limits.
In the case where I = 1, the V-category I is terminal, and so any
V-category D has a unique such P; furthermore, the composite IP
is then the terminal object of [D, V], and so this does agree with our
earlier definition.
Example 5.13. For the trivial model structure on V, no longer assuming
that I is terminal, a V-category K will have enough cofibrantlyweighted
limits if and only if it has all limits weighted by functors of
the form IP : D → V for a small V-category D, a V-functor P : D → I,
and I : I → V the weight picking out the object I ∈ V.
Example 5.14. For the split model structure on V, no longer assuming
that I is terminal, each weight W ∈ [D, V] admits a canonical cofibrant
replacement q: W → W given by the evaluation map
D∈D WD · D(D, −)
q
GG W
To see this, observe that the left hand side is a coproduct of copowers
of representables, and so cofibrant by Proposition 5.2. Furthermore,
each component of q is a split epimorphism with section
WA ∼= WA · I → WA · D(A, A) → ΣD∈DWA · D(D, A)
obtained by first copowering the map I → D(A, A) defining the identity
morphism on A, and then composing this with the coproduct inclusion.
26 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Thus q is a pointwise split epimorphism, and so a trivial fibration in
[D, V]. In the case that W = IP, as in Definition 5.12, each WA is just
I. Thus the right hand side above reduces to D∈D D(D, −), whence
a V-category has enough cofibrantly-weighted limits provided it has
products.
6. The weak adjoint functor theorem in the case I = 1
The key technical step in the proof of Freyd’s general adjoint functor
theorem is to prove that a complete category with a weakly initial set
of objects has an initial object. We shall now show that, on replacing
completeness by homotopical completeness, we can still construct a biinitial
object. It is then straightforward to deduce the weak adjoint
functor theorem.
Theorem 6.1. Suppose that the unit object I of V is terminal. If B
is a V-category with enough cofibrantly-weighted limits, and B0 has a
small weakly initial family of objects, then B has a bi-initial object.
Proof. Let G be the full subcategory of B consisting of the objects
appearing in the small weakly initial family. We write J : G → B for
the inclusion of this small full subcategory. By assumption, B admits
Q-weighted limits for some cofibrant weight Q: G → V for which the
unique map Q → 1 is a trivial fibration. We shall show that the limit
L = {Q, J} is bi-initial in B.
Following the notation of Section 5.2, we obtain morphisms pC : L →
JC for each C ∈ G, such that for f : C → D the triangle
L JC
JD
Jf

pC
GG
pD
11
∼=
commutes up to V-homotopy.
Let B ∈ B. Since G is weakly initial, there exists a C ∈ G and a
morphism f : JC → B, and now f ◦ pC : L → B gives the existence
part of the required property of L.
Suppose now that B ∈ B and that f, g: L → B are two morphisms.
As in Example 5.11, we may form the isoinserter k: K → L of f and g,
and fk ∼= gk. By weak initiality of G there is a morphism u: JC → K
for some C ∈ G. Now
f ◦ k ◦ u ◦ pC
∼= g ◦ k ◦ u ◦ pC
and so it will suffice to show that k ◦ u ◦ pC
∼= 1. Lemma 5.10 gives
v ◦ pC
∼= 1 for any v: JC → L and, in particular, for v = k ◦ u.
WEAK ADJOINT FUNCTOR THEOREMS 27
Theorem 6.2. Suppose that the unit object I of V is terminal. Let
B be a V-category with powers and enough cofibrantly-weighted limits,
and let U : B → A be a V-functor that preserves them. Then U has an
E-weak left adjoint if and only if U0 satisfies the solution set condition.
Proof. Suppose firstly that U has an E-weak left adjoint, and ηA : A →
UA is an E-weak reflection. Then the singleton family consisting of
(A , η) is a solution set for U0. This proves the “only if” direction.
Since I is terminal the unit V-category I is terminal in V-Cat and
hence complete as a V-category. Then for any A ∈ A, it follows from
Proposition 4.1 that the comma category A/U has any type of limit
which B has and U : B → A preserves, and so has enough cofibrantlyweighted
limits. By Proposition 4.5 we know that (A/U)0
∼= A/U0,
which has a weakly initial set of objects since U0 satisfies the solution
set condition. Hence, by Theorem 6.1, A/U has a bi-initial object
η: A → UA . Since B has powers and U preserves them, it follows
from Theorem 4.7 that
B(A , B)
U GG A(UA , UB)
A(η,UB)
GG A(A, UB)
is bi-terminal in V/A(A, UB) — in other words, by Proposition 4.6, it
is shrinkable.
7. V/I-categories and the weak adjoint functor theorem
in the case I = 1
The proof given in the previous section relied on the fact that the
enriched comma categories A/U had enough cofibrantly-weighted limits,
which we were able to prove only in the case where I = 1, since
only then could we be sure that the unit V-category I had enough
cofibrantly-weighted limits. We shall overcome this problem by viewing
A/U as a category enriched in the monoidal category V/I, whose unit
is terminal. To this end, observe that, since the unit I is (trivially) a
commutative monoid in V, the slice category V/I becomes a symmetric
monoidal category, with unit 1: I → I and with the tensor product of
(X, x: X → I) and (Y, y: Y → I) given by (X ⊗Y, x⊗y: X ⊗Y → I).
In particular, the unit is indeed terminal in V/I. The resulting symmetric
monoidal category V/I is also closed, with the internal hom of
(Y, y) and (Z, z) given by the left vertical in the pullback
(Y, y), (Z, z) GG

[Y, Z]
[Y,z]

I j
GG [I, I]
[y,I]
GG [Y, I]
(7.1)
28 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
The category V/I is once again complete and cocomplete. It becomes
a monoidal model category if we define a morphism to be a cofibration,
weak equivalence, or fibration just when its underlying morphism in V
is one.
A category enriched in V/I involves a collection of objects together
with, for each pair X, Y of objects, an object
CX,Y : C(X, Y ) → I
of V/I. Translating through the remaining structure, this is equally to
specify a V-category C together with a functor C : C → I to the unit
V-category. We sometimes write C = (C, C) for such a V/I-enriched
category, and we refer to the functor C as the augmentation.
Proposition 7.1. This correspondence extends to an isomorphism of
categories V/I-Cat ∼= V-Cat/I.
Proof. A V/I-functor F : C → D consists of a V-functor F : C → D
for which the action FC,D : C(C, D) → D(FC, FD) on homs commutes
with the maps into I; equivalently, such that F commutes with the
augmentations.
Example 7.2. The augmentations P : A/U → I and Q: A/A →
I equip the V-categories A/U and A/A with the structure of V/I-
categories.
The next four results extend Propositions 4.3, 4.5, 4.6, and Theorem
4.7 respectively, with essentially the same proof as before in each
case.
Proposition 7.3. The identity morphism on A is a terminal object in
the slice V/I-category A/A.
Proof. From (4.2), for any f : B → A we have the pullback
(A/A)((B, f), (A, 1A)) GG

B(B, A)
B(B,1A)

I
f
GG B(B, A)
and since the pullback of an isomorphism is an isomorphism, the left
vertical map is an isomorphism over I. As such, it is terminal as an
object of V/I.
Proposition 7.4. Let U : B → A and A ∈ A.
(1) Then the V/I-category A/U has underlying category A/U0. Furthermore,
parallel morphisms f, g: (B, b) (C, c) are V/Ihomotopic
if and only if f and g are V-homotopic under A.
WEAK ADJOINT FUNCTOR THEOREMS 29
(2) Then A/A has underlying category A0/A. Furthermore, parallel
morphisms f, g: (B, b) (C, c) are V/I-homotopic if and only
if f and g are V-homotopic over A.
Proof. Since the unit in V/I is 1: I → I, a morphism in (A/U)0 is
given by a map into the pullback
I GG (A/U)((B, b), (C, c)) GG

I
c

B(B, C) U
GG A(UB, UC)
b∗
GG A(A, UC)
whose component in I is the identity — we no longer require the assumption
I = 1 for this to be true. As in the proof of Proposition 4.5
this amounts to a morphism of A/U0. Now a V/I-interval is an interval
in V viewed as a diagram
(I + I, )
(d c)
GG (J, e)
e GG (I, 1)
in V/I. Therefore a (J, e)-homotopy f ∼= g in A/U is specified by a map
from J into the pullback whose component in I is fixed as e: J → I,
and so as in the proof of Proposition 4.5 amounts to a (V, J)-homotopy
h: f ∼= g for which Uh◦b is trivial. The case of A/A follows by duality,
as before.
Proposition 7.5. A morphism f : A → B ∈ C is shrinkable if and
only if it is biterminal in the slice V/I-category C/B.
Proof. The proof of Proposition 4.6 carries over unchanged on replacing
the use of Propositions 4.3 and 4.5 by their respective generalisations,
Propositions 7.3 and 7.4.
Theorem 7.6. Let B be a V-category with powers, and U : B → A
a V-functor which preserves them; if η: A → UA is bi-initial in the
V/I-category A/U then
B(A , B)
U GG A(UA , UB)
A(η,UB)
GG A(A, UB)
is bi-terminal in the V/I-category V/A(A, UB).
Proof. The proof of Theorem 4.7 carries over unchanged on replacing
the use of Proposition 4.5 by the more general Proposition 7.4.
The proof of the next proposition relies on two lemmas which we
defer until after the proof of our main result, Theorem 7.8 below.
30 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Proposition 7.7. If B has enough cofibrantly-weighted limits and U
preserves them, in the V-sense, then A/U has enough cofibrantly-weighted
limits in the V/I-sense.
Proof. By Lemma 7.11 below R(B) has enough cofibrantly-weighted
limits and R(U): R(A) → R(B) preserves them. Therefore, by Proposition
4.1, the comma V/I-category A/R(U) has enough cofibrantlyweighted
limits. By Lemma 7.9 below, A/R(U) is isomorphic to the
V/I-category P : A/U → I, which consequently has enough cofibrantlyweighted
limits too.
Theorem 7.8. Let B be a V-category with powers and enough cofibrantlyweighted
limits, and let U : B → A be a V-functor that preserves them.
Then U has an E-weak left adjoint if and only if U0 satisfies the solution
set condition.
Proof. If U has an E-weak left adjoint, and ηA : A → UA is an E-weak
reflection, then the singleton family consisting of (A , η) is a solution
set for U0. This proves the “only if” direction.
Conversely, by Proposition 7.7 it follows that the V/I-category A/U
has enough cofibrantly-weighted limits. By Proposition 7.4 we know
that the underlying category (A/U)0 of the V/I-category A/U is isomorphic
to A/U0. Since U0 satisfies the solution set condition A/U0,
and so also (A/U)0, have weakly initial sets of objects. Thus, by Theorem
6.1, A/U has a bi-initial object η: A → UA , and now by Theorem
7.6 the morphism
B(A , B)
U GG A(UA , UB)
A(η,UB)
GG A(A, UB) .
is bi-terminal in the V/I-category V/A(A, UB), and so is shrinkable
by Proposition 7.5.
In order to complete the proof of Proposition 7.7, we need to consider
weighted limits in V/I-categories of the form P : A/U → I. First, we
show that these are comma categories in the V/I-enriched sense.
To this end, first observe that the forgetful functor V-Cat/I →
V-Cat has a right adjoint R, sending A to the V/I-category π2 : A ×
I → I.
Lemma 7.9. Given a V-functor U : B → A and an object A ∈ A, the
V/I-category P : A/U → I is isomorphic to the comma V/I-category
A/R(U).
Proof. By Proposition 7.4, an object of the V/I-category A/U is given
by a pair (B, f : I → A(A, UB) ∈ V), which bijectively corresponds to
WEAK ADJOINT FUNCTOR THEOREMS 31
a pair (B, (f, 1): (I, 1) → (A(A, UB) × I, π2) ∈ V/I); that is, to an
object of A/R(U).
Given objects (B, (f, 1)) and (C, (g, 1)) of A/R(U), consider the following
diagram, in which each region is a pullback.
(A/R(U))((B, (f, 1)), (C, (g, 1))) GG

B(B, C) × I
π1
GG
U×1

B(B, C)
U

A(UB, UC) × I
π1
GG
A(f,UB)×1

A(UB, UC)
A(f,UB)

I
(g,1)
GG
g
IIA(A, UC) × I
π1
GG A(A, UC)
We may deduce an isomorphism
(A/R(U))((B, (f, 1)), (C, (g, 1))) ∼= A/U((B, f), (C, g))
in V/I, since the left hand side is defined by the pullback on the left,
and the right hand side is defined by the composite pullback. The
remaining verifications of functoriality are straightforward.
To complete the argument, we need to consider weighted limits in
the V/I-enriched sense. To get started, observe, on comparing (4.2)
and (7.1), that V/I, as a V/I-enriched category, is given by the slice
V-category Q: V/I → I. We denote it by V/I = (V/I, Q).
Let D = (D, D) be a V/I-category. Then a V/I-weight D → V/I is
specified by a commutative triangle as on the left below.
D GG
D
33
V/I GG
Q

V
⇓ ≡
D
W GG
D
00
⇓w
V
I I
GG V I
I
dd
By the universal property of the slice V-category V/I this amounts
to a V-weight W : D → V together with a V-natural transformation
w: W ⇒ ID. We write W = (W, w) for the V/I-weight.
Building on the above description, it is straightforward to see that
the presheaf V/I-category [D, V/I] is given by the slice V-category
[D, V]/ID equipped with its natural projection to I. In particular,
32 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
its homs are defined by the following pullbacks.
[D, V/I](F, G)
U

GG I
f

[D, V](F, G) g∗
GG [D, V](F, ID)
(7.2)
Finally, observe that given a diagram S : D → A and an object
X ∈ A the induced presheaf
A(X, S−): D → V/I
consists of the V-weight A(X, S−): D → V together with the augmentation
A(X, S−) → ID having components AX,SY : A(X, SY ) → I.
Lemma 7.10. Let W = (W, w) be a V/I-weight. If B has W-weighted
limits then R(B) has W-weighted limits; if U : B → A preserves Wweighted
limits then R(U) preserves W-weighted limits.
Proof. Consider a diagram D → R(B) = (B × I, π2), which is necessarily
of the form S = (S, D) for S : D → B. We claim that the limit
{W, S} also has the universal property of {W, S}.
Using the definition of homs in [D, V/I] we have a pullback
[D, V/I](W, R(B)(B, S−)) GG
U

I
w

[D, V](W, B(B, S−) × ID)
(π2)∗
GG [D, V](W, ID)
and so, using the universal property of the product B(B, S−)×ID, an
isomorphism
[D, V/I](W, R(B)(B, S−)) ∼= [D, V](W, B(B, S−)) × I
over I, and now the right hand side is naturally isomorphic to
B(B, {W, S}) × I = R(B)(B, {W, S}),
as required. This proves that {W, S} also has the universal property of
{W, S}. Given this, the corresponding statement about preservation is
straightforward.
Lemma 7.11. If B has enough cofibrantly-weighted limits and U : B →
A preserves them, then R(B) has enough cofibrantly-weighted limits (in
the V/I-enriched sense) and R(U): R(B) → R(A) preserves them.
Proof. Let D be a V/I-category. Re-expressing the definition of enough
cofibrantly-weighted limits in these terms, we see that there exists a
WEAK ADJOINT FUNCTOR THEOREMS 33
weight W = (W, w) with W cofibrant and w: W → ID a trivial fibration,
and such that B has W-weighted limits and U preserves them. By
Lemma 7.10, this implies that R(B) has W-weighted limits and R(U)
preserves them.
Now (ID, 1) is the terminal V/I-weight so that the trivial fibration
w: W → ID equally specifies a trivial fibration w: W → 1. Therefore,
if we can show that W cofibrant implies W cofibrant, we will
have shown that R(B) has enough cofibrantly-weighted limits as a V/Icategory
and that R(U) preserves them.
To this end, suppose that α: F → G is a trivial fibration and consider
the commutative square on the left below.
[D, V/I](W, F)
UW,F

α∗
GG [D, V/I](W, G)
UW,G

GG I
w

[D, V](W, F) α∗
GG [D, V](W, G) g∗
GG [D, V](W, ID)
Pasting this with the pullback square (7.2) defining [D, V/I](W, G) as
above right yields the pullback square defining [D, V/I](W, F). Therefore
the left square above is a pullback in V. Now since W is cofibrant,
the lower α∗ in the left square is a trivial fibration whence so is its
pullback on the top row, as required.
In the case of the ordinary adjoint functor theorem, there is of course
a converse: if a functor has a left adjoint then it satisfies the solution set
condition and preserves any existing limits. In our setting, if there is a
weak left adjoint, then the solution set condition does hold (as stated
in the theorem), but the preservation of enough cofibrantly-weighted
limits need not, as the following example shows.
Example 7.12. Let V = Set with the split model structure, so that
the shrinkable morphisms are just the surjections, and we are dealing
with classical weak category theory. Let G be a non-trivial group, and
SetG
the category of G-sets. The forgetful functor U : SetG
→ Set has
a left adjoint F. Let P : SetG
→ Set be the functor sending a G-set
to its set of orbits. There is a natural transformation π: U → P whose
components are surjective. For each set X, the composite of the unit
X → UFX and π: UFX → PFX exhibits F as a weak left adjoint
to P. But P does not preserve powers (or products): in particular, it
does not preserve the power G2
.
In good cases, however, there is a sort of converse involving preservation
of homotopy limits, as we now describe.
34 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Theorem 7.13. Let B and A be locally fibrant V-categories, and let
U : B → A have a E-weak left adjoint. Let Q: D → V be a cofibrant
weight for which [D, V](Q, −): [D, V] → V sends levelwise shrinkable
morphisms between levelwise fibrant objects to weak equivalences. Then
U preserves homotopy Q-weighted limits.
Proof. The induced maps B(A , B) → A(A, UB) are shrinkable morphisms
between fibrant objects and so are weak equivalences (between
fibrant objects). Suppose that ϕ: Q → B(L, S) exhibits L as the homotopy
weighted limit {Q, S}h, meaning that for each B ∈ B the induced
morphism B(B, L) → [D, V](Q, B(B, S)) is a weak equivalence in V.
Then given A ∈ A there is a commutative diagram
B(A , L) GG

[D, V](Q, B(A , S))

A(A, UL) GG [D, V](Q, A(A, US))
in which the upper horizontal is the weak equivalence expressing the
universal property of L = {Q, S}, the left vertical is a shrinkable morphism
(and so a weak equivalence) expressing the universal property
of A , and the right vertical is given by [D, V](Q, −) applied to a levelwise
shrinkable morphism between levelwise fibrant objects, and so is
a weak equivalence. Then the bottom map is also a weak equivalence,
and so U preserves the homotopy limit.
Example 7.14. In [25], Q-weighted limits were said to be E-stable if
E, seen as a full subcategory of V2
, is closed under Q-weighted limits;
in other words, if [D, V](Q, −): [D, V] → V sends a V-natural transformation
whose components are in E to a morphism in E. If moreover
the shrinkable morphisms are weak equivalences then the hypothesis in
the theorem holds, and so U will preserve homotopy Q-weighted limits.
This is the case for all cofibrant Q in the case of Examples 2.1, 2.2,
and 2.3, but not in Example 2.4.
Example 7.15. If the enriched projective model structure on [D, V]
exists, as is the case for Example 2.4, then once again the hypothesis
holds for all cofibrant Q. For a levelwise shrinkable morphism between
levelwise fibrant objects is a weak equivalence between fibrant objects
in the projective model structure, and so sent by [D, V](Q, −) to a weak
equivalence (between fibrant objects).
WEAK ADJOINT FUNCTOR THEOREMS 35
8. Accessibility and E-weak colimits
In ordinary category theory an accessible category which is complete
is also cocomplete — this provides one simple way to see that algebraic
categories, which are obviously complete, are also cocomplete.
The present section is devoted to generalising this to our setting,
which we do in Theorem 8.9. In Section 9 we shall use Theorem 8.9
to deduce homotopical cocompleteness of various enriched categories
of (higher) categorical structures.
8.1. E-weak colimits. The paper [25] introduced the notion of E-weak
colimit for any class of morphisms E. We shall use this in our current
setting where E consists of the shrinkable morphisms.
Definition 8.1. Let W : Dop
→ V be a weight, and consider a diagram
S : D → A. A V-natural transformation
W
η
GG A(S−, C)
in [Dop
, V] exhibits C as the E-weak colimit of S weighted by W if the
induced map
A(C, A)
η∗◦A(S−,1)
GG [Dop
, V](W, A(S−, A)) (8.1)
belongs to E for all A ∈ A.
This is equally the assertion that η exhibits C as an E-weak reflection
of W along the V-functor A(S−, 1): A → [Dop
, V] sending A to
A(S−, A).
Example 8.2. For V with the trivial model structure, E-weak colimits
are weighted colimits in the usual sense.
Example 8.3. For V with the split model structure, η: W → A(S−, C)
exhibits C as the E-weak colimit just when the induced map A(C, A) →
[Dop
, V](W, A(S−, A)) is a split epimorphism in V. This implies in particular
that given f : W → A(S−, A) there exists f : C → A such that
the triangle
W
η

f
GG A(S−, A)
A(S−, C)
A(−,f )
UU
(8.2)
commutes. Indeed, when V = Set, it amounts to precisely this condition,
and then if W is the terminal weight, the E-weak colimit reduces
to the ordinary weak (conical) colimit of S.
36 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Example 8.4. In the Cat case, the map
A(C, A) → [Dop
, Cat](W, A(S−, A))
is asked to be a surjective equivalence. So given f : W → A(S−, A) we
have a factorisation as in (8.2) above; and furthermore, given α: f ⇒
g ∈ [Dop
, Cat](W, A(S−, A)) there exists a unique α : f ⇒ g such
that A(S−, α ) ◦ η = α. For instance, the E-weak coequaliser
X
f
GG
g
GG Y
e GG C
satisfies e ◦ f = e ◦ g and has the following properties:
(1) if h: Y → D satisfies h◦f = h◦g, there exists h : C → D such
that h ◦ e = h;
(2) if moreover k: Y → D satisfies k ◦ f = k ◦ g, and θ: h ⇒ k
satisfies θ ◦f = θ ◦g, there exists a unique θ : h ⇒ k such that
θ ◦ e = θ.
Let us compare E-weak colimits with the better known notion of
bicolimits — given W and S as before, the W-weighted bicolimit C =
W ∗b S [20] is defined by a pseudonatural transformation η: W →
A(S−, C) such that the induced map
A(C, A) → Ps(Dop
, Cat)(W, A(S−, A))
is an equivalence of categories, where Ps(Dop
, Cat) denotes the 2category
of 2-functors, pseudonatural transformations and modifications
from Dop
to Cat. To make the comparison, recall from [5,
Remark 3.15] that the identity on objects inclusion of [Dop
, Cat] in
Ps(Dop
, Cat) has a left adjoint (−) called the pseudomorphism classifier,
with counit qW : W → W. The morphism qW : W → W is in
fact a cofibrant replacement of the weight W in the projective model
structure on [Dop
, Cat]: see [24, Sections 5.9 and 6]. Given the iso-
morphism
[Dop
, Cat](W , A(S−, A)) ∼= Ps(Dop
, Cat)(W, A(S−, A)),
the bicolimit of S weighted by W equally amounts to a 2-natural transformation
η: W → A(S−, C) for which the induced map
A(C, A) → [Dop
, Cat](W , A(S−, A))
is an equivalence of categories for all A. In particular, the E-weak
colimit of S weighted by W is a W-weighted bicolimit of S, but satisfies
the stronger condition that genuine factorisations, as in (8.2), exist.
Thus any 2-category admitting E-weak colimits also admits bicolimits
(with a stronger universal property), but also admits some E-weak
WEAK ADJOINT FUNCTOR THEOREMS 37
colimits, such as E-weak coequalisers, whose defining weights are not
cofibrant, and so do not correspond to any bicolimit.
Example 8.5. In the SSet-case, the map
A(C, A) → [Dop
, SSet](W, A(S−, A))
is a shrinkable morphism and so a weak equivalence (with respect to
the Joyal model structure) with a section.
In the case that W is flexible and A is locally fibrant — as, for instance,
if A is an ∞-cosmos — we can say rather more. For then,
since A(S−, A) is pointwise fibrant in [Dop
, SSet] and W is cofibrant,
the hom [Dop
, SSet](W, A(S−, A)) is also fibrant; that is, a quasicategory.
It follows, by Example 3.8, that the shrinkable morphism
A(C, A) → [Dop
, SSet](W, A(S−, A)) is a surjective equivalence of qua-
sicategories.
For A an ∞-cosmos and W a flexible weight, Riehl and Verity [33]
define the flexibly-weighted homotopy colimit of S weighted by W as
an object C together with a morphism W → A(S−, C) for which the
induced map A(C, A) → [Dop
, SSet](W, A(S−, A)) is an equivalence of
quasicategories for all A. In particular, if A admits E-weak colimits it
admits flexibly-weighted homotopy colimits with the stronger property
that the comparison equivalence of quasicategories is in fact a surjective
equivalence.
8.2. Accessible V-categories with enough cofibrantly-weighted
limits have E-weak colimits. In the present section we suppose that
(the unenriched category) V0 is locally presentable. It follows by [21,
Proposition 2.4] that there is a regular cardinal λ0 such that V0 is locally
λ0-presentable, the unit object I is λ0-presentable, and the tensor
product of two λ0-presentable objects is λ0-presentable. Moreover, the
corresponding statements will remain true for any regular λ ≥ λ0.
Notation 8.6. We let λ0 denote a fixed regular cardinal as in the previous
paragraph. Whenever we consider λ-presentability or λ-accessibility
in the V-enriched context, we shall always suppose that λ is a regular
cardinal and λ ≥ λ0.
If H is a small category we can speak of conical H-shaped colimits
in C: these are H-shaped colimits in C0 which are preserved by
C(−, C)0 : C0 → Vop
0 for each C ∈ C. In particular we can speak of
λ-filtered colimits in C, corresponding to H-shaped colimits for all λfiltered
categories H.
38 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
An object A in C is said to be λ-presentable if C(A, −): C → V
preserves λ-filtered colimits. We say that C is λ-accessible2
if it admits
λ-filtered colimits and a set G of λ-presentable objects such that each
object of C is a λ-filtered colimit of objects in G. It follows, arguing
as usual, that the full subcategory of λ-presentable objects in C is
essentially small and we denote by J : Cλ → C a small skeletal full
subcategory of λ-presentables. It follows from [18, Theorem 5.19] that
J is dense, and from [18, Theorem 5.29] that J then exhibits C as
the free completion of Cλ under λ-filtered colimits. (Indeed, the λaccessible
V-categories are equally the free completions of small Vcategories
under λ-filtered colimits.)
Notation 8.7. Let D: A → C be a V-functor with small domain.
We write ND for the induced V-functor C → [Aop
, V] sending an object
C ∈ C to the presheaf C(D−, C) which in turn sends A ∈ A
to C(DA, C). Other authors have written C(D, 1) or ˜D for ND; our
notation is designed to remind that this is a generalised (N)erve.
Let C be a λ-accessible V-category, and J : Cλ → C the inclusion of
the full sub-V-category of λ-presentable objects. Then J is dense, so
that the associated functor NJ : C → [Cop
λ , V] is fully faithful.
Proposition 8.8. If C is λ-accessible, with powers and enough cofibrantlyweighted
limits, then NJ : C → [Cop
λ , V] admits an E-weak left adjoint.
Proof. The composite of NJ with the evaluation functor evX at a λpresentable
object X is the representable C(JX, −). Each such Vfunctor
preserves λ-filtered colimits; so, since the evaluation functors
are jointly conservative, NJ also preserves λ-filtered colimits. Now
[Cop
λ , V] is locally λ-presentable as a V-category by [19, Examples 3.4].
In particular, since (NJ )0 : C0 → [Cop
λ , V]0 is a λ-filtered colimit preserving
functor between λ-accessible categories, it satisfies the solution set
condition. The result now follows from our adjoint functor theorem,
Theorem 7.8.
Theorem 8.9. Let C be an accessible V-category with powers and
enough cofibrantly-weighted limits. Then C admits all E-weak colim-
its.
Proof. Given Proposition 8.8, this follows directly from Proposition
4.3 of [25], the argument of which we repeat here for convenience. Let
D: A → C with A small. We must show that ND : C → [Aop
, V] has
an E-weak left adjoint; then the value of this adjoint at W ∈ [Aop
, V]
2A different notion of enriched accessibility was given in [6].
WEAK ADJOINT FUNCTOR THEOREMS 39
will give the E-weak colimit of D weighted by W. First suppose that
C is λ-accessible and consider the dense inclusion J : Cλ → C. We then
have the following diagram.
A
D GG C
ND 99
NJ
GG [Cop
λ , V]
N(NJ D)=[Cop
λ ,V](NJ D−,1)

[Aop
, V]
Since NJ is fully faithful, there are natural isomorphisms
NNJ DNJ C ∼= [Aop
, V](NJ D−, NJ C) ∼= C(D−, C) ∼= NDC
and so the triangle commutes up to isomorphism. Thus it will suffice
to show that NJ and NNJ D have E-weak left adjoints. The first of these
does so by Proposition 8.8, while the second has a genuine left adjoint,
namely the weighted colimit functor NJ D −.
We follow [25] in saying that W-weighted limits are E-stable if the
enriched full subcategory E → V2
is closed under W-weighted limits.
Similarly, we say that enough cofibrantly-weighted limits are E-stable if
E has, and the inclusion E → V2
preserves, enough cofibrantly-weighted
limits. In that case we can use results from [25] to prove a converse to
the previous theorem.
Theorem 8.10. Suppose that the shrinkable morphisms in V are closed
in V2
under enough cofibrantly-weighted limits, and let C be an accessible
V-category. Then the following are equivalent.
(i) C admits E-stable limits.
(ii) C admits powers and enough cofibrantly-weighted limits.
(iii) C admits E-weak colimits.
Proof. Since shrinkable morphisms are always closed under powers,
powers are E-stable. Since also enough cofibrantly-weighted limits are
E-stable by assumption, (1 =⇒ 2). The implication (2 =⇒ 3) holds
by Theorem 8.9. For (3 =⇒ 1), suppose that C is λ-accessible and
consider the dense inclusion J : Cλ → C. Then NJ : C → [Cop
λ , V] is
fully faithful and has an E-weak left adjoint sending W to the E-weak
colimit of J weighted by W. Furthermore, since each morphism of E is
a split epimorphism, the representable V(I, −): V0 → Set sends morphisms
of E to surjections: in the language of [25] this says that I is
E-projective. Using Propositions 6.1 and 2.6 of [25] we deduce that C
is closed in [Cop
λ , V] under E-stable limits, as required.
40 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
8.3. Recognising accessible enriched categories. Examples of accessible
V-categories C are much easier to identify in the case that C
admits powers by a suitable strong generator G of V0. As observed in
[18, Section 3.8], if the V-category C has powers, the difference between
conical colimits in C and conical colimits in the underlying category C0
disappears. But since the map C(colimi Di, C) → limi C(Di, C) expressing
the universal property of a conical colimit will be invertible if
and only if the induced V0(G, C(colimi Di, C)) → V0(G, limi C(Di, C))
is so for each G ∈ G, it suffices for C to have powers by objects in
G. Building on this well-known argument, the following proposition
enables us to recognise accessible V-categories by looking at their underlying
categories.
Recall that λ0 satisfies the standing assumptions of Notation 8.6.
Proposition 8.11. Let V0 be locally presentable, let G be a strong
generator of V0, and let λ ≥ λ0 be a regular cardinal such that each
G ∈ G is λ-presentable in V0. Then for any V-category C with powers
by objects in G, the following are equivalent:
(1) C is λ-accessible as a V-category;
(2) C0 is λ-accessible and C(C, −)0 : C0 → V0 is a λ-accessible functor,
for each C in some strong generator of C0;
(3) C0 is λ-accessible and (G −)0 : C0 → C0 is a λ-accessible
functor, for each G ∈ G.
Moreover, for such λ and C, an object C ∈ C is λ-presentable if and
only if it is λ-presentable in C0.
Proof. Because of the powers, C has λ-filtered colimits if and only if
C0 does so, and a class H of objects of C generates C under λ-filtered
colimits if and only if it generates C0 under λ-filtered colimits. So the
only possible difference between λ-accessibility of C and λ-accessibility
of C0 lies in the possible difference between λ-presentability in C and
λ-presentability in C0.
Since I is λ-presentable in V0, if C is λ-presentable in C then the
composite
C0
C(C,−)0
GG V0
V0(I,−)
GG Set
preserves λ-filtered colimits, but this composite is C0(C, −), and so C is
λ-presentable in C0. Thus (Cλ)0 ⊆ (C0)λ; while if C is λ-accessible as a
V-category, then C0 is an accessible ordinary category, and (1) implies
(2).
For an arbitrary G-powered V-category C, however, a λ-presentable
object of C0 might fail to be λ-presentable in C.
WEAK ADJOINT FUNCTOR THEOREMS 41
For the remaining implications, consider the diagram
C0
(G −)0
GG
C(C,−)0

C0
C0(C,−)

V0
V0(G,−)
GG Set
(8.3)
in which C ∈ C, G ∈ G, and so the lower horizontal preserves λ-filtered
colimits.
Suppose that (2) holds, so that there is a strong generator H for C0
consisting of objects which are λ-presentable in C (and so also in C0).
As C ranges through H, the lower composite V0(G, C(C, −)0) in (8.3)
preserves λ-filtered colimits, while the C0(C, −) preserve and jointly
reflect them. Thus (G −)0 preserves them and (3) holds.
Now suppose that (3) holds. If C is λ-presentable in C0, then the
upper composite C0(C, G −) in (8.3) preserves λ-filtered colimits,
while the V0(G, −) preserve and jointly reflect them, thus C(C, −)0
also preserves them, and C is λ-presentable in C. Thus (C0)λ ⊆ (Cλ)0
and (1) follows.
In practice, it is often convenient to lift enriched accessibility from
one (typically locally presentable) V-category to another using the fol-
lowing.
Corollary 8.12. Let A and C be V-categories with powers by objects
in G and let U : A → C be a conservative V-functor preserving them.
Suppose that C is an accessible V-category, A0 is an accessible category,
and U0 : A0 → C0 is an accessible functor. Then in fact A is accessible
as a V-category.
Proof. Consider the following commutative diagram
A0
(G −)0
GG
U0

A0
U0

C0
(G −)0
GG C0
where G ∈ G.
The categories A0 and C0 are both accessible, U0 is accessible by
assumption, and the lower horizontal by Proposition 8.11 and the fact
that C is an accessible V-category. By the uniformization theorem
[1, Theorem 2.19] for accessible categories, we can choose λ ≥ λ0
42 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
such that each of the aforementioned categories and functors is λaccessible.
Since U0 is conservative, the upper horizontal is also λaccessible.
Therefore A is a λ-accessible V-category by Proposition 8.11
once again.
9. Examples and applications
To summarise, we have two main results. First, our weak adjoint
functor theorem, and second, our result on the E-weak cocompleteness
of accessible V-categories with sufficient limits. In this final section we
illustrate the scope of these results by describing what they capture
in our various settings, with a particular emphasis on 2-categories and
∞-cosmoi.
9.1. The classical case. In the case of Set with the trivial model
structure, a category A has powers and enough cofibrantly-weighted
limits just when it is complete. The shrinkable morphisms in Set
are the bijections, and these are of course closed in Set2
under (all)
limits. Therefore our weak adjoint functor theorem specialises exactly
to the general adjoint functor theorem of Freyd [27]. Theorem 8.10
becomes the well-known result that an accessible category is complete
if and only if it is cocomplete: see for example [1, Corollary 2.47].
Since preservation of homotopy limits is just preservation of limits in
this case, Theorem 7.13 is just the elementary fact that right adjoints
preserve limits.
For general V equipped with the trivial model structure, a V-category
A has powers and enough cofibrantly-weighted limits just when it is
complete, now in the sense of weighted limits, and the shrinkable morphisms
are again the isomorphisms. Therefore our weak adjoint functor
theorem specialises exactly to the adjoint functor theorem for enriched
categories. Theorem 7.13 says that right adjoints preserve limits,
and Theorem 8.10 yields the well-known result that an accessible
V-category is complete if and only if it is cocomplete.
9.2. Ordinary weakness. In the case of Set with the split model
structure we know from Example 5.6 that for a category A to have
enough cofibrantly-weighted limits it suffices that it admit products.
Furthermore the shrinkable morphisms are the surjections, and these
are closed under products. Since powers are products in the Setenriched
setting our weak adjoint functor theorem thus yields the weak
adjoint functor theorem of Kainen [16]. Theorem 8.10 yields the well
known result that an accessible category has products if and only if it
has weak colimits: see for example [1, Theorem 4.11].
WEAK ADJOINT FUNCTOR THEOREMS 43
For general V equipped with the split model structure, our results
appear to be completely new. In this setting a V-category A has powers
and enough cofibrantly-weighted limits provided that it has powers and
products, and the shrinkable morphisms are the split epimorphisms,
which are of course closed under powers and products. Using this, our
main results (in slightly weakened form) are:
• Let B be a V-category with products and powers, and let U : B →
A be a V-functor that preserves them. Then U has a (split
epi)-weak left adjoint if and only if it satisfies the solution set
condition.
• An accessible V-category has products and powers if and only
if it has (split epi)-weak colimits.
9.3. 2-category theory. One of our guiding topics is that of 2-category
theory, understood as Cat-enriched category theory, and where Cat is
equipped with the canonical model structure.
As recalled in Example 5.7, in this case the cofibrantly-weighted
limits are precisely the flexible limits of [4], and any 2-category with
flexible limits has powers. The shrinkable morphisms are the retract
equivalences, and these are closed in Cat2
under cofibrantly-weighted
limits as observed, for example, in [25, Section 9].
Our primary applications will be in the accessible setting which we
describe now. Since each accessible functor between accessible categories
satisfies the solution set condition, our weak adjoint functor in
this setting gives the first part of the following, the second part of which
is the instantiation of (part of) Theorem 8.10.
• Let U : B → A be an accessible 2-functor between accessible
2-categories. If B has flexible limits and U preserves them then
it has an E-weak left adjoint.
• An accessible 2-category has flexible limits if and only if it has
E-weak colimits.
The first result is new; the second is part of Theorem 9.4 of [25]. The
utility of these results lies in the fact that many, though not all, 2categories
of pseudomorphisms are in fact accessible with flexible limits.
For instance, the 2-category of monoidal categories and strong
monoidal functors is accessible (although, as recalled in Section 5, the
2-category of strict monoidal categories and strong monoidal functors
is not [8, Section 6.2]). The difference between the two cases is that
the definition of strict monoidal category involves equations between
objects whereas that of monoidal category does not — in terms of the
associated 2-monads this corresponds to the fact that the 2-monad for
44 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
strict monoidal categories is not cofibrant (=flexible) whereas that for
monoidal categories is. One result generalising this is Corollary 7.3 of
[8], which asserts that if T is a finitary flexible 2-monad on Cat then
the 2-category T-Algp of strict algebras and pseudomorphisms is accessible
with flexible limits and filtered colimits, and these are preserved
by the forgetful functor to Cat. It now follows from our theorem that
any such 2-category admits E-weak colimits, and in particular bicolimits,
and also that if f : S → T is a morphism of such 2-monads, then
the induced map T-Algp → S-Algp has an E-weak left adjoint, and so
a left biadjoint. These results concerning bicolimits and biadjoints are
not new, being special cases of the results of Section 5 of [5].
However, as shown in [8, Section 6.5], many structures beyond the
scope of two-dimensional monad theory are also accessible. For instance,
the 2-category of small regular categories and regular functors
is accessible with flexible limits and filtered colimits preserved by the
forgetful 2-functor to Cat. Similar results hold for Barr-exact categories,
coherent categories, distributive categories, and so on. It follows
from the above theorems that the 2-categories of such structures
admit E-weak colimits, and so bicolimits, and this is the simplest proof
of bicocompleteness in these examples that we know. Again, for such
structures the weak adjoint functor theorem provides an easy technique
for constructing E-weak left adjoints, and so biadjoints, to forgetful 2functors
between them.
Theorem 7.13 asserts that a 2-functor with an E-weak left adjoint
preserves bilimits — since such 2-functors are right biadjoints, this is
a special case of the well known fact that right biadjoints preserve
bilimits.
9.4. Simplicial enrichment and accessible ∞-cosmoi. Our second
main motivation concerns the ∞-cosmoi of Riehl and Verity [35].
Here the base for enrichment is SSet equipped with the Joyal model
structure. In this setting, for a simplicially enriched category B to
admit powers and enough cofibrantly-weighted limits it suffices, as observed
in Example 5.8, that it admit flexible limits in the sense of
[35]. As observed in Example 3.8, every shrinkable morphism is a weak
equivalence which has a section.
In this example it can be shown that the shrinkable morphisms are
not closed in SSet2
under cofibrantly-weighted limits, so we have to
content ourselves with Theorem 8.9 rather than Theorem 8.10. On
the other hand, enriched projective model structures do exist, so Theorem
7.13 does imply that E-weak right adjoints preserve cofibrantlyweighted
homotopy limits.
WEAK ADJOINT FUNCTOR THEOREMS 45
Specialising our main results, we then obtain:
• Let B be a SSet-category with flexible limits, and let U : B → A
be a V-functor which preserves them. Then U has an E-weak
left adjoint provided that it satisfies the solution set condition.
• If A is an accessible SSet-category with flexible limits then it
has E-weak colimits.
We wish to examine these results further in the setting of ∞-cosmoi.
By definition, an ∞-cosmos is a simplicially enriched category A equipped
with a class of morphisms called isofibrations satisfying a number of
axioms — see Chapter 1 of [35]. For the purposes of the present paper,
the reader need only know that the homs of an ∞-cosmos are quasicategories
and that it admits flexible limits, in the sense of Examples 5.8.
The idea is that the objects of an ∞-cosmos are ∞-categories, broadly
interpreted, and that the axioms for an ∞-cosmos provide what is
needed to define and work with key structures — such as limits and
adjoints — that arise in the ∞-categorical world. For instance one can
define what it means for an object of an ∞-cosmos to have limits. This
is model-independent, in the sense that one recovers the usual notion
of quasicategory with limits or complete Segal space with limits on
choosing the appropriate ∞-cosmos.
The natural structure preserving morphisms between ∞-cosmoi are
called cosmological functors, and cosmological functors preserve flexible
limits. By an accessible ∞-cosmos, we mean an ∞-cosmos which is
accessible as a simplicial category3
, whilst a cosmological functor is
accessible if it is so as a simplicial functor.
Our main insight here is that many of the ∞-cosmoi that arise in
practice are accessible, as are the cosmological functors between them.
Thus the following specialisations of our main results are broadly applicable.
We remind the reader that in this context the relevant morphisms
of E are certain surjective equivalences of quasicategories — see
Examples 3.8.
• Let U : B → A be an accessible cosmological functor between
accessible ∞-cosmoi. Then U has an E-weak left adjoint.
• If A is an accessible ∞-cosmos then it has E-weak colimits, and
in particular flexibly-weighted homotopy colimits (see Example
8.5.)
The first examples of accessible ∞-cosmoi come from the following
result, the first part of which is due to Riehl and Verity [35].
3There are further conditions that one could ask for, such as the accessibility of
the class of isofibrations, but we shall not consider these in the present paper.
46 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
Proposition 9.1. (1) Let A be a model category that is enriched
over the Joyal model structure on simplicial sets, and in which
every fibrant object is cofibrant. Then the simplicial subcategory
Afib spanned by its fibrant objects is an ∞-cosmos, and is closed
in A under flexible limits.
(2) If A is furthermore a combinatorial model category then Afib
is an accessible ∞-cosmos and the inclusion Afib → A is an
accessible embedding.
Proof. Part (1) holds by Proposition E.1.1 of [35]. In Part (2), A0 is
locally presentable and A is cocomplete as a V-category by assumption.
Since for each X ∈ SSet the functor (X −)0 : A0 → A0 has left
adjoint (X.−)0 given by taking copowers (or tensors) it is accessible.
Therefore A is accessible as a SSet-category by Proposition 8.11. Since
Afib is closed in A under flexible limits, it is closed under powers.
Since A is combinatorial, (Afib)0 = Inj(J) → A0 where J is the set of
generating trivial cofibrations in A. Now by Theorem 4.8 of [1] Inj(J)
is accessible and accessibly embedded, so by Corollary 8.12 we conclude
that Afib is accessible and therefore an accessible ∞-cosmos.
As explained in Appendix E of [35], one obtains as instances of the
above the ∞-cosmoi of quasicategories, complete Segal spaces, Segal
categories and Θn-spaces. Since all of the defining model structures
are combinatorial, it follows furthermore that each of these ∞-cosmoi
is accessible.
The above ∞-cosmoi can be regarded as basic. In future work, we
aim to vastly extend the scope of these examples by establishing that
accessible ∞-cosmoi (satisfying natural additional properties) are stable
under a variety of constructions, such as the passage from an ∞cosmos
A to the ∞-cosmos Cart(A) of cartesian fibrations therein.
For (not necessarily accessible) ∞-cosmoi this was done in Chapter 6
of [35], and it remains to add accessibility to the mix. (The corresponding
stability results for the appropriate class of accessible 2-categories
have been established in [8].)
In the present paper we content ourselves with describing two examples
of accessible ∞-cosmoi built from the quasicategories example: the
∞-cosmoi of quasicategories with a class of limits, and the ∞-cosmos
of cartesian fibrations of quasicategories. In order to construct these
∞-cosmoi systematically, we shall make use of the generalised sketch
categories C|C of Makkai [29], which we now recall.
Given a category C and object C ∈ C we form the category C|C,
an object of which consists of an object A ∈ C together with a subset
AC ⊆ C(C, A), which we often refer to as a marking. A morphism
WEAK ADJOINT FUNCTOR THEOREMS 47
f : (A, AC) → (B, BC) consists of a morphism f : A → B in C such
that if x ∈ AC then f ◦x ∈ BC. The forgetful functor C|C → C functor
has a left adjoint equipping A with the empty marking (A, ∅).
The following straightforward result is established in Item (9) of
Section (1) of [29], except that Makkai works with locally finitely presentable
categories. In its present form, it is a special case of Proposition
2.2 of [13].
Proposition 9.2. Let C be locally presentable and C ∈ C. Then C|C
is also locally presentable and the forgetful functor C|C → C preserves
filtered colimits.
If J is a set of morphisms in C|C then we may form the category
Inj(J ) of J -injectives in C|C. We follow Makkai’s terminology in calling
such a category Inj(J ) a doctrine in C.
Corollary 9.3. Let C be locally presentable, with C ∈ C and J a set
of morphisms in C|C. Then Inj(J ) is accessible, and the composite
forgetful functor U : Inj(J ) → C is accessible.
Proof. By the preceding proposition C|C is locally presentable whence,
by Theorem 4.8 of [1], the full subcategory Inj(J ) → C|C is accessible
and accessibly embedded. The composite Inj(J ) → C|C → C is
accessible since its two components are.
We now combine this with simplicial enrichment.
Corollary 9.4. Let A be an ∞-cosmos and U : A → C a conservative
simplicially enriched functor preserving flexible limits to a locally presentable
simplicially enriched category. Suppose that U0 : A0 → C0 has
the form Inj(J ) → C|C0 → C0 for some C ∈ C and set J of morphisms
in C|C0. Then A is accessible as a simplicial category, and so is an
accessible ∞-cosmos.
Proof. Since flexible limits include powers, this follows immediately
from Corollaries 8.12 and 9.3.
In the following two examples, we consider the ∞-cosmoi QCatD of
quasicategories with D-limits, and Cart(QCat) of cartesian fibrations
respectively.
These are shown to be ∞-cosmoi in Propositions 6.4.11 and 6.4.12,
respectively, of [35]; moreover the forgetful simplicial functors QCatD →
SSet and Cart(QCat) → SSet2
preserve flexible limits and are conservative.
To prove that they are accessible ∞-cosmoi, it suffices by
Corollary 9.4 to describe the categories (QCatD)0 and Cart(QCat)0
as doctrines, respectively, in SSet0 and in (SSet2
)0.
48 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
9.4.1. Quasicategories with limits of a given class. To begin with, we
treat the simpler case of quasicategories with a terminal object. To
this end, consider ∆0
|SSet, whose objects are pairs (X, U) where X
is a simplicial set and U ⊆ X0 a distinguished subset of marked 0simplices,
and whose morphisms are simplicial maps preserving marked
0-simplices. We shall describe the category QCatT of quasicategories
admitting a terminal object and morphisms preserving them as the
small injectivity class in ∆0
|SSet consisting of those (X, U) for which
X is a quasicategory and U ⊆ X0 is the set of terminal objects in X.
To begin with, we view the inner horn inclusions
{Λk[n]: Λn
k → ∆n
| 0 < k < n} (9.1)
as morphisms of ∆0
|SSet in which no 0-simplex is marked in their
source or target — then (X, U) is injective with respect to the inner
horns just when X is a quasicategory. Now recall (see Definition 4.1
of [14] for the dual case of an initial object), that a 0-simplex a of a
quasicategory X is terminal if the solid part of each diagram
∂∆n
jn

f
GG X
∆n
UU (9.2)
in which f : ∂∆n
→ X has value a at the final vertex n ∈ ∂∆n
admits a
filler. Accordingly, if (X, U) is a quasicategory with marked 0-simplices,
then each marked 0-simplex is terminal precisely if the solid part of each
diagram
(∂∆n
, {n})
jn

GG (X, U)
(∆n
, {n})
SS
(9.3)
in ∆0
|SSet admits a filler. The existence of a terminal object can then
be expressed via injectivity of (X, U) with respect to the morphism
(∅, ∅) → (∆0
, {0})
where the unique 0-simplex 0 of ∆0
is marked. Combining this morphism
with the inner horn inclusions (9.1) and the marked boundary
inclusions (9.3) an injective object (X, U) consists of a quasicategory
X together with a non-empty subset of terminal objects in X. If we
stopped here, the evident full inclusion QCatT → ∆0
|SSet would not
however be essentially surjective on objects; for this we require our
injectives (X, U) to have the property that U consists of all terminal
objects in X. Since each terminal object is equivalent to any other,
WEAK ADJOINT FUNCTOR THEOREMS 49
it therefore suffices to add the repleteness condition that each object
equivalent to one in U belongs to U. To this end, we consider the
nerve J of the free isomorphism which has two 0-simplices 0 and 1.
The repleteness condition is captured by the morphism
(J, {0})
id GG (J, {0, 1})
whose underlying simplicial map is the identity.
It is straightforward to extend this example to quasicategories with
limits of shape D for D a general simplicial set. To see this, recall
the join : SSet × SSet → SSet of simplicial sets. As explained in
[14], there is a natural inclusion D → A D, whereby for fixed D,
we obtain a functor − D: SSet → D/SSet with this value at A.
This has a right adjoint D/SSet → SSet which sends t: D → X to a
simplicial set X/t. Now the limit of t is defined to be a terminal object
∆0
→ X/t. By adjointness, this amounts to a morphism
∆0
D → X
extending t along the inclusion j : D → ∆0
D, which should be thought
of as cones over t, satisfying lifting properties obtained, by adjointness,
from those of (9.2).
We work this time with the locally finitely presentable category (∆0
D)|SSet, whose objects (X, U) are simplicial sets X equipped with
a set U of marked cones ∆0
D → X, and whose morphisms are
simplicial maps preserving marked cones. There is a full embedding
(QCatD) → (∆0
D)|SSet sending a quasicategory with limits of
shape D to its underlying simplicial set with all limit cones marked.
Now, given (X, U) ∈ (∆0
D)|SSet, the condition that a cone of U is
a limit cone amounts to asking that each
(∂∆n
D, {n D})
jn D

GG (X, U)
(∆n
D, {n D})
RR (9.4)
has a filler. Similarly, injectivity with respect to the inclusion
j : (D, ∅) → (∆0
D, {id})
asserts that each diagram t: D → X admits a limit, with limit cone in
U. The repleteness condition capturing the fact that each limit cone
belongs to U is expressed by injectivity with respect to
(J D, {0 D})
id GG (J D, {0 D, 1 D}) ,
50 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
while finally we equip the inner horns
{Λk[n]: Λn
k → ∆n
: 0 < k < n}
with the empty markings to encode the fact that the X in (X, U) is a
quasicategory.
9.4.2. Cartesian fibrations. An inner fibration p: X → Y of simplicial
sets is a morphism having the right lifting property with respect to the
inner horn inclusions. Such an inner fibration is said to be a cartesian
fibration if each 1-simplex f : x → py in Y admits a lifting along p to a
p-cartesian 1-simplex f : x → y ∈ X, where a morphism g: a → b ∈ X
is said to be p-cartesian if each diagram
∆1 GG
i 33
g
@@
Λn
n

GG X
p

∆n
bb
GG Y
where i includes the vertices of ∆1
as the vertices n−1, n ∈ ∆n
, admits
a filler as depicted. A commutative square is said to be a morphism
of cartesian fibrations if it preserves cartesian 1-simplices. As such, we
obtain a category Cart(SSet) of cartesian fibrations.
Let SSet2
denote the arrow category. Then an object of (id: ∆1
→
∆1
)|SSet2
is a morphism p: X → Y of simplicial sets together with a
subset U ⊆ X1 of marked 1-simplices, whilst a morphism is a commutative
square whose domain component preserves marked 1-simplices;
we write p: (X, U) → Y for such an object. We wish to describe
Cart(SSet) as a small injectivity class in id∆1 |SSet2
; that is, we shall
describe a set of morphisms J such that p: (X, U) → Y is J -injective
if and only if p is a cartesian fibration with U the set of all p-cartesian
1-simplices. Note that p is an inner fibration just when it is injective
in SSet2
with respect to each square
Λn
k

GG ∆n
1

∆n
1
GG ∆n
with 0 < k < n. (Here we are employing the trick (see, for instance,
Lemma 1 of [7]) that g has the right lifting property with respect to f
if and only if g is injective in the arrow category with respect to the
map f → 1cod(f) determined by f and 1cod(f)). Thus equipping the
above squares with the empty markings captures as injectives those
WEAK ADJOINT FUNCTOR THEOREMS 51
p: (X, U) → Y with p an inner fibration. To express the requirement
that the elements of U be cartesian 1-simplices we use injectivity with
respect to the squares
(Λn
n, n − 1 → n)

GG (∆n
, n − 1 → n)
1

∆n
1
GG ∆n
whose sources have the unique 1-simplex n − 1 → n marked. The
existence of cartesian liftings is expressed by injectivity against
(∆0
, ∅)
d0

d0
GG (∆1
, 0 → 1)
1

∆1
1
GG ∆1
.
In order to ensure that each cartesian 1-simplex belong to U we add a
repleteness condition based on the following lemma.
Lemma 9.5. Let p: X → Y be a cartesian fibration. Let f : a → b and
g: c → b be 1-simplices in X, with g a cartesian lifting of (pf : pa →
pb, b) so that we obtain a 2-simplex
a
f 11
α GG c
g

b
where pα: pa → pa is degenerate. Then f : a → b is cartesian if and
only if α is an equivalence.
Proof. The degeneracy pα is an equivalence. Thus, by Lemma 2.4.1.5
of [26], α is an equivalence if and only it is cartesian. And now since g
is cartesian, by Proposition 2.4.1.7 of [26], α is cartesian if and only if
f is.
Given this, consider the pushout below left
∆1

d2
GG ∆2

J GG (∆2
)0∼=1
0
00
GG 1

2
which is the generic 2-simplex with 0 → 1 an equivalence. There is a
unique morphism (∆2
)0∼=1 → ∆1
sending the equivalence 0 → 1 to the
52 JOHN BOURKE, STEPHEN LACK, AND LUK´AˇS VOKˇR´INEK
degeneracy on 0, and 2 to 1. Then injectivity of p: (X, U) → Y against
the square below left
((∆2
)0∼=1, 1 → 2)

GG ((∆2
)0∼=1, {1 → 2, 0 → 2})
1

∆1
1
GG ∆1
a
f 11
α GG c
g

b
asserts: for any 2-simplex in X as on the right above in which g ∈ U
and α is an equivalence sent by p to a degeneracy, the 1-simplex f is in
U. Since each morphism of U is cartesian this implies, by Lemma 9.5,
that all cartesian morphisms belong to U. In this way, we obtain the
category of cartesian fibrations Cart(SSet) → (id: ∆1
→ ∆1
)|SSet2
as the full subcategory of injectives.
Finally, to encode the full subcategory Cart(QCat) → Cart(SSet)
of cartesian fibrations between quasicategories we need to further encode
that the target Y of p: (X, U) → Y is a quasicategory — of
course it then follows that X is also a quasicategory, since p is an inner
fibration. To this end, we add the injectivity condition
(∅, ∅)

1 GG (∅, ∅)
p

Λk
n
GG ∆n
for each inner horn.
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Department of Mathematics and Statistics, Masaryk University,
Kotl´aˇrsk´a 2, Brno 61137, Czech Republic
E-mail address: bourkej@math.muni.cz
Department of Mathematics and Statistics, Macquarie University
NSW 2109, Australia
E-mail address: steve.lack@mq.edu.au
Department of Mathematics and Statistics, Masaryk University,
Kotl´aˇrsk´a 2, Brno 61137, Czech Republic
E-mail address: koren@math.muni.cz