Elements of monoidal topology⋆
Lecture 2: properties of the category (T, V )-Cat
Sergejs Solovjovs
Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU)
Kam´yck´a 129, 16500 Prague - Suchdol, Czech Republic
Abstract
This lecture proves that the category (T, V )-Cat is a topological construct (providing the explicit form of
the initial structures), describes its induced preorder, and shows, how to obtain functors (T1, V )-Cat →
(T2, V )-Cat as well as (T, V1)-Cat → (T, V2)-Cat for different monads and different quantales, respectively.
1. (T, V )-Cat is a topological construct
1.1. Eilenberg-Moore algebras versus (T, V )-categories
Definition 1.
(1) A lax extension ˆT to V -Rel of a functor T on Set is flat provided that ˆT1X = T1X for every set X.
(2) A lax extension ˆT to V -Rel of a monad T = (T, m, e) on Set is flat provided that the lax extension ˆT
of T is flat.
Lemma 2. Every lax extension ˆT satisfies the following:
ˆT(s · f) = ˆTs · Tf and ˆT(f◦
· r) = (Tf)◦
· ˆTr (1.1)
for every map X
f
−→ Y and every V -relations Y
s
// Z, Z
r
// Y.
Proof. First, observe that ˆTs·Tf ⩽ ˆTs· ˆTf ⩽ ˆT(s·f)
1T X ⩽(T f)◦
·T f
⩽ ˆT(s·f)·(Tf)◦
·Tf ⩽ ˆT(s·f)· ˆT(f◦
)·Tf ⩽
ˆT(s · f · f◦
) · Tf
f·f◦
⩽1Y
⩽ ˆTs · Tf, which implies ˆT(s · f) = ˆTs · Tf.
Second, observe that Tf · ˆT(f◦
·r) ⩽ ˆTf · ˆT(f◦
·r) ⩽ ˆT(f ·f◦
·r)
f·f◦
⩽1Y
⩽ ˆTr implies ˆT(f◦
·r)
1T X ⩽(T f)◦
·T f
⩽
(Tf)◦
· Tf · ˆT(f◦
· r) ⩽ (Tf)◦
· ˆTr, i.e., ˆT(f◦
· r) ⩽ (Tf)◦
· ˆTr. Moreover, (Tf)◦
· ˆTr ⩽ ˆT(f◦
) · ˆTr ⩽ ˆT(f◦
· r),
i.e., (Tf)◦
· ˆTr ⩽ ˆT(f◦
· r). Altogether, one obtains that ˆT(f◦
· r) = (Tf)◦
· ˆTr. □
⋆This lecture course was supported by the ESF Project No. CZ.1.07/2.3.00/20.0051 “Algebraic methods in Quantum Logic”
of the Masaryk University in Brno, Czech Republic.
Email address: solovjovs@tf.czu.cz (Sergejs Solovjovs)
URL: http://home.czu.cz/solovjovs (Sergejs Solovjovs)
Preprint submitted to the Masaryk University in Brno July 1, 2022
Theorem 3. A lax extension ˆT is flat iff the following diagrams commute:
Set
(−)◦

T // Set
(−)◦

V -Rel
ˆT
// V -Rel
Setop
(−)◦

T op
// Setop
(−)◦

V -Rel
ˆT
// V -Rel
(namely, ˆTf = Tf and ˆT(f◦
) = (Tf)◦
for every map X
f
−→ Y ).
Proof. The sufficiency is clear. For the necessity, notice that given a map X
f
−→ Y , it follows that
ˆTf = ˆT(1Y · f)
Lemma 2
= ˆT1Y · Tf = Tf and ˆT(f◦
) = ˆT(f◦
· 1Y )
Lemma 2
= (Tf)◦
· ˆT1Y = (Tf)◦
. □
Example 4.
(1) The identity monad I on V -Rel is a flat lax extension of the identity monad I on Set.
(2) The lax extension ˆβ of the ultrafilter monad β on Set is flat, since given a set X, βX

ˆβ1X
// βX is defined
by x ˆβ1X y iff for every A ∈ x and every B ∈ y, there exist x ∈ A and y ∈ B such that x (1X)◦ y iff for
every A ∈ x and every B ∈ y, there exist x ∈ A and y ∈ B such that x = y iff for every A ∈ x and every
B ∈ y, A B ̸= ∅ iff x = y (since x and y ultrafilters).
(3) The lax extensions ˇP, ˆP of the powerset monad P on Set are non-flat, since given a set X, PX

ˇP 1X
// PX
is defined by A ˇP1X B iff for every x ∈ A there exists y ∈ B such that x (1X)◦ y iff for every x ∈ A there
exists y ∈ B such that x = y iff A ⊆ B (PX 
ˆP 1X
// PX is defined by A ˆP1X B iff for every y ∈ B there
exists x ∈ A such that x (1X)◦ y iff for every y ∈ B there exists x ∈ A such that x = y iff B ⊆ A).
(4) The largest lax extension T⊤
of a monad T on Set is (in general) non-flat, since given a set X,
TX

ˆT ⊤
1X
// TX is defined by ˆT⊤
1X(x, y) = ⊤V for every x, y ∈ TX, which provides the identity on
TX in V -Rel iff TX is at most a singleton and k = ⊤V .
Theorem 5. Every flat lax extension ˆT of a monad T on Set has a full embedding SetT   E // (T, V )-Cat,
which is given by E((X, a)
f
−→ (Y, b)) = (X, a)
f
−→ (Y, b).
Proof. Given a T-algebra (X, a), it follows that a · eX = 1X and a · Ta = a · mX. Since ˆT is flat,
a· ˆTa = a·mX, and therefore, (X, a) is a (T, V )-category. To show that the embedding E is full, notice that
given a (T, V )-functor E(X, a)
f
−→ E(Y, b), the inequality f · a ⩽ b · Tf yields that the graph of the first map
is contained in the graph of the second map. Sharing the same domain, the maps thus must coincide. □
Remark 6. By Theorem 5, a flat lax extension ˆT of a monad T gives the category (T, V )-Cat, which (in
general) is larger than its respective Eilenberg-Moore category SetT
. Since the latter category is monadic
(or, more generally, essentially algebraic), one could ask about the nature of the former category.
1.2. (T, V )-Cat is a topological construct
1.2.1. Topological categories
Definition 7. A functor X
F
−→ Y is called faithful provided that Ff ̸= Fg for every X-morphisms
X
f
//
g
// Y such that f ̸= g.
2
Example 8. The powerset functor P on Set is faithful, since given maps X
f
//
g
// Y such that f ̸= g, there
exists x ∈ X with f(x) ̸= g(x) and then Pf({x}) = {f(x)} ̸= {g(x)} = Pg({x}), which implies Pf ̸= Pg.
Definition 9.
(1) Given a category X, a concrete category over X is a pair (A, U), where A is a category and A
U
−→ X is
a faithful functor. The functor U is called the forgetful (or underlying) functor of the concrete category,
and X is called the base category for (A, U).
(2) A concrete category over Set is called a construct.
Example 10. There exist the constructs (Prost, U) of preordered sets, (QPMet, U) of quasi-pseudo-metric
spaces, (Top, U) of topological spaces, (App, U) of approach spaces and (Cls, U) of closure spaces.
Definition 11.
(1) A source in a category X is a pair (X, (fi)i∈I), which contains an X-object X, and a family (fi)i∈I of
X-morphisms X
fi
−→ Xi indexed by a class I.
(2) Dual concept: a sink in a category X is a pair ((fi)i∈I, X), which contains an X-object X, and a family
(fi)i∈I of X-morphisms Xi
fi
−→ X indexed by a class I.
Remark 12. A more concise notation (X
fi
−→ Xi)I for sources is sometimes used, and dually, for sinks.
Example 13. Products and sums of sets are examples of sources and sinks, respectively.
Definition 14. Let (A, U) be a concrete category over X.
(1) A source (A
fi
−→ Ai)I in A is called (U-)initial provided that for every source (B
gi
−→ Ai)I in A and
every X-morphism UB
h
−→ UA, which makes the diagram
UB
h

Ugi
""
UA
Ufi
// UAi
commute for every i ∈ I, there exists a (necessarily unique) A-morphism B
¯h
−→ A such that U¯h = h.
Dual concept: (U-)final sink.
(2) An A-object A is called indiscrete provided that the empty source (A, ∅) is initial. Dual concept:
discrete object.
Example 15.
(1) A source ((X, ⩽)
fi
−→ (Xi, ⩽i))I in Prost is initial iff the preorder ⩽ on X is defined by x ⩽ y iff
fi(x) ⩽i fi(y) for every i ∈ I. A preordered set (X, ⩽) is indiscrete iff ⩽ = X × X.
(2) A sink ((Xi, ⩽i)
fi
−→ (X, ⩽))I in Prost is final iff the preorder ⩽ on X is generated by the union
( i∈I fi ×fi(⩽i)) ∆X, where ∆X = {(x, x) | x ∈ X}. A preordered set (X, ⩽) is discrete iff ⩽ = ∆X.
Definition 16. Given a functor X
F
−→ Y, an (F-)structured source is a source in Y, which has the form
(Y
fi
−→ FXi)I. Dual concept: (F-)costructured sink.
Definition 17. Let (A, U) be a concrete category over X.
3
(1) An initial lift of a structured source (X
fi
−→ UAi)I is a source (A
¯fi
−→ Ai)I in A, which is initial, and,
moreover, U(A
¯fi
−→ Ai) = X
fi
−→ UAi for every i ∈ I. Dual concept: final lift of a costructured sink.
(2) (A, U) is called topological provided that every structured source has a unique initial lift.
Definition 18.
(1) Given a functor A
G
−→ X, an A-morphism A
f
−→ B is called G-initial (or G-Cartesian) provided that for
every A-morphism C
g
−→ B and every X-morphism GC
h
−→ GA, which makes the diagram
GC
h

Gg
""
GA
Gf
// GB
commute, there exists a unique A-morphism C
¯h
−→ A such that the diagram
C
¯h

g

A
f
// B
commutes and, moreover, G¯h = h. Dual concept: G-final (or G-co-Cartesian) morphism.
(2) A functor A
G
−→ X is called a fibration provided that every G-structured morphism X
f
−→ GB has a Ginitial
lift, i.e., there exists a G-initial morphism A
¯f
−→ B such that G ¯f = f. Dual concept: cofibration.
Remark 19. Forgetful functors of topological categories are particular cases of fibrations.
Theorem 20. Every topological category has unique final lifts of costructured sinks.
Example 21.
(1) The constructs Prost, QPMet, Top, App, and Cls are topological.
(2) The construct Pos of partially ordered sets is not topological. Observe that there exists no initial lift
of, e.g., the unique map {1, 2}
!
−→ U({3}, ⩽), since
U({1, 2}, 1 ⩽ 2)
1{1,2}

U!
((
{1, 2}
!
// U({3}, ⩽)
and U({1, 2}, 2 ⩽ 1)
1{1,2}

U!
((
{1, 2}
!
// U({3}, ⩽)
cannot be both lifted to Pos by the same partial order ⩽ on {1, 2} (1 ⩽ 2 and 2 ⩽ 1 will imply 1 = 2).
1.2.2. Initial structures in the category (T, V )-Cat
Remark 22. The category (T, V )-Cat is a construct, where the forgetful functor (T, V )-Cat
U
−→ Set is
given by U((X, a)
f
−→ (Y, b)) = X
f
−→ Y .
Lemma 23. Given a V -relation Y r
// Z and maps X
f
−→ Y , W
h
−→ Z, it follows that
r · f(x, z) = r(f(x), z) and h◦
· r(y, w) = r(y, h(w)) (1.2)
for every x ∈ X, z ∈ Z, w ∈ W.
4
Proof. Observe that r · f(x, z) = y∈Y f◦(x, y) ⊗ r(y, z) = f(x)=y r(y, z) = r(f(x), y) and h◦
· r(y, w) =
z∈Z r(y, z) ⊗ h◦
(z, w) = z∈Z r(y, z) ⊗ h◦(w, z) = h(w)=z r(y, z) = r(y, h(w)). □
Lemma 24. Given a family of V -relations {X ri
// Y | i ∈ I} and a map Z
f
−→ X, it follows that
(
i∈I
ri) · f =
i∈I
(ri · f), (1.3)
i.e., V -relational composition with maps is distributive over from the right.
Proof. For every z ∈ Z and every y ∈ Y , it follows that ( i∈I ri) · f(z, y)
(1.2)
= ( i∈I ri)(f(z), y) =
i∈I ri(f(z), y)
(1.2)
= i∈I ri · f(z, y) = ( i∈I ri · f)(z, y). □
Theorem 25. The category (T, V )-Cat is a topological construct.
Proof. Given a structured source (X
fi
−→ U(Xi, ai))I, the required initial structure on X is given by
a = i∈I f◦
i · ai · Tfi, or, in pointwise notation,
a(x, x) =
i∈I
ai(Tfi(x), fi(x)) (1.4)
for every x ∈ TX and every x ∈ X.
To show 1X ⩽ a·eX (reflexivity), observe that a·eX = ( i∈I f◦
i ·ai ·Tfi)·eX
(1.3)
= i∈I(f◦
i ·ai ·Tfi ·eX)
(†)
=
i∈I(f◦
i · ai · eXi
· fi)
(††)
⩾ i∈I f◦
i · 1X · fi = i∈I f◦
i · fi ⩾ i∈I 1X ⩾ 1X, where (†) relies on the fact that
1Set
e
−→ T is a natural transformation, and (††) uses the fact that (Xi, ai) is a (T, V )-category for every i ∈ I.
To show a · ˆTa ⩽ a · mX (transitivity), observe that a · ˆTa = ( i∈I f◦
i · ai · Tfi) · ˆT( i∈I f◦
i · ai · Tfi) ⩽
( i∈I f◦
i ·ai ·Tfi)·( i∈I
ˆT(f◦
i ·ai ·Tfi))
(1.1)
= ( i∈I f◦
i ·ai ·Tfi)·( i∈I(Tfi)◦
· ˆTai ·TTfi) ⩽ i∈I(f◦
i ·ai ·Tfi ·
(Tfi)◦
· ˆTai ·TTfi) ⩽ i∈I(f◦
i ·ai ·1T Xi
· ˆTai ·TTfi) = i∈I(f◦
i ·ai · ˆTai ·TTfi)
(†)
⩽ i∈I(f◦
i ·ai ·mXi
·TTfi)
(††)
=
i∈I(f◦
i · ai · Tfi · mX)
(1.3)
= ( i∈I f◦
i · ai · Tfi) · mX = a · mX, where (†) relies on the fact that (Xi, ai) is a
(T, V )-category for every i ∈ I, and (††) employs the fact that TT
m
−→ T is a natural transformation.
To verify that U(X, a)
fj
−→ U(Xj, aj) is a (T, V )-functor, i.e., fj · a ⩽ aj · Tfj, for every j ∈ I, observe
that given j ∈ I, fj · a = fj · ( i∈I f◦
i · ai · Tfi) ⩽ fj · f◦
j · aj · Tfj ⩽ 1Xj
· aj · Tfj = aj · Tfj.
To check that the source ((X, a)
fi
−→ (Xi, ai))I is initial, observe that given any other source ((Y, b)
gi
−→
(Xi, ai))I in (T, V )-Cat and any map Y
h
−→ X such that the triangle
U(Y, b)
h

Ugi
%%
U(X, a)
Ufi
// U(Xi, ai)
(1.5)
commutes for every i ∈ I, it follows that U(Y, b)
h
−→ U(X, a) is a (T, V )-functor, i.e., h · b ⩽ a · Th, since
a · Th = ( i∈I f◦
i · ai · Tfi) · Th
(1.3)
= i∈I(f◦
i · ai · Tfi · Th) = i∈I(f◦
i · ai · T(fi · h))
(1.5)
= i∈I(f◦
i · ai · Tgi)
(†)
⩾
i∈I(f◦
i · gi · b)
(††)
⩾ i∈I h · b ⩾ h · b, where (†) relies on the fact that (Y, b)
gi
−→ (Xi, ai) is a (T, V )-functor for
every i ∈ I, and (††) uses the fact that for every i ∈ I, fi · h = gi implies h = 1X · h ⩽ f◦
i · fi · h = f◦
i · gi. □
5
Remark 26. Despite their existence, there is no convenient formula for the explicit description of final structures
in the category (T, V )-Cat. In particular, given a costructured sink (U(Xi, ai)
fi
−→ X)I in (T, V )-Cat,
the V -relation TX
a
// X defined by a = i∈I fi · ai · (Tfi)◦
, in general, fails to provide a (T, V )-category
structure on X. More precisely, if I = ∅, then a = ⊥V (the constant map with value ⊥V ), and therefore,
a · eX = ⊥V · eX = ⊥V . The condition 1X ⩽ a · eX holds then iff V is a singleton.
Example 27. Formula (1.4) provides the following examples of initial structures:
(1) ((X, ⩽)
fi
−→ (Xi, ⩽i))I in Prost: given x, y ∈ X, x ⩽ y iff fi(x) ⩽i fi(y) for every i ∈ I;
(2) ((X, ρ)
fi
−→ (Xi, ρi))I in QPMet: given x, y ∈ X, ρ(x, y) = supi∈I ρi(fi(x), fi(y));
(3) ((X, a)
fi
−→ (Xi, ai))I in Top: given x ∈ βX and x ∈ X, x a x iff βfi(x) ai fi(x) for every i ∈ I;
(4) ((X, a)
fi
−→ (Xi, ai))I in App: given x ∈ βX and x ∈ X, a(x, x) = supi∈I ai(βfi(x), fi(x));
(5) ((X, c)
fi
−→ (Xi, ci))I in Cls: given A ∈ PX and x ∈ X, x ∈ c(A) iff fi(x) ∈ ci(fi(A)) for every i ∈ I.
Corollary 28. The category (T, V )-Cat is both complete and cocomplete, well-powered and co-well-powered,
and has both a separator and a coseparator. The underlying functor U has both a right- and a left-adjoint.
Remark 29. The constructs Prost, QPMet, Top, App, and Cls have the properties of Corollary 28.
Lemma 30. Given a lax extension ˆT = ( ˆT, m, e) to the category V -Rel of a monad T = (T, m, e) on the
category Set, it follows that ˆT1X = ˆT(e◦
X) · m◦
X for every set X.
Proof. First, observe that ˆT1X = ˆT1X · 1T X = ˆT1X · 1◦
T X
(†)
= ˆT1X · (mX · TeX)◦
= ˆT1X · (TeX)◦
· m◦
X ⩽
ˆT1X · ˆT(e◦
X)·m◦
X ⩽ ˆT(1X ·e◦
X)·m◦
X = ˆT(e◦
X)·m◦
X, i.e., ˆT1X ⩽ ˆT(e◦
X)·m◦
X, where (†) relies on the property
mX · TeX = 1T X of the monad T.
Second, observe that ˆT(e◦
X) · m◦
X = ˆT(e◦
X · 1T X) · m◦
X = ˆT(e◦
X · T1X) · m◦
X ⩽ ˆT(e◦
X · ˆT1X) · m◦
X
Lemma 2
=
(TeX)◦
· ˆT ˆT1X · m◦
X
(†)
⩽ (TeX)◦
· m◦
X · ˆT1X = (mX · TeX)◦
· ˆT1X
(††)
= 1◦
T X · ˆT1X = 1T X · ˆT1X = ˆT1X, i.e.,
ˆT(e◦
X) · m◦
X ⩽ ˆT1X, where (†) follows from the fact that mX · ˆT ˆT1X ⩽ ˆT1X · mX (since m is an oplax
natural transformation) implies ˆT ˆT1X · m◦
X ⩽ m◦
X · mX · ˆT ˆT1X · m◦
X ⩽ m◦
X · ˆT1X · mX · m◦
X ⩽ m◦
X · ˆT1X,
i.e., ˆT ˆT1X · m◦
X ⩽ m◦
X · ˆT1X, and (††) relies on the property mX · TeX = 1T X of the monad T. □
Lemma 31. Given a (T, V )-category (X, a), it follows that e◦
X ⩽ a.
Proof. Observe that 1X ⩽ a · eX implies e◦
X ⩽ a · eX · e◦
X ⩽ a, i.e., e◦
X ⩽ a. □
Lemma 32. Given a (T, V )-category (X, a), it follows that a · ˆT1X = a.
Proof. First, observe that a = a · 1T X = a · T1X ⩽ a · ˆT1X, i.e., a ⩽ a · ˆT1X. Second, observe that
a · ˆT1X
Lemma 30
= a · ˆT(e◦
X) · m◦
X
Lemma 31
⩽ a · ˆTa · m◦
X
(†)
⩽ a · mX · m◦
X ⩽ a, i.e., a · ˆT1X ⩽ a, where (†) relies
on the transitivity property of the (T, V )-category (X, a). □
Theorem 33. Let X be a set.
(1) The discrete (T, V )-category structure on X is given by 1♯
X = e◦
X · ˆT1X. The functor Set
D
−→ (T, V )-Cat,
defined by D(X
f
−→ Y ) = (X, 1♯
X)
f
−→ (Y, 1♯
Y ), is a left adjoint to the forgetful functor (T, V )-Cat
U
−→ Set.
(2) The indiscrete (T, V )-category structure on X is given by the constant map TX × X
⊤V
−−→ V with value
⊤V . The functor Set
I
−→ (T, V )-Cat, defined by I(X
f
−→ Y ) = (X, ⊤V )
f
−→ (Y, ⊤V ), is a right adjoint to
the forgetful functor (T, V )-Cat
U
−→ Set.
6
Proof. For item (1): To show that (X, 1♯
X) is a (T, V )-category, one has to check that 1♯
X is both reflexive
and transitive. To verify 1X ⩽ 1♯
X · eX (reflexivity), observe that 1♯
X · eX = e◦
X · ˆT1X · eX ⩾ e◦
X · T1X · eX ⩾
e◦
X ·eX ⩾ 1X. To verify 1♯
X · ˆT1♯
X ⩽ 1♯
X ·mX (transitivity), observe that 1♯
X · ˆT1♯
X = e◦
X · ˆT1X · ˆT(e◦
X · ˆT1X) ⩽
e◦
X · ˆT(1X · e◦
X · ˆT1X) = e◦
X · ˆT(e◦
X · ˆT1X)
Lemma 2
= e◦
X · (TeX)◦
· ˆT ˆT1X
(†)
⩽ e◦
X · (TeX)◦
· m◦
X · ˆT1X · mX =
e◦
X · (mX · TeX)◦
· ˆT1X · mX
(††)
= e◦
X · (1T X)◦
· ˆT1X · mX = e◦
X · ˆT1X · mX = 1♯
X · mX, where (†) follows from
the fact that mX · ˆT ˆT1X ⩽ ˆT1X ·mX (since m is oplax) implies ˆT ˆT1X ⩽ m◦
X ·mX · ˆT ˆT1X ⩽ m◦
X · ˆT1X ·mX,
i.e., ˆT ˆT1X ⩽ m◦
X · ˆT1X · mX, and (††) relies on the property mX · TeX = 1T X of the monad T.
To show that 1♯
X is the discrete structure on X, one has to check that given a (T, V )-category (Y, b),
every map U(X, 1♯
X)
f
−→ U(Y, b) provides a (T, V )-functor (X, 1♯
X)
f
−→ (Y, b), i.e., f · 1♯
X ⩽ b · Tf. Observe
that f · 1♯
X = f · e◦
X · ˆT1X
(†)
⩽ e◦
Y · Tf · eX · e◦
X · ˆT1X ⩽ e◦
Y · Tf · ˆT1X
Lemma 31
⩽ b · Tf · ˆT1X ⩽ b · ˆTf · ˆT1X ⩽
b · ˆT(f · 1X) = b · ˆTf = b · ˆT(1Y · f)
Lemma 2
= b · ˆT1Y · Tf
Lemma 32
= b · Tf, where (†) relies on the fact that
eY · f = Tf · eX implies f ⩽ e◦
Y · eY · f = e◦
Y · Tf · eX, i.e., f ⩽ e◦
Y · Tf · eX.
As a consequence of the above paragraph, (X, 1♯
X)
f
−→ (Y, 1♯
Y ) is a (T, V )-functor for every map X
f
−→ Y .
For item (2): To show that (X, ⊤V ) is a (T, V )-category, observe that 1X ⩽ ⊤V · eX (reflexivity) is
implied by ⊤V ·eX(x, x) = ⊤V (eX(x), x) = ⊤V ⩾ k for every x ∈ X, and ⊤V · ˆT⊤V ⩽ ⊤V ·mX (transitivity)
is implied by ⊤V ·mX(X, x) = ⊤V (mX(X), x) = ⊤V ≥ ⊤V · ˆT⊤V (X, x) for every X ∈ TTX and every x ∈ X.
To show that ⊤V is the indiscrete structure on X, one has to check that given a (T, V )-category (Y, b),
every map U(Y, b)
f
−→ U(X, ⊤V ) provides a (T, V )-functor (Y, b)
f
−→ (X, ⊤V ), i.e., f · b ⩽ ⊤V · Tf. Observe
that for every y ∈ TY and every x ∈ X, it follows that ⊤V · Tf(y, x) = ⊤V (Tf(y), x) = ⊤V ⩾ f · b(y, x).
As a result of the above paragraph, (X, ⊤V )
f
−→ (Y, ⊤V ) is a (T, V )-functor for every map X
f
−→ Y . □
Example 34. In the construct Top of topological spaces, the discrete (resp. indiscrete) structure on a set
X is given by the powerset PX (resp. the topology {∅, X}).
2. Preordered sets induced by (T, V )-categories
Definition 35. Given concrete categories (A, U) and (B, W) over X, a concrete functor from (A, U) to
(B, W) is a functor A
F
−→ B such that the triangle
A
F //
U
B
W
~~
X
commutes.
Remark 36. Every topological space (X, τ) gives rise to the underlying (or induced) preorder on X, which
is given by x ⩽τ y iff y ∈ cl({x}) (equivalently, for every A ∈ τ, if y ∈ A, then x ∈ A). Observe that the
dual of this preorder is called the specialization preorder of a topological space (the specialization preorder
is a partial order iff (X, τ) is a T0-space, i.e., for every pair of distinct points x, y ∈ X there exists A ∈ τ
containing exactly one of them). A continuous map (X, τ)
f
−→ (Y, σ) provides an order-preserving map
(X, ⩽τ )
f
−→ (Y, ⩽σ). Thus, one gets a concrete functor Top
Ind
−−→ Prost. Since Top is an instance of
(T, V )-Cat, one could ask for a similar functor in case of the latter category.
Theorem 37. Given a (T, V )-category (X, a), the binary relation ⩽a on X, which is defined by x ⩽a y iff
k ⩽ a(eX(x), y), provides a preordered set (X, ⩽a).
7
Proof. Reflexivity follows directly from the property 1X ⩽ a · eX of the (T, V )-category (X, a). To show
transitivity, notice that since e is an oplax natural transformation, the (T, V )-category (X, a) has the property
a ⩽ e◦
X · ˆTa · eT X. (2.1)
Given x, y, z ∈ X such that x ⩽a y and y ⩽a z, it follows that k = k ⊗k ⩽ a(eX(x), y)⊗a(eX(y), z)
(2.1)
⩽ e◦
X ·
ˆTa·eT X(eX(x), y)⊗a(eX(y), z)
(1.2)
= ˆTa(eT X(eX(x)), eX(y))⊗a(eX(y), z) ⩽ a· ˆTa(eT X(eX(x)), z)
a· ˆT a⩽a·mX
⩽
a · mX(eT X(eX(x)), z)
(1.2)
= a(mX · eT X(eX(x)), z)
mX ·eT X =1T X
= a(eX(x), z), and therefore, x ⩽a z. □
Remark 38. Given a (T, V )-category (X, a), the preorder ⩽a on the set X defined in Theorem 37 is called
the underlying preorder induced by a (or the induced preorder for short).
Theorem 39. Every (T, V )-functor (X, a)
f
−→ (Y, b) provides an order-preserving map (X, ⩽a)
f
−→ (Y, ⩽b).
Proof. The (T, V )-functor condition f · a ⩽ b · Tf can be rewritten as a ⩽ f◦
· b · Tf. Given x, z ∈ X
such that x ⩽a z, it follow that k ⩽ a(eX(x), z) ⩽ f◦
· b · Tf(eX(x), z)
(1.2)
= b(Tf · eX(x), f(z))
T f·eX =eY ·f
=
b(eY (f(x)), f(z)), and therefore, f(x) ⩽b f(z). □
Corollary 40. There exists a concrete functor (T, V )-Cat
Ind
−−→ Prost, which is defined by Ind((X, a)
f
−→
(Y, b)) = (X, ⩽a)
f
−→ (Y, ⩽b).
Proof. Follows from Theorems 37, 39. □
Example 41. Corollary 40 provides the following functors:
(1) the identity functor Prost
1Prost
−−−−→ Prost;
(2) the functor QPMet
Ind
−−→ Prost, which is given by Ind(X, ρ) = (X ⩽ρ), where x ⩽ρ y iff ρ(x, y) = 0;
(3) the functor Top
Ind
−−→ Prost, which is given by Ind(X, a) = (X, ⩽a), where x ⩽a y iff ˙x a y iff y ∈ cl({x}),
which is the induced preorder of Remark 36;
(4) the functor App
Ind
−−→ Prost, which is given by Ind(X, δ) = (X, ⩽δ), where x ⩽δ y iff δ(y, {x}) = 0;
(5) the functor Cls
Ind
−−→ Prost, which is given by Ind(X, c) = (X, ⩽c), where x ⩽c y iff y ∈ c({x}).
3. Algebraic functors
Definition 42. Given monads T = (T, m, e) and S = (S, n, d) on a category X, a morphism of monads
S
α
−→ T is a natural transformation S
α
−→ T, which makes the diagrams
SS
n

α◦α // TT
m

S α
// T
1X
ed
~~
S α
// T
(3.1)
commute, where α ◦ α is defined by the diagonal of the commutative diagram
SS
αS
 ""
Sα // ST
αT

TS
T α
// TT.
8
Example 43. There exists a unique monad morphism from the identity monad I on a category X to every
monad T = (T, m, e) on X, which is given by 1X
e
−→ T.
Definition 44. Let ˆS and ˆT be lax extensions to V -Rel of functors S and T on Set, respectively. A morphism
of lax extensions of functors (S, ˆS)
α
−→ (T, ˆT) is an oplax natural transformation ˆS
α
−→ ˆT, which means
SX
αX
//
_ˆSr

⩽
TX
_ ˆT r

SY αY
// TY
(3.2)
for every V -relation X
r
// Y.
Definition 45. Let ˆS and ˆT be lax extensions to V -Rel of monads S = (S, n, d) and T = (T, m, e) on
Set, respectively. A morphism of lax extensions of monads ˆS
α
−→ ˆT is a monad morphism S
α
−→ T, which,
additionally, is a morphism of lax extensions (S, ˆS)
α
−→ (T, ˆT).
Theorem 46. Every morphism of lax extensions of monads ˆS
α
−→ ˆT gives rise to a concrete functor
(T, V )-Cat
Aα
−−→ (S, V )-Cat, which is defined by Aα((X, a)
f
−→ (Y, b)) = (X, a · αX)
f
−→ (Y, b · αY ), and
which, additionally, preserves initial sources.
Proof. To show that (X, a · αX) provides an (S, V )-category, notice that 1X ⩽ a · eX
(3.1)
= a · αX · dX,
and, additionally, a · αX · ˆS(a · αX)
(1.1)
= a · αX · ˆSa · SαX
(3.2)
⩽ a · ˆTa · αT X · SαX
a· ˆT a⩽a·mX
⩽ a · mX · αT X ·
SαX
(3.1)
= a · αX · nX. To show that U(X, a · αX)
f
−→ U(Y, b · αY ) provides an (S, V )-functor, notice that
f · a · αX ⩽ b · Tf · αX
T f·αX =αY ·Sf
= b · αY · Sf.
To show the second statement, notice that given an initial source ((X, a)
fi
−→ (Xi, ai))i∈I in (T, V )-Cat,
by Theorem 25, it follows that a = i∈I f◦
i · ai · Tfi. Applying the functor Aα, one gets an initial source
((X, a · αX)
fi
−→ (Xi, ai · αXi
))i∈I in (S, V )-Cat, since i∈I f◦
i · (ai · αXi
) · Sfi = i∈I f◦
i · ai · (αXi
·
Sfi)
αXi
·Sfi=T fi·αX
= i∈I f◦
i · ai · (Tfi · αX)
(1.3)
= ( i∈I f◦
i · ai · Tfi) · αX = a · αX. □
Remark 47. Aα is called the algebraic functor associated with α.
Theorem 48. Every lax extension ˆT of a monad T = (T, m, e) on Set provides a morphism of lax extensions
of monads I
e
−→ ˆT, and therefore, there exists a concrete functor (T, V )-Cat
Ae
−−→ V -Cat, which is given by
Ae((X, a)
f
−→ (Y, b)) = (X, a · eX)
f
−→ (Y, b · eY ).
Remark 49. Given a (T, V )-category (X, a), (X, a · eX) is called the underlying V -category of (X, a).
Definition 50. Let (A, U) and (B, W) be concrete categories over X. If A
G
−→ B and B
F
−→ A are concrete
functors, then the pair (F, G) is a Galois correspondence (between A and B) provided that UFGA
1UA
−−−→
UA is an A-morphism for every A-object A (namely, there exists an A-morphism FGA
f
−→ A such that
Uf = 1UA), and WB
1W B
−−−→ WGFB is a B-morphism for every B-object B.
Remark 51. If (A, U), (B, W) are concrete categories over X, and A
G
−→ B, B
F
−→ A are concrete functors,
then (F, G) is a Galois correspondence iff there exist concrete (i.e., given by the identity X-morphisms)
natural transformations η and ε such that (η, ε) : F ⊣ G : A −→ B is an adjoint situation.
Theorem 52. If (F, G) is a Galois correspondence, then G preserves initial sources.
9
Theorem 53. The algebraic functor Ae has a concrete left adjoint functor V -Cat
A◦
−−→ (T, V )-Cat defined
by A◦
((X, a)
f
−→ (Y, b)) = (X, e◦
X · ˆTa)
f
−→ (Y, e◦
Y · ˆTb). The adjoint situation A◦
⊣ Ae is concrete (both
its unit and co-unit are given by the identity maps), i.e., provides a Galois correspondence (A◦
, Ae), and
therefore, the functor A◦
preserves final sinks. If the lax extension ˆT of T satisfies the condition
e◦
Y · ˆTr · eX ⩽ r for every V -relation X r
// Y, (3.3)
then A◦
is a full embedding.
Proof. To show that (X, e◦
X · ˆTa) is a (T, V )-category, notice that 1X ⩽ e◦
X ·eX
T 1X ⩽ ˆT 1X
⩽ e◦
X · ˆT1X ·eX
1X ⩽a
⩽
e◦
X · ˆTa · eX, and, moreover, e◦
X · ˆTa · ˆT(e◦
X · ˆTa)
(1.1)
= e◦
X · ˆTa · (TeX)◦
· ˆT ˆTa
1T T X ⩽m◦
X ·mX
⩽ e◦
X · ˆTa · (TeX)◦
·
ˆT ˆTa · m◦
X · mX
ˆT ˆT a·m◦
X ⩽m◦
X · ˆT a
⩽ e◦
X · ˆTa · (TeX)◦
· m◦
X · ˆTa · mX = e◦
X · ˆTa · (mX · TeX)◦
· ˆTa · mX
mX ·T eX =1X
=
e◦
X · ˆTa· ˆTa·mX
ˆT a· ˆT a⩽ ˆT (a·a)
⩽ e◦
X · ˆT(a·a)·mX
a·a⩽a
⩽ e◦
X · ˆTa·mX. To show that U(X, e◦
X · ˆTa)
f
−→ U(Y, e◦
Y · ˆTb) is
a (T, V )-functor, notice that f·e◦
X · ˆTa
f·e◦
X ⩽e◦
Y ·T f
⩽ e◦
Y ·Tf· ˆTa
T f⩽ ˆT f
⩽ e◦
Y · ˆTf· ˆTa
ˆT f· ˆT a⩽ ˆT (f·a)
⩽ e◦
Y · ˆT(f·a)
f·a⩽b·f
⩽
e◦
Y · ˆT(b·f)
(1.1)
= e◦
Y · ˆTb·Tf. For the last statement, notice first that given a V -relation X r
// Y, it follows
that r
1Y ⩽e◦
Y ·eY
⩽ e◦
Y · eY · r
eY ·r⩽ ˆT r·eX
⩽ e◦
Y · ˆTr · eX, which, together with (3.3), implies r = e◦
Y · ˆTr · eX.
To show that A◦
is an embedding, notice that given a V -category (X, a), AeA◦
(X, a) = Ae(X, e◦
X · ˆTa) =
(X, e◦
X · ˆTa·eX) = (X, a). To show that A◦
is full, notice that given a (T, V )-functor (X, e◦
X · ˆTa)
f
−→ (Y, e◦
Y · ˆTb),
f · e◦
X · ˆTa ⩽ e◦
Y · ˆTb · Tf implies f · e◦
X · ˆTa · eX ⩽ e◦
Y · ˆTb · Tf · eX = e◦
Y · ˆTb · eY · f implies f · a ⩽ b · f. □
Remark 54. By Lemma 23, (3.3) is equivalent to ˆTr(eX(x), eY (y))⩽r(x, y) for every V -relation X
r
//Y.
(1) The lax extension of the identity monad I on Set to the identity monad I on V -Rel satisfies (3.3).
(2) The extension ˆβ to Rel (resp. ¯β to P+-Rel) of the ultrafilter monad β on Set satisfies (3.3). Observe
that eX(x) (ˆβr) eY (y) iff ˙x ˆβr ˙y iff for every A ∈ ˙x and every B ∈ ˙y, there exist x′
∈ A and y′
∈ B such
that x′
r y′
iff x r y, since {x} ∈ ˙x and {y} ∈ ˙y.
(3) The extension ˆP to Rel of the powerset monad P on Set satisfies (3.3). Observe that eX(x) ( ˆPr) eY (y)
iff {x} ˆPr {y} iff for every y′
∈ {y} there exists x′
∈ {x} such that x′
r y′
iff x r y.
(4) The largest lax extension T⊤
of a monad T on Set does not satisfy (3.3). Observe that for the V -relation
{∗} r
// {∗} with r(∗, ∗) = ⊥V , it follows that ˆT⊤
r(e{∗}(∗), e{∗}(∗)) = ⊤V > ⊥V = r(∗, ∗), since the
quantale V is assumed to have at least two elements.
Example 55. Theorems 48, 53 and Remark 54 (2) give the next adjoint situation A◦
⊣ Ae : Top −→ Prost.
(1) Ae is the induced preorder functor (Remark 36).
(2) The full embedding Prost   A◦
// Top is the Alexandroff topology functor, i.e., A◦
(X, ⩽) = (X, τ), where
τ = {B ∈ PX | ↓ B = B} with ↓ B = {x ∈ X | x ⩽ y for some y ∈ B}. Observe that given a preordered
set (X, ⩽), A◦
(X, ⩽) = (X, e◦
X · ˆβ ⩽), where for every x ∈ βX and every x ∈ X, x (e◦
X · ˆβ ⩽) x iff
x (ˆβ ⩽) eX(x) (by (1.2)) iff x (ˆβ ⩽) ˙x iff for every B ∈ x and every C ∈ ˙x, there exist y ∈ B and z ∈ C
such that y ⩽ z. Since {x} ∈ ˙x, it follows that for every B ∈ x, there exists y ∈ B such that y ⩽ x.
Thus, given a subset S ⊆ X, for every x ∈ X, x ∈ cl(S) (where cl(S) is the closure of the set S w.r.t.
the topology on X) iff there exists x ∈ βX such that S ∈ x and x (e◦
X · ˆβ ⩽) x (see Lecture 1) iff s ⩽ x
for some s ∈ S, where given s ∈ S such that s ⩽ x, one defines x = ˙s. As a consequence, S is closed iff
cl(S) = S iff S =↑ S with ↑ S = {x ∈ X | s ⩽ x for some s ∈ S}. The open sets are then exactly the
sets of the form B = X\ ↑ S =↓ D for some D ⊆ X, i.e., B =↓ B as defined above.
Observe that the Alexandroff topology has the property that arbitrary intersections of open sets are open.
10
4. Change-of-base functors
Definition 56. A homomorphism of unital quantales (V, ⊗, k)
φ
−→ (W, ⊗, l) is a map V
φ
−→ W such that
(1) φ( A) = φ(A) for every A ⊆ V ;
(2) φ(a ⊗ b) = φ(a) ⊗ φ(b) for every a, b ∈ V ;
(3) φ(k) = l.
Definition 57. A lax homomorphism of unital quantales (V, ⊗, k)
φ
−→ (W, ⊗, l) is a map V
φ
−→ W such that
(1) φ(A) ⩽ φ( A) for every A ⊆ V ;
(2) φ(a) ⊗ φ(b) ⩽ φ(a ⊗ b) for every a, b ∈ V ;
(3) l ⩽ φ(k).
Remark 58. The first condition of the above definition is equivalent to φ being order-preserving.
Definition 59. A quantic nucleus on a quantale (Q, ⊗) is a map Q
j
−→ Q such that
(1) j is order-preserving;
(2) j is increasing, i.e., a ⩽ j(a) for every a ∈ Q;
(3) j is idempotent, i.e., j(j(a)) = j(a) for every a ∈ Q;
(4) j(a) ⊗ j(b) ⩽ j(a ⊗ b) for every a, b ∈ Q.
Example 60. A quantic nucleus on a unital quantale V is a lax homomorphism of V .
Theorem 61. Every lax homomorphism of unital quantales V
φ
−→ W gives a lax functor V -Rel
φ
−→ W-Rel
defined by φ(X r
// Y ) = X φr
// Y, where φr is the composition of the maps X × Y
r
−→ V and V
φ
−→ W.
Proof. By the definition of lax functor, φ should satisfy the following:
(1) φr ⩽ φs for every V -relations X
r
//
s
// Y such that r ⩽ s;
(2) φs · φr ⩽ φ(s · r) for every V -relations X r
// Y and Y s
// Z;
(3) 1X ⩽ φ1X for every set X.
Item (1) (resp. (3)) follows from item (1) (resp. (3)) of Definition 57. To show item (2), notice that
φs · φr(x, z) = y∈Y φr(x, y) ⊗ φs(y, z) = y∈Y φ(r(x, y)) ⊗ φ(s(y, z)) ⩽ y∈Y φ(r(x, y) ⊗ s(y, z)) ⩽
φ( y∈Y r(x, y) ⊗ s(y, z)) = φ(s · r(x, z)) = φ(s · r)(x, z) for every x ∈ X, z ∈ Z. □
Lemma 62. Given a lax homomorphism of unital quantales V
φ
−→ W, maps X
f
−→ Y , S
g
−→ Z, and V relations
Y r
// Z, U s
// X, it follows that
f ⩽ φf, f◦
⩽ φ(f◦
), g◦
· φr · f = φ(g◦
· r · f), f · φs ⩽ φ(f · s), (4.1)
and, moreover, if φ is -preserving, then f ·φs = φ(f ·s), where f, f◦
, and g◦
are considered as W-relations
when appearing on the left-hand side of the above (in)equalities, and as V -relations on the right-hand side.
Proof. Given x ∈ X and w ∈ W, it follows that (φ(g◦
·r·f))(x, w) = φ(g◦
·r·f(x, w))
(1.2)
= φ(r(f(x), g(w)))=
φr(f(x), g(w)) = g◦
· φr · f(x, w). Moreover, given u ∈ U and y ∈ Y , it follows that f · φs(u, y) =
f(x)=y φs(u, x) = f(x)=y φ(s(u, x))
(†)
⩽ φ( f(x)=y s(u, x)) = φ(f · s(u, y)) = φ(f · s)(u, y), in which (†)
turns into “=” provided that φ is -preserving, i.e., f · φs(u, y) = φ(f · s)(u, y). □
11
Definition 63. Given lax extensions ˆT and ˇT of a functor T on Set to the categories V -Rel and W-Rel,
respectively, a lax homomorphism of unital quantales V
φ
−→ W is said to be compatible with the structure
of the lax extensions ˆT and ˇT provided that ˇT(φr) ⩽ φ( ˆTr) for every V -relation r, which means
V -Rel
ˆT //
φ

⩽
V -Rel
φ

W-Rel ˇT
// W-Rel.
(4.2)
Theorem 64. Given lax extensions ˆT and ˇT of a monad T on Set to the categories V -Rel and W-Rel,
respectively, every lax homomorphism of unital quantales V
φ
−→ W, which is compatible with the structure of
the lax extensions, induces a concrete functor (T, V )-Cat
Bφ
−−→ (T, W)-Cat defined by Bφ((X, a)
f
−→ (Y, b)) =
(X, φa)
f
−→ (Y, φb). If φ is injective (resp. a -preserving order-embedding), then Bφ is a (resp. full)
embedding.
Proof. To show that (X, φa) is a (T, W)-category, notice that e◦
X
(4.1)
⩽ φe◦
X
Lemma 31
⩽ φa, i.e., 1X ⩽ φa · eX,
and, moreover, φa · ˇT(φa)
(4.2)
⩽ φa · φ( ˆTa) ⩽ φ(a · ˆTa)
a· ˆT a⩽a·mX
⩽ φ(a · mX)
(4.1)
= φa · mX. To show that
U(X, φa)
f
−→ U(Y, φb) is a (T, W)-functor, notice that f · φa
(4.1)
⩽ φ(f · a)
a·f⩽b·T f
⩽ φ(b · Tf)
(4.1)
= φb · Tf. To
show fullness of Bφ, notice that given a (T, W)-functor Bφ(X, a)
f
−→ Bφ(Y, b), -preservation of φ and the
last statement of Lemma 62 imply that φ(f · a) = f · φa ⩽ φb · Tf
(4.1)
= φ(b · Tf), and thus, f · a ⩽ b · Tf. □
Remark 65. Bφ is called the change-of-base functor associated to φ.
Definition 66. Given partially ordered sets (X, ⩽), (Y, ⩽) and order-preserving maps (X, ⩽)
f
// (Y, ⩽),
g
oo
f is left adjoint to g and g is right adjoint to f (denoted f ⊣ g) provided that 1X ⩽ g · f and f · g ⩽ 1Y .
Remark 67. A right adjoin map preserves all existing , and a left adjoint map preserves all existing .
Theorem 68. Let ˆT and ˇT be lax extensions of a monad T on Set to the categories V -Rel and W-Rel,
respectively, and let V
φ
// W
ψ
oo be lax homomorphisms of unital quantales compatible with the structure of
the lax extensions. If φ ⊣ ψ, then Bφ ⊣ Bψ : (T, W)-Cat −→ (T, V )-Cat (Bφ is a left adjoint to Bψ), and,
moreover, the latter adjoint situation is concrete (both its unit and co-unit are given by the identity maps).
Proof. Given a (T, V )-category (X, a), BψBφ(X, a) = (X, ψφa). Since φ ⊣ ψ, it follows that 1V ⩽ ψ · φ,
and therefore, a ⩽ ψφa. As a consequence, (X, a)
1X
−−→ (X, ψφa) is a (T, V )-functor. Dually, given a (T, W)category
(X, b), (X, φψb)
1X
−−→ (X, b) is a (T, W)-functor. The above two maps provide the unit and the
co-unit of the adjunction Bφ ⊣ Bψ, respectively. For example, for the former statement, observe that given
a (T, V )-functor (X, a)
f
−→ Bψ(Y, b) (which implies f · a ⩽ ψb · Tf), it follows that Bφ(X, a)
f
−→ (Y, b) is a
(T, W)-functor, since f · φa
(4.1)
⩽ φ(f · a) ⩽ φ(ψb · Tf)
(4.1)
= φψb · Tf ⩽ b · Tf, which makes the triangle
(X, a)
f
&&
1X
// BψBφ(X, a)
Bψf

Bψ(Y, b)
commute, and which, moreover, is uniquely determined by the above commutativity property. □
12
Remark 69. One could generalize Theorem 68 in the following way. First, given a monad T on the category
Set, there exists the quasicategory Quant(T), the objects of which are pairs (V, ˆT) comprising a unital
quantale V (with at least two elements) and a lax extension ˆT of the monad T to the category V -Rel,
and whose morphisms (V, ˆT)
φ
−→ (W, ˇT) are lax homomorphisms of unital quantales V
φ
−→ W compatible
with the lax extensions ˆT and ˇT. Since Quant(T) is a partially ordered quasicategory (given Quant(T)morphisms
(V, ˆT)
φ
//
ψ
// (W, ˇT), one defines φ ⩽ ψ iff φ(a) ⩽ ψ(a) for every a ∈ V ), it is a 2-quasicategory
with thin 2-cells. Second, let CAT be the 2-quasicategory of categories and functors. Third, there exists a
2-functor Quant(T)
B
−→ CAT given by B((V, ˆT)
φ
−→ (W, ˇT)) = V -Cat
Bφ
−−→ W-Cat, where Bφ is the functor
of Theorem 64. Observe that given Quant(T)-morphisms (V, ˆT)
φ
//
ψ
// (W, ˇT) such that φ ⩽ ψ, for every
(T, V )-category (X, a), it follows that Bφ(X, a)
1X
−−→ Bψ(X, a) = (X, φa)
1X
−−→ (X, ψa) is a (T, W)-functor
(since 1X · φa = φa ⩽ ψa = ψa · 1T X = ψa · T1X), which is a part of a natural transformation Bφ
α⩽
−−→ Bψ
given by the identity maps. As a consequence, the functor B preserves adjunctions, i.e., if φ ⊣ ψ in terms of
partially ordered sets, then Bφ ⊣ Bψ in terms of categories and functors, which then implies Theorem 68.
Example 70. Given a unital quantale V with at least two elements, there exists the (unique) unital quantale
embedding 2   ι // V given by
ι(a) =
k, a = ⊤
⊥V , a = ⊥,
which has a right adjoint V
p
−→ 2 given by
p(a) =
⊤, k ⩽ a
⊥, otherwise,
and which is a lax homomorphism of unital quantales (for example, to show Definition 57 (1) for p, observe
that p(A) = ⊤ iff there exists a ∈ A such that p(a) = ⊤ iff there exists a ∈ A such that k ⩽ a, which
implies k ⩽ a ⩽ A, which gives p( A) = ⊤). ι has a left adjoint V
o
−→ 2 iff k = ⊤V (for the necessity,
notice that since ι is -preserving by Remark 67, k = ι(⊤) = ι( ∅) = ι(∅) = ⊤V ), which is given by
o(a) =
⊥, a = ⊥V
⊤, otherwise.
Observe that o is -preserving (as a left-adjoint map) and o(k) = ⊤. Thus, o is a lax homomorphism of unital
quantales iff o is a homomorphism of unital quantales (given a, b ∈ V , it follows that o(a ⊗ b) ⩽ o(a) ⊗ o(b),
since o(a ⊗ b) = ⊤ iff a ⊗ b ̸= ⊥V , which implies a ̸= ⊥V and b ̸= ⊥V , which gives o(a) = ⊤ and o(b) = ⊤,
which finally provides o(a) ⊗ o(b) = ⊤) iff a ⊗ b = ⊥V implies a = ⊥V or b = ⊥V for every a, b ∈ V (observe
that then o(a ⊗ b) = ⊥ iff a ⊗ b = ⊥V iff a = ⊥V or b = ⊥V iff o(a) = ⊥ or o(b) = ⊥ iff o(a) ⊗ o(b) = ⊥).
The above maps are compatible with the lax extensions of the identity functor on Set to Rel and V -Rel,
respectively. Theorem 68 provides then the adjunctions
Prost = 2-Cat 
Bι
// V -Cat,
Bp
⊥
kk
Bo
⊥tt
where Bp is the induced preorder functor of Corollary 40, and Bι(X, ⩽) = (X, a), where for every x, y ∈ X,
a(x, y) =
k, x ⩽ y
⊥V , otherwise.
13
Moreover, Bo(X, a) = (X, ⩽), where for every x, y ∈ X, x ⩽ y iff ⊥V < a(x, y).
Remark 71. The adjunction of item (1) of Theorem 33 can be decomposed now as follows:
Set
D
  E //
Prost = 2-Cat
U
⊥oo
  Bι
//
V -Cat
Bp
⊥oo
A◦
//
(T, V )-Cat,
Ae
⊥oo
U
OO
where EX = (X, ∆) = (X, {(x, x) | x ∈ X}) (discrete preorder). Observe that given a set X, it follows that
A◦
BιEX = A◦
Bι(X, ∆) = A◦
(X, ι∆) = A◦
(X, 1X) = (X, e◦
X · ˆT1X) = (X, 1♯
X) = DX.
Problem 72. The adjunction of item (2) of Theorem 33 can be decomposed as follows:
Set
I
  F //
Prost = 2-Cat
U
⊤oo
  Bι
//
V -Cat
Bo
⊤oo
H //
(T, V )-Cat,
K
⊤oo
U
OO
where FX = (X, X × X) (indiscrete preorder), H(X, a) = (X, ⊤V ) (indiscrete (T, V )-category structure),
in which TX × X
⊤V
−−→ V is the constant map with value ⊤V , and K(X, a) = (X, 1X) (identity V -relation).
Observe that given a set X, it follows that HBιFX = HBι(X, X × X) = H(X, ι(X × X))
(†)
= H(X, ⊤V ) =
(X, ⊤V ) = IX, where (†) relies on the fact that the existence of a left adjoint map V
o
−→ 2 to 2   ι // V implies
k = ⊤V . To show that K is left adjoint to H, notice that given a (T, V )-category (X, a), the identity map
X
1X
−−→ X provides an H-universal arrow (X, a)
1X
−−→ HK(X, a) for (X, a), i.e., first, (X, a)
1X
−−→ HK(X, a) =
(X, ⊤V ) is a (T, V )-functor (since a ⩽ ⊤V ), and, second, given a (T, V )-functor (X, a)
f
−→ H(Y, b), there is a
unique V -functor K(X, a) = (X, 1X)
f
−→ (Y, b) (since f · 1X = f = 1Y · f ⩽ b · f), which makes the triangle
(X, a)
f
%%
1X
// HK(X, a)
Hf

H(Y, b)
commute, and which, moreover, is uniquely determined by the above commutativity property.
Remark 73. The diagram
Top = (β, 2)-Cat
Ae⊣

 
Bι
// App = (β, P+)-Cat
Bp
⊥ll
L
⊥ss
Ae⊣

Prost = 2-Cat
?
A◦
OO
 
Bι
// QPMet = P+-Cat
Bp
⊥
kk
Bo
⊥ss ?
A◦
OO
(4.3)
commutes w.r.t. both the solid and the dotted arrows (excluding the dashed ones). The functors of (4.3)
can be described explicitly as follows.
14
(1) The adjunction A◦
⊣ Ae : Top −→ Prost is that of Example 55.
(2) The full embedding Prost   Bι
// QPMet is given by Bι(X, ⩽) = (X, ρ), where for every x, y ∈ X,
ρ(x, y) =
0, x ⩽ y
∞, otherwise.
The functors QPMet
Bo
//
Bp
// Prost are given by Bo,p(X, ρ) = (X, ⩽o,p), where for every x, y ∈ X, x ⩽o y
iff ρ(x, y) < ∞, and x ⩽p y iff ρ(x, y) = 0, respectively.
(3) The full embedding QPMet   A◦
// App is given by A◦
(X, ρ) = (X, δ), where for every x ∈ X, A ∈ PX,
δ(x, A) = inf{ρ(y, x) | y ∈ A}. The functor App
Ae
−−→ QPMet is given by Ae(X, δ) = (X, ρ), where for
every x, y ∈ X, ρ(x, y) = sup{δ(y, A) | x ∈ A}.
(4) The full embedding Top   Bι
// App is given by Bι(X, τ) = (X, δ), where for every x ∈ X, A ∈ PX,
δ(x, A) =
0, x ∈ cl(A)
∞, otherwise.
The functor App
Bp
−−→ Top sends an approach space (X, a) (represented as a (β, P+)-category) to a
topological space, in which an ultrafilter x converges to a point x iff a(x, x) = 0. The unital quantale
homomorphism P+
o
−→ 2 is incompatible with the lax extensions of the ultrafilter monad β to P+-Rel
and Rel, respectively, but still provides a left adjoint functor L to Bι.
Observe that o is compatible with the lax extensions of the ultrafilter monad β to P+-Rel and Rel,
respectively, provided that ˆβ(or) ⩽ o(¯βr) for every V -relation X r
// Y . Also notice that ˆβ(or)(x, y) =
A∈x,B∈y x∈A,y∈B or(x, y) and o(¯βr)(x, y) = o(supA∈x,B∈y infx∈A,y∈B r(x, y)) for every x ∈ βX and
every y ∈ βY (see Lecture 1 for more detail). Take a set X such that there exists a non-principal
ultrafilter x on X, and consider the identity V -relation 1X on X. On the one hand, o(¯β1X)(x, x) =
o(supA,B∈x infx∈A,y∈B(1X)◦(x, y))
(†)
= o(supA,B∈x ⊥V ) = o(⊥V ) = ⊥, where (†) uses the fact that for every
A, B ∈ x, it follows that infx∈A,y∈B(1X)◦(x, y)⩽infx,y∈A B,x̸=y(1X)◦(x, y) = infx,y∈A B,x̸=y ⊥V =
⊥V , since A B ∈ x and therefore, A B has at least two elements (recall that the ultrafilter x is nonprincipal).
On the other hand, ˆβ(o1X)(x, x)= A,B∈x x∈A,y∈B o1X(x, y)⩾ A,B∈x x∈A B o1X(x, x)=
A,B∈x x∈A B o(k) = A,B∈x x∈A B ⊤
(†)
⩾ A,B∈x ⊤ ⩾ ⊤, where (†) uses the fact that for every
A, B ∈ x, it follows that A B ∈ x and thus, A B ̸= ∅. As a consequence, ˆβ(o1X)(x, x) = ⊤ > ⊥ =
o(¯β1X)(x, x), violating the condition of compatibility with the lax extensions.
References
[1] M. M. Clementino and D. Hofmann, Topological features of lax algebras, Appl. Categ. Structures 11 (2003), no. 3, 267–286.
[2] M. M. Clementino, D. Hofmann, and W. Tholen, One setting for all: Metric, topology, uniformity, approach structure,
Appl. Categ. Struct. 12 (2004), no. 2, 127–154.
[3] D. Hofmann, G. J. Seal, and W. Tholen (eds.), Monoidal Topology: A Categorical Approach to Order, Metric and Topology,
Cambridge University Press, 2014.
[4] D. Kruml and J. Paseka, Algebraic and Categorical Aspects of Quantales, Handbook of Algebra (M. Hazewinkel, ed.),
vol. 5, Elsevier, 2008, pp. 323–362.
[5] J. MacDonald and M. Sobral, Aspects of Monads, Categorical Foundations: Special Topics in Order, Topology, Algebra,
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