Elements of monoidal topology
Lecture 5: (T, V )-categories as generalized spaces
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 takes a look at (T, V )-categories as generalized spaces and considers two well-known topological
properties in this new generalized setting, e.g., Hausdorff separation axiom and compactness. In particular,
this lecture provides generalized analogues of the Tychonoff Theorem and the ˇCech–Stone compactification.
1. Hausdorff and compact spaces
Remark 1. Since this lecture considers properties inspired by general topology, given a category (T, V )-Cat,
its objects (resp. morphisms) will be often referred to as (T, V )-spaces (resp. (T, V )-continuous maps).
Definition 2. A topological space (X, τ), where τ is a topology on a set X, is Hausdorff (or T2-space)
provided that for every distinct x, y ∈ X, there exist disjoint elements U, V ∈ τ such that x ∈ U, y ∈ V .
Definition 3. A topological space (X, τ) is compact provided that for every subset {Ui | i ∈ I} ⊆ τ such that
X ⊆ i∈I Ui (an open cover of X), there exists a finite set {i1, . . . , in} ⊆ I such that X ⊆ Ui1
. . . Uin
(a subcover). Briefly speaking, every open cover of the set X has a finite subcover.
Proposition 4. Every topological space (X, τ), where τ is a topology on a set X, has the following properties:
(1) (X, τ) is Hausdorff iff every ultrafilter on X has at most one convergence point;
(2) (X, τ) is compact provided that every ultrafilter on X converges to some point of X.
Remark 5. Lecture 1 described topological spaces as (β, 2)-categories for the ultrafilter monad β and the
two-element unital quantale 2 = ({⊥, ⊤}, ∧, ⊤). In particular, (β, 2)-categories are sets X equipped with a
relation βX a
// X, which satisfies the two properties of a (β, 2)-category. This relation a will be a map
(written x −→ x instead of x a x and meaning “an ultrafilter x converges to a point x”) provided that
(1) for every x1, x2 ∈ X and every y ∈ βX, if y −→ y1 and y −→ y2, then y1 = y2, which means a · a◦
⩽ 1X;
(2) for every x ∈ βX, there exists x ∈ X such that x −→ x, which means 1βX ⩽ a◦
· a.
By Proposition 4, the above item (1) (resp. item (2)) makes the space (X, a) Hausdorff (resp. compact).
Definition 6. A (T, V )-space (X, a) is said to be
(1) Hausdorff provided that a · a◦
⩽ 1X;
(2) compact provided that 1T X ⩽ a◦
· a.
Email address: solovjovs@tf.czu.cz (Sergejs Solovjovs)
URL: http://home.czu.cz/solovjovs (Sergejs Solovjovs)
Preprint submitted to the Czech University of Life Sciences Prague (CZU) March 23, 2022
The full subcategory of (T, V )-Cat of Hausdorff (resp. compact) spaces is denoted (T, V )-CatHaus (resp.
(T, V )-CatComp). The intersection of the above two subcategories is denoted (T, V )-CatCompHaus.
Proposition 7. Given a (T, V )-space (X, a), the following holds:
(1) (X, a) is Hausdorff iff for every x1, x2 ∈ X and every y ∈ TX,
⊥V < a(y, x1) ⊗ a(y, x2) implies x1 = x2, (1.1)
a(y, x1) ⊗ a(y, x2) ⩽ k, (1.2)
where the latter condition is always satisfied in case the quantale V is strictly two-sided (k = ⊤V ).
(2) (X, a) is compact iff for every x ∈ TX,
k ⩽
x∈X
a(x, x) ⊗ a(x, x). (1.3)
Proof. To show item (1), notice that given x1, x2 ∈ X, it follows that (a · a◦
)(x1, x2) = y∈T X a◦
(x1, y) ⊗
a(y, x2) = y∈T X a(y, x1) ⊗ a(y, x2), i.e., (X, a) is Hausdorff provided that
y∈T X
a(y, x1) ⊗ a(y, x2) ⩽ 1X(x1, x2) =
k, x1 = x2
⊥V , otherwise
which is equivalent to conditions (1.1), (1.2).
To show item (2), notice that given x1, x2 ∈ TX, it follows that (a◦
·a)(x1, x2) = x∈X a(x1, x)⊗a◦
(x, x2) =
x∈X a(x1, x) ⊗ a(x2, x), i.e., (X, a) is compact provided that
1T X(x1, x2) =
k, x1 = x2
⊥V , otherwise
⩽
x∈X
a(x1, x) ⊗ a(x2, x) iff k ⩽
x∈X
a(x, x) ⊗ a(x, x).
□
Definition 8. A unital quantale V is said to be superior provided that for every subset {ui | i ∈ I} ⊆ V ,
k ⩽
i∈I
ui ⊗ ui iff k ⩽
i∈I
ui. (1.4)
Example 9.
(1) Every idempotent unital quantale V (i.e., v ⊗ v = v for every v ∈ V ) is superior.
(2) Every frame V , i.e., a complete lattice such that u ∧ ( S) = s∈S u ∧ s for every u ∈ V and every
subset S ⊆ V (namely, finite meets distribute over arbitrary joins), is a superior quantale.
(3) In every strictly two-sided quantale V (k = ⊤V ), k ⩽ i∈I ui ⊗ui implies k ⩽ i∈I ui, since given i ∈ I,
ui ⊗ ui ⩽ ui ⊗ ⊤V = ui. The converse implication is generally not valid.
(4) An example of a non-superior unital quantale provides the standard construction of the free unital
quantale over a monoid. Given a monoid M = (M, ⊗, k), let V be the powerset PM of M with the
operation PM × PM
ˆ⊗
−→ PM defined by U ˆ⊗V = {u ⊗ v | u ∈ U, v ∈ V } and with the unit ˆk = {k}.
Then V = (PM, , ˆ⊗, ˆk) is a unital quantale. Let M be now a group with more than one element, and
let U = {m, m−1
} with m ̸= k. Then, ˆk = {k} ̸⊆ U, but ˆk = {k} ⊆ {m ⊗ m, m−1
⊗ m−1
, k} = U ˆ⊗U.
Remark 10.
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(1) If the unital quantale V is superior, then the compactness condition (1.3) simplifies to
k ⩽
x∈X
a(x, x). (1.5)
(2) In view of Example 9, if V is a frame (for example, the two-element quantale 2 = ({⊥, ⊤}, ∧, ⊤)
of Lecture 1) or V is the extended real half-line P+ = ([0, ∞]op
, +, 0) (recall Lecture 1), then the
compactness condition (1.3) reduces to condition (1.5).
Proposition 11. Given a category V -Cat, the following holds.
(1) A V -space (X, a) is Hausdorff iff a = 1X (a is the discrete V -category structure on X of Lecture 2).
(2) Every V -space (X, a) is compact.
Proof. To show item (1), on the one hand, if (X, a) is Hausdorff, then 1X ⩽ a (since (X, a) is a V -category),
and a◦
= 1X · a◦
⩽ a · a◦
⩽ ((X, a) is Hausdorff) ⩽ 1X implies a = (a◦
)◦
⩽ (1X)
◦
= 1X; and, on the other
hand, if a = 1X, then 1X · (1X)
◦
= 1X · 1X = 1X.
To show item (2), notice that 1X ⩽ a (since (X, a) is a V -category) implies 1X ⩽ a◦
= a◦
·1X ⩽ a◦
·a. □
Example 12.
(1) In view of Proposition 11, given a category V -Cat, V -CatComp = V -Cat, and V -CatHaus is the full
(coreflective) subcategory comprising discrete V -categories in V -Cat.
Since 2-Cat is the category Prost of preordered sets and monotone maps (see Lecture 1), a preordered
set (X, ⩽) is Hausdorff iff “⩽” is given by the equality, and, moreover, every preordered set is compact.
Since P+-Cat is the category QPMet of quasi-pseudo-metric spaces (generalized metric spaces in the
sense of F. W. Lawvere) and non-expansive maps (see Lecture 1), a quasi-pseudo-metric space (X, ρ) is
Hausdorff iff the quasi-pseudo-metric ρ is given by
ρ(x1, x2) =
0, x1 = x2
∞, otherwise,
and, moreover, every quasi-pseudo-metric space is compact.
(2) In the category Top of topological spaces and continuous maps, which is exactly the category (β, 2)-Cat,
Hausdorffness and compactness for (β, 2)-spaces of Definition 6 coincide with their classical topological
analogues of Definitions 2, 3. In particular, (β, 2)-CatHaus is the category Haus of Hausdorff spaces,
and (β, 2)-CatComp is the category Comp of compact spaces.
(3) In the category App of approach spaces and non-expansive maps, which is precisely the category
(β, P+)-Cat, a (β, P+)-space is Hausdorff provided that for every x ∈ βX and every x1, x2 ∈ X, a(x, x) <
∞ and a(x, x2) < ∞ together imply x = y. Moreover, a (β, P+)-space is compact provided that
infx∈X a(x, x) = 0 for every x ∈ βX. Such a property is called 0-compactness in the theory of approach
spaces. In particular, in view of the full embedding Top → App of Lecture 1, an approach space
induced by a topological space is 0-compact iff its underlying topology is compact.
2. An excursus into category theory
Definition 13.
(1) A source (C
fi
−→ Ci)i∈I in a category C is said to be a mono-source provided that for every C-morphisms
A
g
//
h
// C, if fi · g = fi · h for every i ∈ I, then g = h.
(2) Dually, a sink (Ci
fi
−→ C)i∈I in a category C is said to be an epi-sink provided that for every C-morphisms
C
g
//
h
// A, if g · fi = h · fi for every i ∈ I, then g = h.
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Definition 14. Let M be a conglomerate of sources in a category C. A subcategory B of C is said to be
closed under the formation of M-sources provided that whenever (C
fi
−→ Bi)i∈I is a source in M such that
every Bi belongs to B, then C belongs to B. Dually, one defines the closure under the formation of C-sinks.
Definition 15. An epimorphism e of a category C is said to be strong provided that whenever g · e = m · f
with m a C-monomorphism, there exists a C-morphism h such that the diagram
•
f

e // •
f

h

• m
// •
commutes.
Definition 16.
(1) A full subcategory B of a category C is said to be reflective in C provided that the inclusion functor
B   E // C has a left adjoint, i.e., every C-object C has a B-reflection arrow C
rC
−−→ EB, which means
that for every C-morphism C
f
−→ EB′
, there exists a unique B-morphism B
f′
−→ B′
making the triangle
C
rC
//
f
!!
EB
Ef′

EB′
commute.
(2) Let C be a category, and let E be a class of C-morphisms. An isomorphism-closed, full subcategory B
of C is E-reflective in C provided that every C-object has a B-reflection arrow in E. If E is the class of
all (strong) C-epimorphisms, then one uses the term (strongly) epireflective instead of E-reflective.
Proposition 17. Given a category C with a factorization system (E, M) for sources, a full isomorphismclosed
subcategory B of C is E-reflective iff B is closed under the formation of M-sources in C.
Proposition 18. Given a topological construct C, if E is the class of strong C-epimorphisms, and M is
the conglomerate of mono-sources in C, then (E, M) is a factorization system for sources in C.
3. Properties of Hausdorff and compact (T, V )-spaces
Proposition 19.
(1) A source S = (X
fi
−→ Yi)i∈I with I ̸= ∅ in Set is a mono-source iff i∈I f◦
i · fi = 1X.
(2) A sink T = (Xi
gi
−→ Y )i∈I is an epi-sink in Set iff i∈I gi · g◦
i = 1Y .
Proof. For (1), notice that S is a mono-source in Set iff given x, x′
∈ X, “fi(x) = fi(x′
) for every i ∈ I”
is equivalent to “x = x′
”. Further, ( i∈I f◦
i · fi)(x, x′
) = i∈I(f◦
i · fi)(x, x′
) = i∈I(f◦
i · 1Yi
· fi)(x, x′
) =
i∈I 1Yi
(fi(x), fi(x′
)) = k iff fi(x) = fi(x′
) for every i ∈ I; and 1X(x, x′
) = k iff x = x′
.
For (2), notice that T is an epi-sink in Set iff Y = i∈I gi(Xi). Further, given y, y′
∈ Y , it follows that
( i∈I gi ·g◦
i )(y, y′
) = i∈I(gi ·g◦
i )(y, y′
) = i∈I x∈Xi
g◦
i (y, x)⊗gi(x, y′
) = i∈I x∈Xi
gi(x, y)⊗gi(x, y′
) =
i∈I {k | x ∈ Xi : gi(x) = y and gi(x) = y′
} = k iff y = y′
and g−1
i (y) ̸= ∅; and 1Y (y, y′
) = k iff y = y′
. □
Proposition 20. Given (T, V )-spaces (X, a), (Y, b) and a map X
f
−→ Y , the following are equivalent:
4
(1) (X, a)
f
−→ (Y, b) is a (T, V )-continuous map;
(2) a ⩽ f◦
· b · Tf;
(3) f · a · (Tf)
◦
⩽ b.
Proof. (1) ⇒ (2): Since (X, a)
f
−→ (Y, b) is a (T, V )-continuous map,
TX
T f
//
_a

⩽
TY
_ b

X
f
// Y
implies f · a ⩽ b · Tf, which gives a ⩽ f◦
· f · a ⩽ f◦
· b · Tf, since 1X ⩽ f◦
· f.
(2) ⇒ (3): a ⩽ f◦
· b · Tf implies f · a · (Tf)
◦
⩽ f · f◦
· b · Tf · (Tf)
◦
⩽ b, since f · f◦
⩽ 1Y and
Tf · (Tf)
◦
⩽ 1T Y .
(3) ⇒ (1): f · a · (Tf)
◦
⩽ b implies f · a ⩽ f · a · (Tf)
◦
· Tf ⩽ b · Tf, since 1T X ⩽ (Tf)
◦
· Tf. □
Proposition 21.
(1) (T, V )-CatHaus is closed under non-empty mono-sources in (T, V )-Cat. (T, V )-CatHaus is closed under
all mono-sources (and, therefore, is strongly epireflective in (T, V )-Cat) if V is strictly two-sided.
(2) (T, V )-CatComp is closed under those sinks ((Xi, ai)
gi
−→ (Y, b))i∈I in (T, V )-Cat for which (TXi
T gi
−−→
TY )i∈I is an epi-sink in Set.
Proof.
(1) Observe that given a mono-source S = ((X, a)
fi
−→ (Yi, bi))i∈I in (T, V )-Cat with I ̸= ∅ and such that
(Yi, ai) is Hausdorff for every i ∈ I, since the forgetful functor (T, V )-Cat
U
−→ Set has a left adjoint (see
Lecture 2), it preserves mono-sources, i.e., US = (X
fi
−→ Yi)i∈I is a mono-source in Set with I ̸= ∅.
By Proposition 19 (1), it follows that i∈I f◦
i · fi = 1X. Given i ∈ I, since (Yi, ai) is Hausdorff, a · a◦
⩽
(Proposition 20 (2)) ⩽ (f◦
i · bi · Tfi) · (f◦
i · bi · Tfi)
◦
= f◦
i · bi · Tfi · (Tfi)
◦
· b◦
i · fi ⩽ (Tfi · (Tfi)
◦
⩽ 1T Yi
)
⩽ f◦
i ·bi ·b◦
i ·fi ⩽ ((Yi, bi) is Hausdorff) ⩽ f◦
i ·fi. Thus, a·a◦
⩽ i∈I f◦
i ·fi = 1X, i.e., (X, a) is Hausdorff.
If I = ∅, then |X| ⩽ 1, since S is a mono-source. Therefore, Hausdorff condition (1.1) is satisfied.
Moreover, Hausdorff condition (1.2) is also satisfied provided that k = ⊤V (V is strictly two-sided).
The claim on strong epireflectivity follows immediately from Propositions 17, 18.
(2) Notice that given a sink ((Xi, ai)
gi
−→ (Y, b))i∈I in (T, V )-Cat with (Xi, ai) compact for every i ∈ I, and
such that (TXi
T gi
−−→ TY )i∈I is an epi-sink in Set, by Proposition 19 (2), one gets i∈I Tgi ·(Tgi)
◦
= 1Y .
Given i ∈ I, since (Xi, ai) is compact, b◦
· b ⩾ (Proposition 20 (3)) ⩾ (gi · ai · (Tgi)
◦
)
◦
· gi · ai · (Tgi)
◦
=
Tgi ·ai
◦
·gi
◦
·gi ·ai ·(Tgi)
◦
⩾ (gi
◦
·gi ⩾ 1Xi
) ⩾ Tgi ·ai
◦
·ai ·(Tgi)
◦
⩾ ((Xi, ai) is compact) ⩾ Tgi ·(Tgi)
◦
.
As a result, it follows that b◦
· b ⩾ i∈I Tgi · (Tgi)
◦
= 1T Y , i.e., the (T, V )-space (Y, b) is compact. □
Corollary 22.
(1) Given a surjective (T, V )-continuous map (X, a)
g
−→ (Y, B), if (X, a) is compact, then (Y, b) is compact.
(2) If the functor Set
T
−→ Set preserves small coproducts, then (T, V )-CatComp is closed under small episinks
in (T, V )-Cat. The same statement holds for finite coproducts with closure under finite epi-sinks.
Proof.
(1) Since X
g
−→ Y is surjective, it is a retraction in Set, i.e., there exists a map Y
f
−→ X such that g ·f = 1Y .
Every functor preserves retractions, and, therefore, T also does, i.e., Tg is a retraction in Set. Moreover,
every retraction is an epimorphism. Thus, by Proposition 21 (2), (Y, b) must be compact.
5
(2) Given a small epi-sink T = ((Xi, ai)
gi
−→ (Y, b))i∈I in (T, V )-Cat, where small means that I is a set,
since the forgetful functor (T, V )-Cat
U
−→ Set has a right adjoint (see Lecture 2), it preserves epi-sinks,
and, therefore, UT = (Xi
gi
−→ Y )i∈I is a small epi-sink in Set. Thus, forming a coproduct of (Xi)i∈I in
Set, the unique morphism i∈I Xi
g
−→ Y , making the triangle
Xi
µi
//
gi
##
i∈I Xi
g

Y
commute for every i ∈ I (notice that µi are the coproduct injections), is an epimorphism in Set, i.e., g
is surjective. Applying the functor T to the above triangle, one gets
TXi
T µi
//
T gi
%%
T( i∈I Xi)
T g

TY,
where (Tµi)i∈I is an epi-sink, since T preserves small coproducts, and Tg is an epimorphism in Set
by the argument used in item (1) above. Since composition of epi-sinks is an epi-sink, it follows that
(Tgi)i∈I is an epi-sink. To establish the claim of item (2), it remains to apply Proposition 21 (2). □
Example 23.
(1) Compact spaces are closed under finite epi-sinks in Top ∼= (β, V )-Cat, since the functor Set
β
−→ Set
preserves finite coproducts. In particular, one obtains the classical result of general topology that given
an onto continuous map (X, τ)
f
−→ (Y, σ) between topological spaces, if (X, τ) is compact, then (Y, σ)
is compact. However, compact spaces are not closed under countable epi-sinks. As a counterexample,
consider, e.g., the epi-sink (1
n
−→ N)n∈N in Top, where 1 = {∗}, n(∗) = n, and N (the set of natural
numbers) is given the discrete topology. It is easy to see that the space 1 is compact, but N is not.
(2) In a similar way, 0-compact spaces are closed under finite epi-sinks in App ∼= (β, P+)-Cat, but not
under countable epi-sinks (recall Example 12 (3) for the notion of 0-compactness).
Proposition 24. Given a morphism of lax extensions of monads ˆS
α
−→ ˆT, the respective algebraic 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 ), preserves
both Hausdorffness and compactness property.
Proof. Given a Hausdorff (T, V )-space (X, a), to show that the (T, V )-space (X, a·αX) is Hausdorff, notice
that (a · αX) · (a · αX)
◦
= a · αX · α◦
X · a◦
⩽ (αX · α◦
X ⩽ 1T X) ⩽ a · a◦
⩽ 1X.
Given a compact (T, V )-space (X, a), to show that the (T, V )-space (X, a · αX) is compact, notice that
(a · αX)
◦
· (a · αX) = α◦
X · a◦
· a · αX ⩾ (a◦
· a ⩾ 1T X, since (X, a) is compact) ⩾ α◦
X · αX ⩾ 1SX. □
4. Tychonoff Theorem, ˇCech–Stone compactification
Definition 25. A quantale V is said to be lean provided that for every u, v ∈ V , if u ∨ v = ⊤V and
u ⊗ v = ⊥V , then u = ⊤V or v = ⊤V .
Remark 26. Given a strictly two-sided and lean quantale V , ⊤V and ⊥V are the only its complemented
elements, i.e., elements u ∈ V such that there exists v ∈ V with u ∨ v = ⊤V , u ∧ v = ⊥V . Indeed, since V
is strictly two-sided, for every u, v ∈ V , u ⊗ v ⩽ u ⊗ ⊤V = u and u ⊗ v ⩽ ⊤V ⊗ v = v imply u ⊗ v ⩽ u ∧ v.
6
Example 27.
(1) The quantales 2 and P+ are strictly two-sided and lean.
(2) The quantale 3 = ({⊥, k, ⊤}, ⊗, k), where ⊥ < k < ⊤, and the multiplication ⊗ is given by the table
⊗ ⊥ k ⊤
⊥ ⊥ ⊥ ⊥
k ⊥ k ⊤
⊤ ⊥ ⊤ ⊤
is lean but not strictly two-sided.
(3) The quantale 2 × 2 is strictly two-sided but fails to be lean, since for u = (⊤, ⊥) ̸= (⊤, ⊤) and v =
(⊥, ⊤) ̸= (⊤, ⊤), it follows that u ∨ v = (⊤, ⊤) and u ⊗ v = u ∧ v = (⊥, ⊥).
Proposition 28. Let V be a strictly two-sided quantale.
(1) If V is lean, then all maps in V -Rel are Set-maps
(2) If V is commutative and all maps in V -Rel are Set-maps, then V is lean.
Proof. Recall from Lecture 4 that since V -Rel is an ordered category, a V -relation X r
// Y is called
a map provided that there exists a V -relation Y s
// X such that r ⊣ s, i.e., 1X ⩽ s · r and r · s ⩽ 1Y .
Suppose that the quantale V is lean, and let r ⊣ s be valid in V -Rel. If X = ∅, then r is the inclusion
map ∅ → Y . Thus, one can assume the existence of an element x ∈ X. Then, ⊥V < ⊤V = 1X(x, x) ⩽
(s·r)(x, x) = y′∈Y r(x, y′
)⊗s(y′
, x) implies the existence of some y ∈ Y such that r(x, y)⊗s(y, x) = u > ⊥V .
As a result, ⊤V = u ∨ v, where v = y′∈Y \{y} r(x, y′
) ⊗ s(y′
, x). Further, for every y′
∈ Y such that y ̸= y′
,
(r · s)(y, y′
) ⩽ 1Y (y, y′
) = ⊥V implies (r · s)(y, y′
) = ⊥V implies s(y, x) ⊗ r(x, y′
) = ⊥V , and, therefore,
u⊗v = (r(x, y)⊗s(y, x))⊗( y′∈Y \{y} r(x, y′
)⊗s(y′
, x)) = y′∈Y \{y} r(x, y)⊗(s(y, x)⊗r(x, y′
))⊗s(y′
, x) =
y′∈Y \{y} r(x, y)⊗⊥V ⊗s(y′
, x) = ⊥V . Thus, we have found u, v ∈ V such that u∨v = ⊤V and u⊗v = ⊥V .
Since the quantale V is lean, it follows that either u = ⊤V or v = ⊤V . If v = ⊤V , then u = u ⊗ ⊤V =
u ⊗ v = ⊥V , which contradicts the above result u > ⊥V . Therefore, one obtains that u = ⊤V must hold,
which implies v = ⊤V ⊗ v = u ⊗ v = ⊥V , namely, v = ⊥V . It then follows that for every x ∈ X, there
exists exactly one y ∈ Y such that r(x, y) ⊗ s(y, x) > ⊥V , and, moreover, r(x, y) ⊗ s(y, x) = ⊤V . Lastly,
since V is strictly two-sided, it follows that ⊤V = r(x, y) ⊗ s(y, x) ⩽ r(x, y) ∧ s(y, x) (see Remark 26), i.e.,
r(x, y) = ⊤V = s(y, x). Defining y = f(x), one gets a map X
f
−→ Y such that f ⩽ r and f◦
⩽ s. It remains
to show that r ⩽ f, which can be done as follows: r = r·1X ⩽ (1X ⩽ f◦
·f) ⩽ r·f◦
·f ⩽ r·s·f ⩽ 1Y ·f = f.
Suppose now that every map in the ordered category V -Rel is a Set-map. To show that the quantale V is
necessarily lean, notice that given u, v ∈ V such that u∨v = ⊤V and u⊗v = ⊥V , one can set X = {u, v} and
define a V -relation {∗} r
// X by r(∗, x) = x. It appears that r ⊣ r◦
, since, first, (r·r◦
)(u, v) = r◦
(u, ∗)⊗
r(∗, v) = r(∗, u) ⊗ r(∗, v) = u ⊗ v = ⊥V and (r · r◦
)(v, u) = r◦
(v, ∗) ⊗ r(∗, u) = r(∗, v) ⊗ r(∗, u) = v ⊗ u = (V
is commutative) = u⊗v = ⊥V imply r·r◦
⩽ 1X, and, second, u = u⊗⊤V = u⊗(u∨v) = (u⊗u)∨(u⊗v) =
(u ⊗ u) ∨ ⊥V = u ⊗ u and v = ⊤V ⊗ v = (u ∨ v) ⊗ v = (u ⊗ v) ∨ (v ⊗ v) = ⊥V ∨ (v ⊗ v) = v ⊗ v imply
(r◦
· r)(∗, ∗) = x∈X r(∗, x) ⊗ r◦
(x, ∗) = x∈X r(∗, x) ⊗ r(∗, x) = (r(∗, u) ⊗ r(∗, u)) ∨ (r(∗, v) ⊗ r(∗, v)) =
(u ⊗ u) ∨ (v ⊗ v) = u ∨ v = ⊤V implies 1{∗} ⩽ r◦
· r. Since r ⊣ r◦
, r should be a map X
f
−→ Y in Set, i.e.,
f(∗) = u or f(∗) = v, which then gives the desired ⊤V = r(∗, u) = u or ⊤V = r(∗, v) = v. □
Proposition 29. If V is a strictly two-sided and lean quantale, then (T, V )-CatCompHaus is the full subcategory
of SetT
of T-algebras (X, a) such that a · ˆTa = a · mX. If ˆT is flat, then (T, V )-CatCompHaus = SetT
.
Proof. Given a compact Hausdorff (T,V)-space (X, a), it follows that 1T X ⩽ a◦
· a and a · a◦
⩽ 1X, and,
therefore, (T, V )-relation TX
a
// X is a map in the ordered category V -Rel. Since the quantale V
is strictly two-sided and lean, (T, V )-relation a is a Set-map TX
a
−→ X by Proposition 28 (1). Thus, the
7
defining two conditions of the (T, V )-space (X, a), i.e., V -relational inequalities a · Ta ⩽ a · ˆTa ⩽ a · mX
and 1X ⩽ a · eX between Set-maps must be equalities. Similarly, the defining condition f · a ⩽ b · Tf of a
(T, V )-continuous map (X, a)
f
−→ (Y, b) must be an equality provided that both (X, a) and (Y, b) are compact
Hausdorff.
Moreover, recall from Lecture 2 that 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). As a consequence, every
T-algebra (X, a) has the property a · ˆTa = a · mX, i.e., (T, V )-CatCompHaus = SetT
. □
Example 30. Since the lax extension of the ultrafilter monad β of Lecture 1 is flat, Proposition 29 implies,
in particular, the classical result (β, 2)-CatCompHaus = Setβ
, i.e., the category of compact Hausdorff spaces
is exactly the category of Eilenberg-Moore algebras for the ultrafilter monad on Set.
Theorem 31 (Tychonoff Theorem). Let V be a strictly two-sided and lean quantale, and let the extension
of the monad T to the category V -Rel be flat. Given a set-indexed family of compact Hausdorff
(T,V)-spaces ((Xi, ai))i∈I, the product i∈I(Xi, ai) in (T, V )-Cat is compact Hausdorff.
Proof. Since (Xi, ai) is compact Hausdorff for every i ∈ I, by Proposition 29, one has a set-indexed
family of SetT
-objects ((Xi, ai))i∈I. The Eilenberg-Moore algebra structure on the product i∈I(Xi, ai) =
( i∈I Xi, a) in SetT
is given by the unique map TX
a
−→ X making the diagram
T( i∈I Xi)
T πi
//
a

TXi
ai

i∈I Xi πi
// Xi
commute for every i ∈ I (notice that πi are the product projections). Since every product is a mono-source,
Proposition 19 (1) implies i∈I π◦
i · πi = 1 i∈I Xi
. Thus, a = 1 i∈I Xi
· a = ( i∈I π◦
i · πi) · a = (a and πi are
Set-maps and, therefore, act on the two-element quantale {⊥V , k}) = i∈I(π◦
i · πi · a) = i∈I(π◦
i · ai · Tπi),
which is exactly the product structure on i∈I Xi formed in the category (T, V )-Cat (recall from Lecture 2
that the category (T, V )-Cat is a topological construct). Thus, the product i∈I(Xi, ai) in (T, V )-Cat
belongs to SetT
, and, therefore, by Proposition 29, i∈I(Xi, ai) is compact Hausdorff. □
Definition 32. Given a functor A
G
−→ B, a G-solution set for a B-object B is a set L of A-objects such
that for every B-morphism B
f
−→ GA, there exists L ∈ L, a B-morphism B
h
−→ GL, and an A-morphism
L
g
−→ A such that the triangle
B
h //
f
!!
GL
Gg

GA
commutes.
Theorem 33 (Adjoint Functor Theorem). Given a functor A
G
−→ B, where A is a complete category,
G has a left adjoint iff it satisfies the following two conditions:
(1) G preserves small limits;
(2) every B-object B has a G-solution set.
Proposition 34. Given a complete category A, a functor A
G
−→ B preserves small limits (namely, limits
of diagrams I
D
−→ A, where I is a small category) iff it preserves small products and equalizers.
8
Remark 35. Let T = (T, m, e) be a monad on the category Set.
(1) A SetT
-object (X, a) is non-trivial provided that the set X has more than one element.
(2) If there exists at least one non-trivial T-algebra, then the unit X
eX
−−→ TX is injective for every set X.
(3) There exist exactly two trivial monads on Set (admitting only trivial T-algebras), i.e., the monad sending
every set to a singleton 1, and the monad sending the empty set to itself and all the other sets to 1.
Theorem 36 (ˇCech–Stone compactification). Let V be a strictly two-sided and lean quantale, and let
the extension of the monad T = (T, m, e) to the category V -Rel be flat. Then (T, V )-CatCompHaus is reflective
in (T, V )-CatHaus, and (T, V )-CatHaus is strongly epireflective in (T, V )-Cat.
Proof. By Definition 16, the category (T, V )-CatCompHaus is reflective in the category (T, V )-CatHaus
provided that the inclusion functor (T, V )-CatCompHaus
  E // (T, V )-CatHaus has a left adjoint. Since
(T, V )-CatCompHaus = SetT
by Proposition 29, one considers the inclusion SetT   E // (T, V )-CatHaus. In
view of Theorem 33 and Proposition 34, it will be enough to show, first, that E preserves products and
equalizers, and, second, that every T-algebra (X, a) has an E-solution set.
Start by considering the inclusion SetT   E′
// (T, V )-Cat. In view of Theorem 31, it preserves small
products. To show that E′
preserves equalizers, notice that given T-homomorphisms (X, a)
f
//
g
// (Y, b),
an equalizer of f, g in SetT
is given by an equalizer Z
i
→ X of f, g in Set, where Z = {x ∈ X | f(x) = g(x)}
and i is the inclusion, equipped with a T-algebra structure c on Z, i.e., a map TZ
c
−→ Z making the diagram
TZ
T i //
c

TX
a

Z
i
// X
commute, namely, i · c = a · Ti. Since the map i is injective, Proposition 19 (1) gives i◦
· i = 1Z, and then
c = 1Z · c = i◦
· i · c = i◦
· a · Ti, which is exactly the equalizer structure on Z formed in the category
(T, V )-Cat (recall from Lecture 2 that the category (T, V )-Cat is a topological construct). Thus, the
inclusion E′
preserves equalizers. As a result, the above inclusion E preserves both small products and
equalizers, since (T, V )-CatHaus is closed in (T, V )-Cat under small mono-sources by Proposition 21 (1).
Given a Hausdorff (T, V )-space (X, a), in order to construct an E-solution set for (X, a), consider a
(T, V )-continuous map (X, a)
f
−→ E(Y, b).
Take the least T-subalgebra of (Y, b) containing M = f(X), which can be obtained as follows. Let
M
i
→ Y be the inclusion map, and consider the next commutative diagram
M
i //
eM

Y
eY

1Y
!!
TM
T i
// TY
b
// Y,
where the left-hand (resp. right-hand) side commutes, since e is the unit of the monad T (resp. (Y, b) is a
T-algebra). Denoting h = b · Ti, one gets a factorization M
i
→ Y = M
eM
−−→ TM
h
−→ Y , where eM is injective
for non-trivial monads by Remark 35. In case of any of the two trivial monads of Remark 35 (3), the set M
9
has at most one element, i.e, the map eM is injective as well. Consider the following commutative diagram
TTM
T T i
//
mM

T h
))
TTY
T b
//
mY

TY
b

TM
T i //
h
66TY
b // Y,
where the left-hand (resp. right-hand) rectangle commutes, since m is the multiplication of the monad T
(resp. (Y, b) is a T-algebra). One obtains a T-homomorphism (TM, mM )
h
−→ (Y, b) (straightforward computations
show that given a set X, (TX, mX) is a T-algebra). Consider a factorization TM
h
−→ Y = TM
h
−→
h(TM)
j
→ Y in Set, where h is the restriction of the map h to h(TM), and j is the inclusion map. Since
SetT
is monadic over Set, this factorization can be lifted to SetT
as (TM, mM )
h
−→ (Y, b) = (TM, mM )
h
−→
(h(TM), c)
j
→ (Y, b). The desired least T-subalgebra of (Y, b) containing M is then (h(TM), c).
Since h(TM) contains M, there exists a factorization X
f
−→ Y = X
f
−→ h(TM)
j
→ Y in Set, where f is
the restriction of the map f to h(TM). Consider the following diagram
TX
T f
//
a

T f
%%
TY
b

T(h(TM))
T j
99
c

h(TM)
j
%%
X
f
//
f
99
Y,
where the two triangles and the right-hand rectangle commute, and the outer rectangle has the property
f · a ⩽ b · Tf (since f is a (T, V )-continuous map). Thus, j · f · a ⩽ j · c · Tf, which implies f · a ⩽ c · Tf,
since j is injective, and, therefore, j◦
· j = 1h(T M). As a consequence, one gets a commutative triangle
(X, a)
f
//
f
&&
E(h(TM), c)
j

E(Y, b)
in the category (T, V )-CatHaus. Moreover, since the restriction of the map X
f
−→ Y to M provides a surjective
map X
ˆf
−→ M, which is a retraction in Set, TX
T ˆf
−−→ TM should be also a retraction in Set, namely, a
surjective map. As a consequence, one obtains a surjective map TX
T ˆf
−−→ TM
h
−→ h(TM), which implies that
the cardinality of the set h(TM) does not exceed the cardinality of the set TX.
As a consequence of the above, a solution set for (X, a) can be given by a representative system of
non-isomorphic T-algebras (Z, c), the cardinalities of which do not exceed that of TX. □
10
References
[1] J. Ad´amek, H. Herrlich, and G. E. Strecker, Abstract and Concrete Categories: the Joy of Cats, Repr. Theory Appl. Categ.
17 (2006), 1–507.
[2] R. B¨orger, Coproducts and ultrafilters, J. Pure Appl. Algebra 46 (1987), 35–47.
[3] R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, 1989.
[4] D. Hofmann, G. J. Seal, and W. Tholen (eds.), Monoidal Topology: A Categorical Approach to Order, Metric and Topology,
Cambridge University Press, 2014.
[5] R. Lowen, Index Analysis. Approach Theory at Work, London: Springer, 2015.
[6] M. C. Pedicchio and W. Tholen (eds.), Categorical Foundations: Special Topics in Order, Topology, Algebra, and Sheaf
Theory, Cambridge Univesity Press, 2004.
[7] K. I. Rosenthal, Quantales and Their Applications, Pitman Research Notes in Mathematics, vol. 234, Addison Wesley
Longman, 1990.
11