Elements of monoidal topology⋆
Lecture 3: a generalization of the Kuratowski-Mrówka theorem
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 shows an example of the application of the theory of (T, V )-categories and (T, V )-functors to
general topology, i.e., a particular generalization of the Kuratowski-Mrówka theorem on the equivalence
between the concepts of proper (or stably closed) and perfect map.
1. Classical Kuratowski-Mrówka theorem
1.1. Proper and perfect maps
Definition 1. Let (X, τ)
f
−→ (Y, σ) be a continuous map between topological spaces.
(1) f is closed provided that the image under f of every closed set in (X, τ) is closed in (Y, σ).
(2) f is proper provided that for every topological space (Z, ϱ), the map X × Z
f×1Z
−−−−→ Y × Z is closed.
Remark 2. Every proper map is closed (take Z to be a singleton topological space in Definition 1 (2)).
Theorem 3. Given a continuous map (X, τ)
f
−→ (Y, σ) between topological spaces, equivalent are:
(1) f is proper;
(2) for every continuous map (Z, ϱ)
g
−→ (Y, σ), the map X ×Y Z
pZ
−−→ Z, which is defined by the pullback
X ×Y Z
pX

pZ
// Z
g

X
f
// Y,
is closed.
⋆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
Proof.
(1) ⇒ (2): Given a continuous map (Z, ϱ)
g
−→ (Y, σ) and its respective pullback
X ×Y Z
pX

pZ
// Z
g

X
f
// Y,
one can construct the following two diagrams (where πX, πY , πZ are product projections)
X ×Y Z
pX
{{
⟨pX ,pZ ⟩

pZ
##
X X × ZπX
oo
πZ
// Z
Z
g
||
⟨g,1Z ⟩

1Z
##
Y Y × ZπY
oo
πZ
// Z,
which then provide the diagram
X ×Y Z
⟨pX ,pZ ⟩

pZ
// Z
⟨g,1Z ⟩

X × Z
f×1Z
// Y × Z,
(1.1)
where the two vertical arrows are given by injective maps. To show that diagram (1.1) commutes, observe
that for every (x, z) ∈ X×Y Z, on the one hand, (f ×1Z)·⟨pX, pZ⟩(x, z) = (f ×1Z)(pX(x, z), pZ(x, z)) = (f ×
1Z)(x, z) = (f(x), z), and, on the other hand, ⟨g, 1Z⟩·pZ(x, z) = ⟨g, 1Z⟩(z) = (g(z), z). Since (x, z) ∈ X×Y Z,
it follows that f(x) = g(z), which then implies (f(x), z) = (g(z), z).
We are going to show that X ×Y Z is a closed subset of X × Z. Since the map (X, τ)
f
−→ (Y, σ) is proper
and therefore, closed by Remark 2, it follows that f(X) is a closed subset of Y , and thus, S := g−1
(f(X)) is a
closed subset of Z. Given (x, z) ∈ (X ×Z)\(X ×Y Z), it follows that z ̸∈ S and therefore, z ∈ Z\S =: U ∈ ϱ.
Thus, (x, z) ∈ π−1
Z (U) =: W, where W is an open subset of X × Z. If (x′
, z′
) ∈ W (X ×Y Z), then z′
∈ U
and f(x′
) = g(z′
), namely, z′
∈ Z\S and z′
∈ S, which is a contradiction. As a consequence, W is an open
subset of X × Z containing (x, z) and, moreover, disjoint from the set X ×Y Z.
Since X ×Y Z is a closed subset of X × Z, the inclusion X ×Y Z   ⟨pX ,pZ ⟩
// X × Z is a closed map.
Moreover, since the map (X, τ)
f
−→ (Y, σ) is proper, X × Z
f×1Z
−−−−→ Y × Z is a closed map as well. Thus, the
left-hand path in diagram (1.1) is a closed map, and thus, the right-hand path is a closed map as well. Since
the inclusion Z  ⟨g,1Z ⟩
// Y × Z is clearly injective, it follows that X ×Y Z
pZ
−−→ Z is a closed map (given a
closed subset P ⊆ X ×Y Z, since ⟨g, 1Z⟩(pZ(P)) is closed, pZ(P) = (⟨g, 1Z⟩)−1
(⟨g, 1Z⟩(pZ(P))) is closed).
(2) ⇒ (1): Observe that given a topological space (Z, ϱ), it follows that
X × Z
πX

f×1Z
// Y × Z
πY

X
f
// Y
is a pullback. □
Remark 4. Theorem 3 motivates the terminology stably closed w.r.t. proper maps.
Definition 5.
2
(1) Given a topological space (X, τ), a subset S ⊆ X is said to be compact provided that for every family
{Ui | i ∈ I} ⊆ τ such that S ⊆ i∈I Ui there exists a finite subfamily {Ui1
, . . . , Uin
} ⊆ {Ui | i ∈ I} such
that S ⊆
n
j=1
Uij
(in other words, every open cover of S has a finite subcover).
(2) A topological space (X, τ) is said to be compact provided that its underlying set X is compact.
Remark 6. Some authors call the property of Definition 5 quasi-compactness. A quasi-compact topological
space (X, τ) is then said to be compact provided that it is additionally Hausdorff or T2-space, namely, for
every distinct points x1, x2 ∈ X there exist U1, U2 ∈ τ such that x1 ∈ U1, x2 ∈ U2 and U1 U2 = ∅.
Definition 7. A continuous map (X, τ)
f
−→ (Y, σ) between topological spaces is called perfect provided that
f is closed, and for every y ∈ Y , the fibre f−1
(y) is a compact subset of X.
1.2. Kuratowski-Mrówka theorem and its generalization
Theorem 8 (Kuratowski-Mrówka). Given a topological space (X, τ), equivalent are:
(1) (X, τ) is compact;
(2) for every topological space (Y, σ), the projection X × Y
πY
−−→ Y is closed.
Proof. (1) ⇒ (2) : K. Kuratowski. (2) ⇒ (1) : S. Mrówka. □
Corollary 9. Given a topological space (X, τ), the unique continuous map (X, τ)
!X
−→ 1 (where 1 = {∗}) is
perfect iff it is proper.
Proof. Observe that the map (X, τ)
!X
−→ 1 is proper iff for every topological space (Z, ϱ), the projection
map X × Z
πZ
−−→ Z, which is defined by the pullback
X × Z
πX

πZ
// Z
!Z

X
!X
// 1,
is closed (by Theorem 3) iff for every topological space (Z, ϱ), the projection map X × Z
πZ
−−→ Z is closed iff
the space (X, τ) is compact (by Theorem 8) iff the map (X, τ)
!X
−→ 1 is perfect (notice that the unique map
(X, τ)
!X
−→ 1 is clearly closed). □
Theorem 10 (Bourbaki). A continuous map between topological spaces is perfect iff it is proper.
Remark 11. Since the category Top of topological spaces is an instance of the categories (T, V )-Cat, one
could ask about the analogues of Theorems 8, 10 for the latter category.
2. Proper maps in the category (T, V )-Cat
2.1. Categorical preliminaries
Remark 12. Every category (T, V )-Cat has the following two properties.
(1) The terminal object in (T, V )-Cat is given by (1, ⊤), where ⊤(x, ∗) = ⊤V for every x ∈ T1 (observe that
one takes the initial (T, V )-category structure on a terminal object in Set w.r.t. the empty source).
3
(2) The (T, V )-category structure d on the pullback of (T, V )-functors (X, a)
f
−→ (Z, c) and (Y, b)
g
−→ (Z, c)
(X ×Z Y, d)
pX

pY
// (Y, b)
g

(X, a)
f
// (Z, c)
(2.1)
is given by d = (p◦
X · a · TpX) ∧ (p◦
Y · b · TpY ), or, in pointwise notation, d(z, (x, y)) = a(TpX(z), x) ∧
b(TpY (z), y) for every z ∈ T(X ×Z Y ), x ∈ X, y ∈ Y (observe that one takes the initial (T, V )category
structure on the set X ×Z Y w.r.t. the source (U(X, a)
pX
←−− X ×Z Y
pY
−−→ U(Y, b)), where
(T, V )-Cat
U
−→ Set is the forgetful functor).
Definition 13. A lax extension ˆT to V -Rel of a functor T on Set is called left-whiskering provided that
ˆT(f · r) = Tf · ˆTr for every V -relation X 1r
// Y and every map Y
f
−→ Z.
Remark 14.
(1) A lax extension ˆT to V -Rel of a functor T on Set satisfies Tf · ˆTr ⩽ ˆT(f · r) for every V -relation
X 1r
// Y and every map Y
f
−→ Z, since Tf · ˆTr ⩽ ˆTf · ˆTr ⩽ ˆT(f · r).
(2) Recall from Lecture 2 that a lax extension ˆT to V -Rel of a functor T on Set is always right-whiskering,
i.e., satisfies ˆT(s · f) = ˆTs · Tf for every map X
f
−→ Y and every V -relation Y 1s
// Z.
Example 15.
(1) The lax extension ˆP to Rel of the powerset functor P on Set is left-whiskering. Observe that given
a relation X
1r
// Y and a map Y
f
−→ Z, for every A ∈ PX and every C ∈ PZ, it follows that
A ˆP(f · r) C iff for every z ∈ C there exists x ∈ A such that x (f · r) z iff for every z ∈ C, there exist
x ∈ A and y ∈ Y such that x r y and y f◦ z iff for every z ∈ C, there exist x ∈ A and y ∈ Y such that
x r y and f(y) = z iff there exists B ∈ PY such that for every y ∈ B there exists x ∈ A such that x r y
and f(B) = C iff there exists B ∈ PY such that A ˆPr B and f(B) = C iff there exists B ∈ PY such
that A ˆPr B and B (Pf)◦ C iff A (Pf · ˆPr) C.
(2) The lax extension ˆβ (resp. ¯β) to Rel (resp. P+-Rel) of the ultrafilter functor β on Set is left-whiskering.
Definition 16. A functor T on Set is said to be taut provided that it preserves pullbacks of monomorphisms
along arbitrary maps, namely, if
X ×Y Z
pX

pZ
// Z
g

X
f
// Y,
is a pullback and X
f
−→ Y is a monomorphism, then
T(X ×Y Z)
T pX

T pZ
// TZ
T g

TX
T f
// TY
is a pullback.
4
Example 17.
(1) The powerset functor P on Set is taut. Observe first that monomorphisms in Set are precisely the
injective maps, which are preserved by the powerset functor P (notice that an injective map X
f
−→ Y
with X ̸= ∅ is a section, and sections are preserved by every functor). Given a pullback
X ×Y Z
pX

pZ
// Z
g

X
f
// Y,
(2.2)
with a monomorphism X
f
−→ Y , there exists a map P(X ×Y Z)
h
−→ PX ×P Y PZ defined by the
commutative diagram
P(X ×Y Z)
P pX
$$
P pZ
))
h
((
PX ×P Y PZ
pP X

pP Z
// PZ
P g

PX
P f
// PY.
We show that h is a bijective map. Since (2.2) is a pullback, pZ is a monomorphism and therefore, PpZ
is a monomorphism. Thus, h is a monomorphism, i.e., injective. To show that h is surjective, notice
that given (A, C) ∈ PX ×P Y PZ, f(A) = Pf(A) = Pf · pP X(A, C) = Pg · pP Z(A, C) = Pg(C) = g(C).
Let D = (A × C) (X ×Y Z). To show that h(D) = (A, C), observe that h(D) = (PpX(D), PpZ(D)) =
(pX(D), pZ(D)). Clearly, pX(D) ⊆ A and pZ(D) ⊆ C. Given a ∈ A, since f(A) = g(C), there exists
c ∈ C such that f(a) = g(c), which implies (a, c) ∈ D, which gives a ∈ pX(D). As a consequence,
A ⊆ pX(D), which implies A = pX(D). Similarly, C = pZ(D). Thus, h(D) = (pX(D), pZ(D)) = (A, C).
(2) The ultrafilter functor β on Set is taut.
Lemma 18. Taut functors preserve monomorphisms.
Proof. Observe that a map X
f
−→ Y is a monomorphism iff the diagram
X
1X

1X
// X
f

X
f
// Y
is a pullback. □
Remark 19. The property of being taut can be defined for a functor T on an arbitrary category C.
Remark 20. From now on, assume that the lax extension ˆT to V -Rel of a monad T on Set satisfies the
following three conditions:
(T) T is taut;
(W) ˆT is left-whiskering;
5
(N) ˆT
m◦
−−→ ˆT ˆT is natural, which means that the diagram
TX 1
m◦
X
//
•ˆT r

TTX
• ˆT ˆT r

TY
1
m◦
Y
// TTY
commutes for every V -relation X 1r
// Y.
Remark 21. Observe that given a lax extension ˆT to V -Rel of a monad T on Set, since ˆT ˆT
m
−→ ˆT is an
oplax natural transformation (recall Lecture 1), it follows that
TTX
mX
//
•ˆT ˆT r

⩽
TX
• ˆT r

TTY mY
// TY
for every V -relation X 1r
// Y. Thus, mY · ˆT ˆTr ⩽ ˆTr · mX implies ˆT ˆTr · m◦
X ⩽ m◦
Y · mY · ˆT ˆTr · m◦
X ⩽
m◦
Y · ˆTr · mX · m◦
X ⩽ m◦
Y · ˆTr, i.e., ˆT ˆTr · m◦
X ⩽ m◦
Y · ˆTr. As a consequence, one obtains that ˆT
m◦
−−→ ˆT ˆT is
a natural transformation iff
TX
1m◦
X
//
•ˆT r

⩽
TTX
• ˆT ˆT r

TY
1
m◦
Y
// TTY
for every V -relation X
1r
// Y.
Example 22.
(1) The lax extension ˆP to Rel of the powerset monad P on Set satisfies the conditions of Remark 20. To
show condition (N), observe that for every V -relation X 1r
// Y, given A ∈ PX and B ∈ PPY , on the
one hand, A (m◦
Y · ˆPr) B iff A ( ˆPr) mY (B) iff A ( ˆPr) B iff for every y ∈ B there exists x ∈ A such
that x r y, and, on the other hand, A ( ˆP ˆPr · m◦
X) B iff there exists A ∈ PPX such that A m◦
X A and
A ( ˆP ˆPr) B iff there exists A ∈ PPX such that mX(A) = A and A ( ˆP ˆPr) B iff there exists A ∈ PPX
such that A = A and for every B ∈ B there exists A′
∈ A such that A′
( ˆPr) B iff there exists
A ∈ PPX such that A = A and for every B ∈ B there exists A′
∈ A such that for every y ∈ B
there exists x′
∈ A′
such that x′
r y. In view of Remark 21, one has to show that m◦
Y · ˆPr ⩽ ˆP ˆPr · m◦
X.
Observe that if A (m◦
Y · ˆPr) B, then taking A = {A} ∈ PPX, one gets A = A and for every B ∈ B
there exists A′
= A ∈ A such that for every y ∈ B ⊆ B there exists x ∈ A′
= A such that x r y.
(2) The lax extension ˆβ (resp. ¯β) to Rel (resp. P+-Rel) of the ultrafilter monad on Set satisfies the
conditions of Remark 20.
Theorem 23. There exists a functor (T, V )-Cat
G
−→ V -Cat, which is given by G((X, a)
f
−→ (Y, b)) =
(TX, ˆa)
T f
−−→ (TY,ˆb), where ˆa = ˆTa · m◦
X.
6
Proof. To show that (TX, ˆa) is a V -category, notice that, firstly, 1T X = T1X ⩽ ˆT1X ⩽ ˆT(a · eX) =
ˆTa·TeX
(†)
⩽ ˆTa·m◦
X = ˆa, where (†) uses the fact that mX ·TeX = 1T X implies TeX ⩽ m◦
X ·mX ·TeX = m◦
X.
Secondly, a · ˆTa · m◦
X ⩽ a · mX · m◦
X ⩽ a gives ˆTa · ˆT ˆTa · (TmX)◦
⩽ ˆTa · ˆT ˆTa · ˆTm◦
X ⩽ ˆT(a · ˆTa · m◦
X) ⩽ ˆTa,
and therefore, ˆa · ˆa = ˆTa · m◦
X · ˆTa · m◦
X
(N)
= ˆTa · ˆT ˆTa · m◦
T X · m◦
X = ˆTa · ˆT ˆTa · (mX · mT X)◦ mX ·mT X =mX ·T mX
=
ˆTa · ˆT ˆTa · (mX · TmX)◦
= ˆTa · ˆT ˆTa · (TmX)◦
· m◦
X ⩽ ˆTa · m◦
X = ˆa.
To show that (TX, ˆa)
T f
−−→ (TY,ˆb) is a V -functor, notice that f · a ⩽ b · Tf gives a ⩽ f◦
· b · Tf,
and therefore, ˆTa ⩽ ˆT(f◦
· b · Tf) = (Tf)◦
· ˆTb · TTf, which then yields Tf · ˆa = Tf · ˆTa · m◦
X ⩽
Tf · (Tf)◦
· ˆTb · TTf · m◦
X
T f·(T f)◦
⩽1T Y
⩽ ˆTb · TTf · m◦
X
T T f·m◦
X ⩽m◦
Y ·T f
⩽ ˆTb · m◦
Y · Tf = ˆb · Tf. □
Proposition 24. Given a lax extension ˆT to V -Rel of a monad T = (T, m, e) on Set, the natural transformation
1Set
e
−→ T provides a natural transformation Ind
e
−→ G, where (T, V )-Cat
Ind
−−→ Prost, Ind((X, a)
f
−→
(Y, b)) = (X, ⩽a)
f
−→ (Y, ⩽b) (with x ⩽a x′
iff k ⩽ a(eX(x), x′
)) is the induced preorder functor, and Prost
is considered as a full subcategory of V -Cat w.r.t. the full embedding Prost   Bι
// V -Cat (cf. Lecture 2).
Proof. It will be enough to show that given a (T, V )-category (X, a), (Ind(X, a) = (X, ⩽a))
eX
−−→ (G(X, a) =
(TX, ˆa)) is a (T, V )-functor, namely,
X
eX
//
•⩽a

⩽
TX
• ˆa

X eX
// TX.
Given x ∈ X, x ∈ TX, on the one hand, eX· ⩽a (x, x) = x′∈X ⩽a (x, x′
) ⊗ (eX)◦(x′
, x) = {k | x′
∈ X such
that k ⩽ a(eX(x), x′
) and eX(x′
) = x} =
k, there exists x′
∈ X with k ⩽ a(eX(x), x′
) and eX(x′
) = x
⊥V , otherwise,
and, on the other hand, ˆa·eX(x, x) = ˆa(eX(x), x) = ˆTa·m◦
X(eX(x), x) = X∈T T X m◦
X(eX(x), X)⊗ ˆTa(X, x) =
X∈T T X(mX)◦(X, eX(x))⊗ ˆTa(X, x) = mX (X)=eX (x)
ˆTa(X, x) ⩾ ˆTa(eT X ·eX(x), x), since mX(eT X ·eX(x)) =
(mX · eT X) · eX(x)
mX ·eT X =1T X
= 1T X · eX(x) = eX(x). If eX· ⩽a (x, x) = k, then there exists x′
∈ X such
that k ⩽ a(eX(x), x′
) and eX(x′
) = x, which implies ˆTa(eT X · eX(x), x) = ˆTa(eT X · eX(x), eX(x′
)) =
ˆTa · eT X(eX(x), eX(x′
))
eX ·a⩽ ˆT a·eT X
⩾ eX · a(eX(x), eX(x′
))=e◦
X · eX · a(eX(x), x′
)
1X ⩽e◦
X ·eX
⩾ a(eX(x), x′
)⩾k. □
Remark 25. Notice that if X
eX
−−→ TX is injective for every set X, then Ind is a subfunctor of G, i.e., G is
an extension of the induced preorder from the underlying set X of a (T, V )-category (X, a) to the set TX.
Example 26.
(1) If T is the identity monad on Set, then V -Cat
G
−→ V -Cat is the identity functor.
(2) For the lax extension ˆβ to Rel of the ultrafilter monad β on Set, the functor Top
G
−→ Prost is defined
by G((X, τ)
f
−→ (Y, σ)) = (βX, ⩽)
βf
−−→ (βY, ⩽), where for every x, z ∈ βX, x ⩽ z iff z τ ⊆ x. In
particular, given principal ultrafilters ˙x, ˙y ∈ βX, it follows that ˙x ⩽ ˙y iff y ∈ cl{x}. In other words,
since the principal ultrafilter natural transformation 1Set
e
−→ β has injective components X
e
−→ βX for
every set X, one obtains that G is an extension of the induced preorder from the underlying set X of a
topological space (X, τ) to the set of ultrafilters on X (cf. Remark 25).
2.2. Algebraic preliminaries
Definition 27. Let (V, , ⊗) be a quantale.
(1) V is called strictly two-sided provided that (V, ⊗, ⊤V ) is a monoid.
7
(2) V is called cartesian closed provided that a ∧ ( B) = b∈B(a ∧ b) for every a ∈ V , B ⊆ V .
Remark 28. Observe that a quantale V is cartesian closed iff its underlying partially ordered set is a frame,
namely, a complete lattice, in which finite meets distribute over arbitrary joins.
Theorem 29. Given a unital quantale V , equivalent are:
(1) V is cartesian closed;
(2) the left Frobenius law
f · ((f◦
· r) ∧ s) = r ∧ (f · s) (F)
holds in V -Rel for every triangle of the form
Z
 
rbs
~~
X
f
// Y.
(3) the right Frobenius law
(r ∧ (s · f)) · f◦
= (r · f◦
) ∧ s
holds in V -Rel for every triangle of the form
X
f
//
¡
r
Y
a
s
~~
Z.
Proof.
(1) ⇒ (2): Given y ∈ Y and z ∈ Z, on the one hand, f · ((f◦
· r) ∧ s)(z, y) = f(x)=y((f◦
· r) ∧ s)(z, x) =
f(x)=y((f◦
· r)(z, x) ∧ s(z, x)) = f(x)=y(r(z, f(x)) ∧ s(z, x)) = f(x)=y(r(z, y) ∧ s(z, x)); on the other
hand, (r ∧ (f · s))(z, y) = r(z, y) ∧ ( f(x)=y s(z, x))
(1)
= f(x)=y(r(z, y) ∧ s(z, x)).
(2) ⇒ (1): Given a ∈ V and B ⊆ V , consider the triangle
1
~
r

ds

B
!B
// 1,
where s(∗, b) = b for every b ∈ B, and r(∗, ∗) = a. It then follows that b∈B(a ∧ b) = b∈B((a ⊗ k) ∧ b) =
b∈B((r(∗, ∗)⊗!◦
B(∗, b)) ∧ b) = b∈B((!◦
B · r)(∗, b) ∧ s(∗, b)) ⊗ k = b∈B((!◦
B · r) ∧ s)(∗, b) ⊗ (!B)◦(b, ∗) =
!B · ((!◦
B · r) ∧ s)(∗, ∗)
(2)
= (r ∧ (!B · s))(∗, ∗) = r(∗, ∗) ∧ (!B · s)(∗, ∗) = a ∧ ( b∈B s(∗, b) ⊗ (!B)◦(b, ∗)) =
a ∧ ( b∈B b ⊗ k) = a ∧ ( B), i.e., a ∧ ( B) = b∈B(a ∧ b). □
Remark 30. From now on, V stands for a cartesian closed, strictly two-sided quantale.
Example 31. The quantales 2 and P+ satisfy the conditions of Remark 30.
8
2.3. Proper (T, V )-functors and their properties
Definition 32. Given a topological space (X, τ) and an ultrafilter x ∈ βX, an element x ∈ X is a limit of
x (x converges to x) provided that x contains every U ∈ τ such that x ∈ U. lim x is the set of limits of x.
Theorem 33. Given topological spaces (X, τ) and (Y, σ), a continuous map X
f
−→ Y is proper iff for every
ultrafilter x ∈ βX and every y ∈ lim βf(x), there exists x ∈ lim x such that f(x) = y.
Remark 34. Representing the category Top as (β, 2)-Cat, one gets that a (β, 2)-functor (X, a)
f
−→ (Y, b) is
proper iff b(βf(x), y) ⩽ f(x)=y a(x, x) for every x ∈ βX, y ∈ Y iff b · βf(x, y) ⩽ f · a(x, y) for every x ∈ βX,
y ∈ Y iff b · βf ⩽ f · a in Rel iff f · a = b · βf in Rel (f · a ⩽ b · βf is the definition of (β, 2)-functors).
Definition 35. A (T, V )-functor (X, a)
f
−→ (Y, b) is proper provided that the diagram
TX
•a

T f
// TY
• b

X
f
// Y
commutes, i.e., f · a = b · Tf.
Example 36.
(1) Prost: an order-preserving map (X, ⩽X)
f
−→ (Y, ⩽Y ) is proper iff f· ⩽X = ⩽Y ·f iff for every x ∈ X
and every y ∈ Y such that f(x) ⩽ y, there exists z ∈ X such that x ⩽ z and f(z) = y.
(2) QPMet: a non-expansive map (X, ρ)
f
−→ (Y, ϱ) is proper iff ϱ(f(x), y) = inf{ρ(x, z) | z ∈ X and f(z) =
y} for every x ∈ X, y ∈ Y .
(3) Top: Definition 35 gives precisely the proper maps of Definition 1 (2).
(4) App: a non-expansive map (X, δ)
f
−→ (Y, σ) is proper iff supf−1(B)∈x σ(y, B) = inff(x)=y supA∈x δ(x, A)
for every x ∈ βX, y ∈ Y .
(5) Cls: a continuous map (X, c)
f
−→ (Y, d) is proper iff for every A ∈ PX, y ∈ Y such that y ∈ d(f(A)),
there exists x ∈ X such that x ∈ c(A) and f(x) = y iff d(f(A)) ⊆ f(c(A)) for every A ∈ PX.
Theorem 37. Proper maps are stable under pullbacks in (T, V )-Cat.
Proof. Notice that given a pullback diagram
X ×Z Y
pX

pY
// Y
g

X
f
// Z
in Set, it follows that
g◦
· f = pY · p◦
X, (2.3)
since given x ∈ X and y ∈ Y , g◦
· f(x, y) = f◦(x, g(y)) =
k, f(x) = g(y)
⊥V , otherwise
=
k, (x, y) ∈ X ×Z Y
⊥V , otherwise
=
(x′,y′)∈X×Z Y pX((x′
, y′
), x)⊗pY ((x′
, y′
), y) = (x′,y′)∈X×Z Y p◦
X(x, (x′
, y′
))⊗pY ((x′
, y′
), y) = pY ·p◦
X(x, y).
Consider now diagram (2.1), in which f is proper. To show that pY is proper, notice that b · TpY =
(b∧b)·TpY
b⩽g◦
·c·T g
⩽ ((g◦
·c·Tg)∧b)·TpY = (g◦
·c·Tg·TpY )∧(b·TpY ) = (g◦
·c·T(g·pY ))∧(b·TpY )
g·pY =f·pX
=
(g◦
· c · T(f · pX)) ∧ (b · TpY ) = (g◦
· c · Tf · TpX) ∧ (b · TpY )
c·T f=f·a
= (g◦
· f · a · TpX) ∧ (b · TpY )
(2.3)
=
(pY · p◦
X · a · TpX) ∧ (b · TpY )
(F)
= pY · ((p◦
X · a · TpX) ∧ (p◦
Y · b · TpY )) = pY · d. □
9
Definition 38. Given a (T, V )-functor (X, a)
f
−→ (Y, b), the fibre of f on y is the pullback (f−1
(y), ˜a)
!f−1(y)
−−−−→
(1, 1♯
) of f along the (T, V )-functor (1, 1♯
)
y
−→ (Y, b), where 1♯
= e◦
1 · ˆT11 is the discrete structure on 1, i.e.,
(f−1
(y), ˜a)
_
if−1(y)

!f−1(y)
// (1, 1♯
)
y

(X, a)
f
// (Y, b),
(2.4)
where ˜a = (i◦
f−1(y) · a · Tif−1(y)) ∧ (!◦
f−1(y) · e◦
1 · ˆT11 · T!f−1(y)), or, in pointwise notation, ˜a(x, x) = a(x, x) ∧
ˆT11(T!f−1(y)(x), e1(∗)) for every x ∈ T(f−1
(y)) and every x ∈ f−1
(y).
Theorem 39. A (T, V )-functor (X, a)
f
−→ (Y, b) is proper iff all its fibres are proper, and the V -functor
(TX, ˆa)
T f
−−→ (TY,ˆb) is proper.
Proof. For the necessity, notice that Theorem 37 provides properness of fibres. To show the second claim,
notice first that for every lax extension ˆT to V -Rel of a monad T on Set, and every set X, one can obtain
ˆT1X = ˆT(e◦
X) · m◦
X (2.5)
(see Lecture 2 for more detail). Then ˆb·Tf = ˆTb·m◦
Y ·Tf ⩽ ˆTb·m◦
Y · ˆTf
(N)
= ˆTb· ˆT ˆTf ·m◦
X ⩽ ˆT(b· ˆTf)·m◦
X
(†)
⩽
ˆT(b·Tf)·m◦
X
b·T f=f·a
= ˆT(f ·a)·m◦
X
(W)
= Tf · ˆTa·m◦
X = Tf ·ˆa, where (†) uses the fact that b· ˆTf = b· ˆT(1Y ·f) =
b · ˆT1Y · Tf
(2.5)
= b · ˆT(e◦
Y ) · m◦
Y · Tf
e◦
Y ⩽b
⩽ b · ˆTb · m◦
Y · Tf
b· ˆT b⩽b·mY
⩽ b · mY · m◦
Y · Tf
mY ·m◦
Y ⩽1T Y
⩽ b · Tf.
The sufficiency can be shown as follows. Firstly, notice that a sink (Xi
fi
−→ X)I in Set is an epi-sink iff
i∈I
fi · f◦
i = 1X. (2.6)
Secondly, notice that b = b · 1◦
T Y = b · (mY · eT Y )◦
= b · e◦
T Y · m◦
Y
b·e◦
T Y ⩽e◦
Y · ˆT b
⩽ e◦
Y · ˆTb · m◦
Y = e◦
Y · ˆb. It
follows then that b · Tf ⩽ e◦
Y · ˆb · Tf
ˆb·T f=T f·ˆa
= e◦
Y · Tf · ˆa = e◦
Y · Tf · ˆTa · m◦
X = 1Y · e◦
Y · Tf · ˆTa · m◦
X
(†)
=
( y∈Y y ·y◦
)·e◦
Y ·Tf · ˆTa·m◦
X = ( y∈Y y ·y◦
·e◦
Y ·Tf)· ˆTa·m◦
X = ( y∈Y y ·(eY ·y)◦
·Tf)· ˆTa·m◦
X
eY ·y=T y·e1
=
( y∈Y y · (Ty · e1)◦
· Tf) · ˆTa · m◦
X = ( y∈Y y · e◦
1 · (Ty)◦
· Tf) · ˆTa · m◦
X = r1, where (†) uses that fact
that (1
y
−→ Y )Y is an epi-sink in Set. Since the underlying Set-diagram of (2.4) is a pullback along the
monomorphism 1
y
−→ Y , by (T),
T(f−1
(y))
T if−1(y)

T !f−1(y)
// T1
T y

TX
T f
// TY
is a pullback as well. Similar to (2.3), (Ty)◦
· Tf = T!f−1(y) · (Tif−1(y))◦
, and then r1 = ( y∈Y y · e◦
1 ·
T!f−1(y) · (Tif−1(y))◦
) · ˆTa · m◦
X = ( y∈Y y · e◦
1 · T11 · T!f−1(y) · (Tif−1(y))◦
) · ˆTa · m◦
X ⩽ ( y∈Y y · e◦
1 · ˆT11 ·
T!f−1(y)·(Tif−1(y))◦
)· ˆTa·m◦
X = r2. Given y ∈ Y , properness of the fibres of f implies that the (T, V )-functor
(f−1
(y), ˜a)
!f−1(y)
−−−−→ (1, 1♯
) is proper, i.e., 1♯
·T!f−1(y) =!f−1(y)·˜a. Definition 38 implies that e◦
1 · ˆT11·T!f−1(y) =
1♯
· T!f−1(y) =!f−1(y) · ˜a =!f−1(y) · ((i◦
f−1(y) · a · Tif−1(y)) ∧ (!◦
f−1(y) · e◦
1 · ˆT11 · T!f−1(y))) ⩽!f−1(y) · i◦
f−1(y) · a ·
10
Tif−1(y), and thus, r2 ⩽ ( y∈Y y·!f−1(y) · i◦
f−1(y) · a · Tif−1(y) · (Tif−1(y))◦
) · ˆTa · m◦
X
T if−1(y)·(T if−1(y))◦
⩽1T X
⩽
( y∈Y y·!f−1(y) · i◦
f−1(y) · a) · ˆTa · m◦
X
y·!f−1(y)=f·if−1(y)
= ( y∈Y f · if−1(y) · i◦
f−1(y) · a) · ˆTa · m◦
X
if−1(y)·i◦
f−1(y)
⩽1X
⩽
( y∈Y f · a) · ˆTa · m◦
X ⩽ f · a · ˆTa · m◦
X
a· ˆT a⩽a·mX
⩽ f · a · mX · m◦
X
mX ·m◦
X ⩽1T X
⩽ f · a. □
Theorem 40. If ˆT
e◦
−→ 1V -Rel is a natural transformation, then every (T, V )-functor has proper fibres.
Proof. Given a (T, V )-functor (X, a)
f
−→ (Y, b), one has to show that the diagram
T(f−1
(y))
T !f−1(y)
//
•˜a

T1
• 1♯

f−1
(y)
!f−1(y)
// 1
commutes. The condition of the theorem implies commutativity of the diagram
T(f−1
(y))
•e◦
f−1(y)

ˆT !f−1(y)
// T1
ˆT 11
//
•e◦
1

TX
• e◦
1

f−1
(y)
!f−1(y)
// 1
11
// 1.
Given x ∈ T(f−1
(y)), it follows then that 1♯
·T!f−1(y)(x, ∗) = e◦
1 · ˆT11 ·T!f−1(y)(x, ∗) ⩽ e◦
1 · ˆT11 · ˆT!f−1(y)(x, ∗) =
!f−1(y) · e◦
f−1(y)(x, ∗) = f(x)=y e◦
f−1(y)(x, x)
e◦
f−1(y)
⩽˜a
⩽ f(x)=y ˜a(x, x) =!f−1(y) · ˜a(x, ∗). □
Corollary 41. If ˆT
e◦
−→ 1V -Rel is a natural transformation, then a (T, V )-functor (X, a)
f
−→ (Y, b) is proper
iff the V -functor (TX, ˆa)
T f
−−→ (TY,ˆb) is proper.
Proof. Follows from Theorems 39, 40. □
Remark 42. Observe that given a lax extension ˆT to V -Rel of a monad T on Set, since 1V -Rel
e
−→ ˆT is an
oplax natural transformation (recall Lecture 1), it follows that
X
eX
//
•r

⩽
TX
• ˆT r

Y eY
// TY
for every V -relation X 1r
// Y. Thus, eY ·r ⩽ ˆTr·eX implies r·e◦
X ⩽ e◦
Y ·eY ·r·e◦
X ⩽ e◦
Y · ˆTr·eX ·e◦
X ⩽ e◦
Y · ˆTr,
i.e., r · e◦
X ⩽ e◦
Y · ˆTr. As a consequence, one obtains that ˆT
e◦
−→ 1V -Rel is a natural transformation iff
TX 1e◦
X
//
•ˆT r

⩽
X
• r

TY 1
e◦
Y
// Y
for every V -relation X 1r
// Y.
11
Remark 43.
(1) The lax extension ˆP of the powerset monad P on Set to Rel fails to satisfy the condition of Corollary 41.
Observe that following Remark 42, it is enough to a find a relation X 1r
// Y such that e◦
Y · ˆPr ̸⩽ r·e◦
X.
Given A ∈ PX and y ∈ Y , on the one hand, A (e◦
Y · ˆPr) y iff A ( ˆPr) eY (y) iff A ˆPr {y} iff there exists
x ∈ A such that x r y (recall Lecture 1), and, on the other hand, A (r · e◦
X) y iff there exists x ∈ X such
that A e◦
X x and x r y iff there exists x ∈ X such that eX(x) = A and x r y iff there exists x ∈ X such
that A = {x} and x r y. If X = Y = {0, 1} and r = {(0, 0), (1, 1)} ⊆ X × Y , then X (e◦
Y · ˆPr) 0 (since
0 r 0), but X and 0 fail to be in relation r · e◦
X since {0} ̸= {0, 1}.
(2) The lax extension ˆβ (resp. ¯β) of the ultrafilter monad β on Set to Rel (resp. P+-Rel) fails to satisfy
the condition of Corollary 41.
(3) There exist monads on Set, whose lax extensions satisfy the condition of Corollary 41.
2.4. Compact (T, V )-categories
Definition 44. A (T, V )-category (X, a) is said to be compact provided that the unique (T, V )-functor
(X, a)
!X
−→ (1, ⊤) is proper.
Example 45. Given a compact (T, V )-category (X, a), it follows that !X · a = ⊤ · T!X, or, in pointwise
notation, x∈X a(x, x) = ⊤V for every x ∈ TX.
(1) Prost: a preordered set (X, ⩽) is compact provided that for every y ∈ X, there exists x ∈ X such that
y ⩽ x, which is always true (choose x = y).
(2) QPMet: a quasi-pseudo-metric space (X, ρ) is compact provided that infx∈X ρ(y, x) = 0 for every
y ∈ X, which is always true (choose x = y).
(3) Top: a topological space (X, τ) is compact provided that every ultrafilter on X has a limit point, which
is precisely the standard definition of compactness of topological spaces.
(4) App: an approach space (X, δ) is compact provided that infx∈X supA∈x δ(x, A) = 0 for every x ∈ βX.
(5) Cls: a closure space (X, c) is compact provided that c(A) ̸= ∅ for every A ∈ PX. Observe that given
A ∈ PX, it follows that A ⊆ c(A), which implies that c(A) ̸= ∅ provided that A ̸= ∅. Thus, a closure
space (X, c) is compact iff c(∅) ̸= ∅. It then follows that a closure space induced by a topological space
is never compact, since ∅ is closed (c(∅) = ∅) in every topological space.
Remark 46. If T1 ∼= 1, then the (T, V )-category (1, 1♯
) (which is additionally a separator in (T, V )-Cat)
coincides with the terminal object (1, ⊤) (since 1♯
(∗, ∗) = e◦
1 · ˆT11(∗, ∗) = ˆT11(∗, e1(∗)) = ˆT11(∗, ∗) ⩾
11(∗, ∗) = ⊤V ), and therefore, it follows that (X, a) is compact iff the only fibre of (X, a)
!X
−→ (1, ⊤) is
proper, since the respective fibre is then given by the pullback
(X, a)
1X

!X
// (1, ⊤)
11

(X, a)
!X
// (1, ⊤).
Example 47.
(1) For the powerset functor P on Set, P1 = {∅, {∗}} ̸∼= {∗} = 1.
(2) For the ultrafilter functor β on Set, β1 ∼= 1.
Theorem 48. If (X, a) is a compact (T, V )-category, then the fibre of the (T, V )-functor (X, a)
!X
−→ (1, ⊤)
is proper. If the two structures 1♯
and ⊤ on 1 coincide, then the converse is true.
12
Proof. Follows from Theorem 37 and the arguments of Remark 46. □
Corollary 49. Suppose that ⊤ is the discrete structure on 1. Given a (T, V )-functor (X, a)
f
−→ (Y, b),
equivalent are:
(1) f is proper;
(2) the V -functor (TX, ˆa)
T f
−−→ (TY,ˆb) is proper and f has compact fibres.
Proof. Follows from Theorems 39, 48. □
Corollary 50. If ⊤ is the discrete structure on 1, and ˆT
e◦
−→ 1V -Rel is a natural transformation, then every
(T, V )-category is compact.
Proof. Recall Theorem 40. □
Theorem 51. If the lax extension ˆT to V -Rel of a functor T on Set is flat, then ⊤ = 1♯
iff T1 ∼= 1.
Proof. The sufficiency is clear. For the necessity, notice that given x ∈ T1, it follows that ⊤V = ⊤(x, ∗) =
1♯
(x, ∗) = e◦
1 · ˆT11(x, ∗)
ˆT is flat
= e◦
1 · T11(x, ∗) = e◦
1(x, ∗), and therefore, x = e1(∗). □
3. Closed maps in the category (T, V )-Cat
Lemma 52. A continuous map (X, τ)
f
−→ (Y, σ) between topological spaces is closed iff cl(f(A)) ⊆ f(cl(A))
for every A ⊆ X.
Proof.
⇒: Given a subset A ⊆ X, if f is closed, then f(cl(A)) is closed. Thus, A ⊆ cl(A) implies f(A) ⊆ f(cl(A))
implies cl(f(A)) ⊆ cl(f(cl(A))) = f(cl(A)), i.e., cl(f(A)) ⊆ f(cl(A)).
⇐: Given a subset A ⊆ X, since A ⊆ f−1
(f(A)) ⊆ f−1
(cl(f(A))) and f−1
(cl(f(A))) is closed,
cl(A) ⊆ f−1
(cl(f(A))) and then f(cl(A)) ⊆ f(f−1
(cl(f(A)))) ⊆ cl(f(A)), i.e., f(cl(A)) ⊆ cl(f(A)). Thus,
f(cl(A)) = cl(f(A)) by the assumption of the lemma. If A is closed, then f(A) = f(cl(A)) = cl(f(A)). □
Lemma 53. Given a topological space (X, τ) and A ⊆ X, it follows that cl(A) = x∈βX and A∈x lim x.
Definition 54. A (T, V )-functor (X, a)
f
−→ (Y, b) is closed provided that for every A ⊆ X,
f · a · TiA·!◦
T A = b · Tf · TiA·!◦
T A, (3.1)
where A   iA
// X is the inclusion map and TA
!T A
−−→ 1 is the unique map.
Remark 55. Observe that given a V -relation TX 1r
// X, for every subset A ⊆ X, the composite V relation
1
1!◦
T A
// TA
  T iA
// TX
1r
// X in pointwise notation provides r·TiA·!◦
T A(∗, x) = y∈T A!◦
T A(∗, y)⊗
(r · TiA)(y, x) = y∈T A(!T A)◦(y, ∗) ⊗ ( x∈T X(TiA)◦(y, x) ⊗ r(x, x)) = y∈T A r(TiA(y), x) for every x ∈ X.
Lemma 56. Given a (T, V )-functor (X, a)
f
−→ (Y, b), the following are equivalent:
(1) f is closed;
(2) b · Tf · TiA·!◦
T A ⩽ f · a · TiA·!◦
T A for every A ⊆ X.
Proof. Recall that since (X, a)
f
−→ (Y, b) is a (T, V )-functor, it follows that f · a ⩽ b · Tf. □
13
Example 57. Let (X, a)
f
−→ (Y, b) be a (T, V )-functor. Given A ⊆ X, denote by A
f
−→ f(A) the restriction
of f to A and f(A), respectively. Commutativity of the diagram
1
1!◦
T A
//
§
!◦
T (f(A)) ##
TA
T iA
//
T f

TX
T f

T(f(A))
T if(A)
// TY
and Lemma 56 replace (3.1) with b · Tif(A)·!◦
T (f(A)) ⩽ f · a · TiA·!◦
T A, which, in pointwise notation, provides
y∈T (f(A))
b(Tif(A)(y), y) ⩽
x∈T A f(x)=y
a(TiA(x), x) (3.2)
for every y ∈ Y . In some particular cases, (3.2) can be rewritten as follows.
(1) Prost: an order-preserving map (X, ⩽)
f
−→ (Y, ⩽) is closed iff for every x ∈ X and every y ∈ Y such
that f(x) ⩽ y, there exists z ∈ X such that x ⩽ z and f(z) = y.
(2) QPMet: a non-expansive map (X, ρ)
f
−→ (Y, ϱ) is closed iff inf{ρ(x, z) | z ∈ X and f(z) = y} ⩽
ϱ(f(x), y) for every x ∈ X, y ∈ Y .
(3) Top: one gets precisely the result of Lemma 52.
(4) App: a non-expansive map (X, δ)
f
−→ (Y, σ) is closed iff inff(x)=y δ(x, A) ⩽ σ(y, f(A)) for every A ⊆ X.
(5) Cls: a continuous map (X, c)
f
−→ (Y, d) is closed iff {d(C) | C ⊆ f(A)} ⊆ {f(c(B)) | B ⊆ A} for every
A ∈ PX iff d(f(A)) ⊆ f(c(A)).
Theorem 58. Every proper (T, V )-functor is closed.
Proof. Follows directly from the definition of the two concepts. □
Theorem 59. Suppose every (T, V )-category (X, a) has the property that given x ∈ TX, there exists A ⊆ X
such that
x ∈ TiA(TA) and a · TiA·!◦
T A ⩽ a · x, where x is considered as a map 1
x
−→ TX. (3.3)
Then every closed (T, V )-functor (X, a)
f
−→ (Y, b) is proper.
Proof. To show that b · Tf ⩽ f · a, notice that given x ∈ TX and y ∈ Y , b · Tf(x, y)
(†)
= b · Tf(TiA(z), y) =
b·Tf ·TiA(z, y) ⩽ w∈T A b·Tf ·TiA(w, y) = b·Tf ·TiA·!◦
T A(∗, y)
(††)
= f ·a·TiA·!◦
T A(∗, y)
(†††)
⩽ f ·a·x(∗, y) =
f · a(x, y), where (†) (resp. († † †)) relies on the left-hand (resp. right-hand) side of (3.3), and (††) uses the
closedness of the (T, V )-functor (X, a)
f
−→ (Y, b). □
Lemma 60. The categories V -Cat and (P, 2)-Cat (for the lax extension ˆP to Rel of the powerset monad
P on Set) satisfy condition (3.3).
Proof. The case of V -Cat is clear (given y ∈ X, take the singleton set A = {y}). To show condition (3.3)
for the category (P, 2)-Cat, recall that every (P, 2)-category (X, a) can be equivalently described as a closure
space (X, c), in which, given A ∈ PX and x ∈ X, x ∈ c(A) iff A a x. Therefore, if B ⊆ A ∈ PX, then B a x
implies x ∈ c(B) implies x ∈ c(A) implies A a x. As a result, given A ∈ PX, for every x ∈ X, it follows that
a · PiA·!◦
P A(∗, x) = B∈P A a · PiA(B, x) = B∈P A a(PiA(B), x) = B∈P A a(B, x) ⩽ a(A, x)=a · A(∗, x). □
14
Corollary 61. The concepts of proper and closed map are equivalent in the categories V -Cat, (P, 2)-Cat.
Thus, for every (T, V )-functor (X, a)
f
−→ (Y, b), the V -functor (TX, ˆa)
T f
−−→ (TY,ˆb) is proper iff it is closed.
Definition 62. A monad T on the category Set is said to be non-trivial provided that it admits EilenbergMoore
algebras, whose underlying sets have more than one element.
Proposition 63. Let T be non-trivial, let T∅ = ∅, and let ˆT be flat. If every (T, V )-category (X, a) satisfies
condition (3.3), then T is isomorphic to the identity functor on Set.
Proof. Given a set X, the assumption on non-triviality of T and [7, Subsection 3.1] together imply that
the map X
eX
−−→ TX is injective. We show that the map is also surjective.
Since ˆT is flat, the discrete (T, V )-category structure on X is provided by 1♯
X = e◦
X · ˆT1X = e◦
X. Given
x ∈ TX, there exists A ⊆ X, which satisfies condition (3.3) w.r.t. e◦
X. Since x ∈ TiA(TA), A ̸= ∅ (by the
assumption of the proposition), and therefore, there exists x ∈ A. One gets then that k ⩽ e◦
X(eX(x), x) ⩽
y∈T A e◦
X · TiA(y, x) = e◦
X · TiA·!◦
T A(∗, x)
(3.3)
⩽ e◦
X · x(∗, x) = e◦
X(x, x), which yields the desired eX(x) = x. □
Remark 64. Notice that while the ultrafilter monad β on Set has the property β∅ = ∅, the powerset
monad P on Set satisfies the converse condition P∅ ̸= ∅. In particular, the category (β, 2)-Cat (for the
lax extension ˆβ of the ultrafilter monad β) does not satisfy condition (3.3).
Definition 65. Given a topological category (A, U) over X, an A-morphism A
f
−→ B is said to be an
embedding provided that f is initial, and its underlying X-morphism UA
Uf
−−→ UB is a monomorphism.
Remark 66. A (T, V )-functor (X, a)
f
−→ (Y, b) is an embedding provided that the map X
f
−→ Y is injective
and a = f◦
· b · Tf.
Theorem 67. If (X, a)
f
−→ (Y, b) is an embedding (T, V )-functor, then f is closed iff f is proper.
Proof. The sufficiency follows from Theorem 58. For the necessity, we notice that f · a·!◦
T X = b · Tf·!◦
T X
since f is closed (take A = X in Definition 54), and also fix x0 ∈ TX. For every y ∈ f(X), it follows
that f · a(x0, y) = f(x)=y a(x0, x)
(†)
= a(x0, f−1
(y))
(†)
= f◦
· b · Tf(x0, f−1
(y)) = b · Tf(x0, f(f−1
(y))) =
b·Tf(x0, y), where (†) relies on the embedding assumption. For every y ̸∈ f(X), it follows that b·Tf(x0, y) ⩽
x∈T X b(Tf(x), y) = b · Tf·!◦
T X(∗, y)
(††)
= f · a·!◦
T X(∗, y) = x∈T X f(x)=y a(x, x) = ⊥V , which yields the
desired b · Tf(x0, y) = ⊥V = f · a(x0, y), where (††) relies on the above property of closed maps. □
Remark 68. The result of Theorem 67 extends the classical one in the category Top, which states that
the embedding assumption makes the concepts of closedness and properness equivalent.
4. Generalized Kuratowski-Mrówka theorem
Remark 69. Given a (T, V )-category (X, a) and x ∈ TX, define Y = X {w}, and let a V -relation
TY
1b
// Y be given by
b(y, y) =
⊤V , y = eY (y) or (y = TiX(x) and y = w)
⊥V , otherwise.
Below, sufficient conditions are provided for the above construction to define a (T, V )-category (Y, b).
Definition 70.
15
(1) A V -relation X
1r
// Y is said to have finite fibres provided that the set r◦
(y) = {x ∈ X | ⊥V < r(x, y)}
is finite for every y ∈ Y .
(2) A lax natural transformation 1V -Rel
e
−→ ˆT is said to be finitely (−)◦
-strict provided that the diagram
TX 1
e◦
X
//
•ˆT r

X
• r

TY
1
e◦
Y
// Y
commutes for every V -relation X 1r
// Y with finite fibres.
Example 71.
(1) The lax natural transformation 1Rel
e
−→ ˆβ (resp. 1V -Rel
e
−→ ¯β) of the extension ˆβ (resp. ¯β) to Rel (resp.
P+-Rel) of the ultrafilter monad β on Set is finitely (−)◦
-strict.
(2) The lax natural transformation 1Rel
e
−→ ˆP of the extension ˆP to Rel of the powerset monad P on Set
fails to be finitely (−)◦
-strict (cf. Remark 43 (1)).
Remark 72. The V -relation TY
1b
// Y of Remark 69 has finite fibres.
Theorem 73. If ˆT is flat and 1V -Rel
e
−→ ˆT is finitely (−)◦
-strict, then (Y, b) is a (T, V )-category.
Proof. The definition of the map b gives 1Y ⩽ b·eY . The condition b· ˆTb ⩽ b·mY can be shown as follows.
Given Y ∈ TTY and y ∈ Y , one gets that b · ˆTb(Y, y) = y∈T Y
ˆTb(Y, y) ⊗ b(y, y) and b · mY (Y, y) =
b(mY (Y), y). If there exists y ∈ TY such that ⊥V < ˆTb(Y, y) ⊗ b(y, y) (otherwise, the claim is clear), then
b(y, y) = ⊤V , and therefore, y = eY (y) or (y = TiX(x) and y = w).
If y = eY (y), then ˆTb(Y, y) ⊗ b(y, y) = ˆTb(Y, eY (y)) = e◦
Y · ˆTb(Y, y). Since the V -relation b has finite
fibres, apply finite (−)◦
-strictness of e and get e◦
Y · ˆTb = b · e◦
T Y . As a consequence, e◦
Y · ˆTb(Y, y) =
b·e◦
T Y (Y, y)
(†)
⩽ b·mY (Y, y) = b(mY (Y), y), where (†) uses the fact that mY ·eT Y = 1T Y implies e◦
T Y ⩽ mY .
If y = TiX(x) and y = w, then ˆTb(Y, y) ⊗ b(y, y) = ˆTb(Y, TiX(x)) = (TiX)◦
· ˆTb(Y, x) = ˆT(i◦
X · b)(Y, x).
Since for every z ∈ TY and every x ∈ X,
i◦
X · b(z, x) = b(z, iX(x)) =
⊤V , z = eY · iX(x)
⊥V , otherwise
= (eY · iX)◦
(z, x),
it follows that ˆT(i◦
X · b) = ˆT(eY · iX)◦
= (TeY · TiX)◦
, since ˆT is flat. Moreover, ⊥V < (TeY · TiX)◦
(Y, x)
implies TeY · TiX(x) = Y. As a result, b(mY (Y), y) = b(mY · TeY · TiX(x), w) = b(TiX(x), w) = ⊤V . □
Remark 74. The (T, V )-category (Y, b) constructed in Remark 69 is called the test structure for x.
Theorem 75 (Generalized Kuratowski-Mrówka theorem). Let ˆT be flat and let 1V -Rel
e
−→ ˆT be finitely
(−)◦
-strict. Given a (T, V )-category (X, a), the following are equivalent:
(1) (X, a) is compact;
(2) for every (T, V )-category (Z, c), the projection (X, a) × (Z, c)
πZ
−−→ (Z, c) is closed.
Proof.
16
(1) ⇒ (2): Since (X, a) is compact, then (X, a)
!X
−→ (1, ⊤) is proper, and therefore, its pullback along
every (T, V )-functor is proper by Theorem 37. In particular, the pullback
(X, a) × (Z, c)
πX

πZ
// (Z, c)
!Z

(X, a)
!X
// (1, ⊤)
provides the proper map (X, a) × (Z, c)
πZ
−−→ (Z, c), which is then necessarily closed by Theorem 58.
(2) ⇒ (1): One has to show that the diagram
TX
T !X
//
•a

T1
• ⊤

X
!X
// 1
commutes. Given x ∈ TX, there exists the respective test structure (Y, b), constructed in Theorem 73.
Moreover, one has the following diagram
TX 1a
// X
1 1
!◦
T X
// TX
T iX
++
T ⟨1X ,iX ⟩
//
T 1X
33
T ⟨1X ,iX ⟩
// T(X × Y )
⩾
⩽T πX
OO
T πY

1c
// X × Y
πX
OO
πY

TY
1
b
// Y,
where the triangles are commutative, whereas the rectangles are lax commutative. It follows then that
⊤ · T!X(x, ∗) = ⊤V = b(TiX(x), w) ⩽
z∈T X
b(TiX(z), w) = b · TiX·!◦
T X(∗, w) =
b · TπY · T⟨1X, iX⟩·!◦
T X(∗, w)
(†)
= πY · c · T⟨1X, iX⟩·!◦
T X(∗, w) =
x∈X z∈T X
c(T⟨1X, iX⟩(z), (x, w))
(††)
=
x∈X z∈T X
((π◦
X · a · TπX) ∧ (π◦
Y · b · TπY ))(T⟨1X, iX⟩(z), (x, w)) =
x∈X z∈T X
a(TπX · T⟨1X, iX⟩(z), πX(x, w)) ∧ b(TπY · T⟨1X, iX⟩(z), πY (x, w)) =
x∈X z∈T X
a(z, x) ∧ b(TiX(z), w)
(†††)
=
x∈X
a(x, x) ∧ b(TiX(x), w) =
x∈X
a(x, x) =!X · a(x, ∗),
where (†) uses the assumption on closedness, (††) relies on the construction of pullbacks in the category
(T, V )-Cat given in Remark 12 (2), whereas († † †) uses the fact that if TiX(z) = eY (w) for some z ∈ TX,
then, since X   iX
// Y has finite fibres, and, moreover, ˆT is flat, the diagram
TX 1e◦
X
//
T iX

X
iX

TY 1
e◦
Y
// Y
17
commutes, which gives ⊤V = e◦
Y · TiX(z, w) = iX · e◦
X(z, w), and therefore, there exists x ∈ X such that
iX(x) = w, which is a contradiction. □
5. Generalized Bourbaki theorem
Theorem 76 (Generalized Bourbaki theorem). Let T1 ∼= 1, let ˆT be flat, and let 1V -Rel
e
−→ ˆT be
finitely (−)◦
-strict. Given a (T, V )-functor (X, a)
f
−→ (Y, b), the following are equivalent:
(1) f is proper;
(2) every pullback of f is closed, and (TX, ˆa)
T f
−−→ (TY,ˆb) is closed;
(3) all fibres of f are compact, and (TX, ˆa)
T f
−−→ (TY,ˆb) is closed.
Proof.
(1) ⇒ (2): Follows from Theorems 39, 37 and 58.
(2) ⇒ (3): Follows from Theorem 75 and the assumption T1 ∼= 1 (and therefore, 1♯
= ⊤), through the
composition of the pullbacks
(f−1
(y), ˜a) × (Z, c)
πf−1(y)

πZ
// (Z, c)
!Z

(f−1
(y), ˜a)
_
if−1(y)

!f−1(y)
// (1, ⊤)
y

(X, a)
f
// (Y, b)
for every (T, V )-category (Z, c).
(3) ⇒ (1): Follows from Corollaries 49, 61. □
Remark 77.
(1) Without the assumption T1 ∼= 1, stably closed maps need not be proper.
(2) It is unclear, whether the condition “(TX, ˆa)
T f
−−→ (TY,ˆb) is closed” can be removed from Theorem 76 (2),
and also, whether it can be replaced by the condition “(X, a)
f
−→ (Y, b) is closed” in Theorem 76 (3).
Example 78.
(1) By Corollary 50, every object of the category Prost (resp. QPMet) is compact. By Corollary 61,
proper and closed maps in the category Prost (resp. QPMet) are equivalent concepts.
(2) In Top, one gets the above-mentioned Kuratowski-Mrówka and Bourbaki theorems.
(3) In App, one gets the results from the theory of approach spaces of [3].
(4) In case of the powerset functor P on Set, it follows that P1 ̸∼= 1, 1Rel
e
−→ ˆP is not finitely (−)◦
-strict
(Example 71), and ˆP is not flat (Lecture 2). Thus, Theorems 75, 76 are not applicable to the category
Cls. Corollary 61 though shows that the concepts of proper and closed map in Cls are equivalent.
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