Elements of monoidal topology⋆
Lecture 1: (T, V )-categories and (T, V )-functors
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 introduces monoidal topology in the form of (T, V )-categories and (T, V )-functors for a monad T
on Set and a unital quantale V , and shows that the examples of these structures include preordered sets, as
well as quasi-pseudo-metric, topological, approach, and closure spaces, together with their respective maps.
1. Categorical preliminaries
1.1. Monads and their algebras
Definition 1. A category X is a sextuple (O, M, dom, cod, 1, ·), where O is a class of objects (denoted
X, Y, Z, etc.), M is a class of morphisms (denoted f, g, h, etc.), M
dom //
cod
// O (domain and codomain) and
O
1
−→ M (identity morphisms denoted 1X, 1Y , 1Z, etc.) are maps, and · is a partial binary operation on M
(composition) such that g ·f is defined iff cod f = dom g. Given X, Y ∈ O and f ∈ M, one uses the notation
X
f
−→ Y as a shorthand for “dom f = X and cod f = Y ”. Additionally, one assumes the following axioms:
(1) for every object X, X
1X
−−→ X;
(2) for every objects X and Y , the family X(X, Y ) := {f ∈ M | X
f
−→ Y } is a set (hom-set);
(3) h · (g · f) = (h · g) · f for every morphisms X
f
−→ Y , Y
g
−→ Z and Z
h
−→ W;
(4) 1Y · f = f = f · 1X for every morphism X
f
−→ Y .
Example 2. There exist the categories Set of sets and maps, Top of topological spaces and continuous
maps, Met of metric spaces and non-expansive maps, Pos of partially ordered sets and monotone maps.
Definition 3. For every category X = (O, M, dom, cod, 1, ·), there exists the opposite (or dual) category
of X, namely, the category Xop
= (O, M, cod, dom, 1, ∗), in which ∗ is defined by f ∗ g = g · f. In other
words, X and Xop
have the same objects and morphisms, whereas the domain and the codomain maps are
switched, and the composition laws are the “opposites” of each other.
⋆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
Proposition 4. For every category X, (Xop
)
op
= X.
Definition 5. A functor F from a category X to a category Y is a pair of maps OX
FO
−−→ OY and MX
FM
−−→
MY (both denoted F), which satisfy the following axioms:
(1) F(X
f
−→ Y ) = FX
F f
−−→ FY for every X-morphism X
f
−→ Y ;
(2) F(g · f) = Fg · Ff for every X-morphisms X
f
−→ Y and Y
g
−→ Z;
(3) F1X = 1F X for every X-object X.
Example 6.
(1) Given a category X, there exists the identity functor 1X on X defined by 1X(X
f
−→ Y ) = X
f
−→ Y for
every X-morphism X
f
−→ Y .
(2) There exist the forgetful functors Top
|−|
−−→ Set, Met
|−|
−−→ Set, and Pos
|−|
−−→ Set, as well as the
powerset functor Set
P
−→ Set defined by P(X
f
−→ Y ) = PX
P f
−−→ PY , where PX = {A | A ⊆ X} and
Pf(A) = f(A) = {f(x) | x ∈ A} for every map X
f
−→ Y .
Remark 7. Functors can be composed componentwise as pairs of maps. The composition of two functors
X
F
−→ Y and Y
G
−→ Z is often written GF instead of G · F.
Definition 8. A natural transformation α from a functor X
F
−→ Y to a functor X
G
−→ Y is a map OX
α
−→ MY
such that FX
αX
−−→ GX for every X ∈ OX, and, additionally, the diagram
FX
F f

αX
// GX
Gf

FY αY
// GY
commutes for every X-morphism X
f
−→ Y .
Example 9. There exists a natural transformation 1Set
e
−→ P, with X
eX
−−→ PX given by eX(x) = {x}.
Remark 10. Given functors and a natural transformation as in the diagram W
K // X
F
&&
G
88 α Y
H // Z,
there exist the following natural transformations (whiskering by a functor from the left or right):
(1) HF
Hα
−−→ HG given by HFX
(Hα)X
−−−−→ HGX := HFX
H(αX )
−−−−→ HGX;
(2) FK
αK
−−→ GK given by FKW
(αK)W
−−−−−→ GKW := FKW
αKW
−−−→ GKW.
Definition 11. A monad T on a category X is a triple (T, m, e), where X
T
−→ X is a functor, and TT
m
−→ T,
1X
e
−→ T are natural transformations, which make the diagrams
TTT
mT

T m // TT
m

TT m
// T
T
1T
!!
eT // TT
m

T
T eoo
1T
}}
T
commute.
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Example 12. There exists the powerset monad P = (P, m, e) on Set, with X
eX
−−→ PX given by eX(x) = {x}
and PPX
mX
−−→ PX given by mX(A) = A.
Remark 13. Monads on a category X are precisely the monoids in the strict monoidal category XX
(see
Lecture 4 for more detail on monoidal categories).
Definition 14. Given a monad T on a category X, a T-algebra (or Eilenberg-Moore algebra) is a pair (X, a),
where X is an X-object, and TX
a
−→ X is an X-morphism, which makes the diagrams
TTX
T a

mX
// TX
a

TX a
// X
X
1X
!!
eX
// TX
a

X
commute. A T-homomorphism (X, a)
f
−→ (Y, b) is an X-morphism X
f
−→ Y , which makes the diagram
TX
a

T f
// TY
b

X
f
// Y
commute. XT
is the category of T-algebras and T-homomorphisms (the Eilenberg-Moore category of T).
Example 15. SetP
is isomorphic to the category Sup of -semilattices and -preserving maps.
1.2. Quantale-valued relations
Definition 16. A -semilattice is a partially ordered set (V, ⩽), which has arbitrary joins (denoted ).
Remark 17. Every -semilattice (V, ⩽) is a complete lattice, in which ⊥V := ∅ (the smallest element)
and ⊤V := ∅ (the largest element).
Definition 18. A quantale V is a -semilattice, which is equipped with an associative binary operation
V × V
⊗
−→ V (multiplication) such that a ⊗ ( B) = b∈B a ⊗ b and ( B) ⊗ a = b∈B(b ⊗ a) for every
a ∈ V and every B ⊆ V . A quantale V is said to be
(1) unital provided that its multiplication has a unit k;
(2) commutative provided that a ⊗ b = b ⊗ a for every a, b ∈ V .
Example 19. There exists the two-element unital quantale 2 = ({⊥, ⊤}, ∧, ⊤). The extended real half-line
[0, ∞] gives a unital quantale P+ = ([0, ∞]op
, +, 0).
Remark 20. Every unital quantale is a strict monoidal closed category (see Lecture 4 for more detail on
monoidal categories).
Remark 21. Given a set X, there is a one-to-one correspondence between subsets of X and maps X −→ 2.
For a subset S ⊆ X, one defines X
χS
−−→ 2 (characteristic map of S) by χS(x) = ⊤ iff x ∈ S, and vice versa.
Definition 22. A relation r from a set X to a set Y is a map X × Y
r
−→ 2 (denoted X
r
// Y ). Given
x ∈ X and y ∈ Y , one uses x r y as a shorthand for “r(x, y) = ⊤”. The opposite (or dual) of a relation
X r
// Y is the relation Y r◦
// X defined by y r◦
x iff x r y.
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Definition 23. Rel is the category, whose objects are sets, and whose morphisms are relations X
r
// Y.
Composition of relations X r
// Y and Y s
// Z is defined by x (s·r) z iff there exists y ∈ Y such that
x r y and y s z. Given a set X, the identity 1X is the diagonal {(x, x) | x ∈ X}.
Remark 24. Rel is an involutive quantaloid (hom-sets are -semilattices w.r.t. the inclusion order, and
composition preserves in both variables; cf. Lecture 4). Additionally, Rel is isomorphic to the Kleisli
category of the powerset monad P on Set (see Lecture 7 for more detail on the Kleisli category of a monad).
Definition 25. Given a unital quantale V , a V -relation r from a set X to a set Y is a map X × Y
r
−→ V
(denoted X r
// Y ). The opposite (or dual) of a V -relation X r
// Y is the V -relation Y r◦
// X
defined by r◦
(y, x) = r(x, y).
Definition 26. Given a unital quantale V , V -Rel is the category, whose objects are sets, and whose
morphisms are V -relations X
r
// Y. Composition of V -relations X
r
// Y and Y
s
// Z is defined
by (s · r)(x, z) = y∈Y r(x, y) ⊗ s(y, z). Given a set X, the identity 1X is defined by
1X(x, y) =
k, x = y
⊥V , otherwise.
Remark 27. 2-Rel is isomorphic to Rel. V -Rel is a quantaloid (with hom-set given by pointwise
evaluation), which is involutive (w.r.t. the dual relation operation (−)◦
) iff V is commutative (one should
observe that given V -relations X r
// Y and Y s
// Z, it follows that (s·r)◦
= r◦
·s◦
iff y∈Y r(x, y)⊗
s(y, z) = (s · r)(x, z) = (s · r)◦
(z, x) = (r◦
· s◦
)(z, x) = y∈Y s◦
(z, y) ⊗ r◦
(y, x) = y∈Y s(y, z) ⊗ r(x, y) for
every pair (z, x) ∈ Z × X; cf. Lecture 4). Moreover, considering V as a quantaloid with one object 1 (also
thought of as a singleton set 1 = {∗}), we get a full quantaloid embedding V   E // V -Rel, which is given
by E(1
a
−→ 1) = 1 a
// 1, where 1 × 1
a
−→ V is the map with value a. Additionally, V -Rel is isomorphic
to the Kleisli category w.r.t. the V -powerset monad PV on Set (an extension of the powerset monad P),
whose Eilenberg-Moore category is the category V -Mod of left unital V -modules.
Remark 28. To avoid trivial cases, suppose that V has at least two elements (k ̸= ⊥V ). Then there exists
a non-full embedding Set   (−)◦
// V -Rel, which takes a map X
f
−→ Y to a relation X f◦
// Y given by
f◦(x, y) =
k, f(x) = y
⊥V , otherwise.
For the sake of simplicity, one identifies a map X
f
−→ Y and its respective relation X f◦
// Y, employing
the notation f for both. It is easy to see that 1X ⩽ f◦
· f and f · f◦
⩽ 1Y .
1.3. Lax extension of monads
Definition 29. Given a unital quantale V and a functor Set
T
−→ Set, a lax extension V -Rel
ˆT
−→ V -Rel of
T to V -Rel is a pair of maps OV -Rel
ˆTO
−−→ OV -Rel, MV -Rel
ˆTM
−−→ MV -Rel (both denoted ˆT), which satisfy
the following axioms:
(1) ˆT(X r
// Y ) = TX ˆT r
// TY for every V -relation X r
// Y ;
4
(2) ˆTr ⩽ ˆTs for every V -relations X
r
//
s
// Y such that r ⩽ s;
(3) ˆTs · ˆTr ⩽ ˆT(s · r) for every V -relations X r
// Y and Y s
// Z;
(4) Tf ⩽ ˆTf and (Tf)◦
⩽ ˆT(f◦
) for every map X
f
−→ Y .
Example 30. The identity functor on V -Rel is a lax extension of the identity functor on Set. The powerset
functor Set
P
−→ Set has lax extensions Rel
ˇP //
ˆP
// Rel, where, given a relation X r
// Y,
(1) A ˇPr B iff for every x ∈ A there exists y ∈ B such that x r y;
(2) A ˆPr B iff for every y ∈ B there exists x ∈ A such that x r y.
Every functor T on Set has the largest lax extension ˆT⊤
to V -Rel, where, given a V -relation X
r
// Y,
ˆT⊤
r(x, y) = ⊤V for every x ∈ TX and every y ∈ TY .
Definition 31. Given a unital quantale V and a monad T on Set, a lax extension ˆT of T to V -Rel is a
triple ( ˆT, m, e), where ˆT is a lax extension of T to V -Rel, and ˆT ˆT
m
−→ ˆT, 1V -Rel
e
−→ ˆT are oplax natural
transformations, which means
TTX
mX
//
_ˆT ˆT r

⩽
TX
_ ˆT r

TTY mY
// TY
and X
eX
//
_r

⩽
TX
_ ˆT r

Y eY
// TY
for every V -relation X r
// Y.
Example 32. The identity monad I on V -Rel is a lax extension of the identity monad I on Set. The lax
extensions ˇP and ˆP of the powerset functor P provide lax extensions of the powerset monad P on Set to
Rel. Every monad T on Set has the largest lax extension T⊤
to V -Rel, which is given by ˆT⊤
.
2. (T, V )-categories and (T, V )-functors, and their examples
2.1. (T, V )-categories and (T, V )-functors
Definition 33. Suppose V is a unital quantale, and ˆT is a lax extension of a monad T on Set to V -Rel. A
(T, V )-category (or (T, V )-algebra, or (T, V )-space, or lax algebra) is a pair (X, a), which comprises a set X
and a V -relation TX
a
// X such that
TTX
mX
//
_ˆT a

⩽
TX
_ a

TX

a
// X
(transitivity) and X
1X
⩽
!!
eX
// TX
_ a

X
(reflexivity).
A (T, V )-functor (or lax homomorphism) (X, a)
f
−→ (Y, b) is a map X
f
−→ Y such that
TX
T f
//
_a

⩽
TY
_ b

X
f
// Y.
(T, V )-Cat is the category of (T, V )-categories and (T, V )-functors.
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Definition 34. The category (I, V )-Cat is denoted V -Cat, whose objects (resp. morphisms) are called
V -categories (resp. V -functors).
Remark 35. There is an analogy between T-algebras and T-homomorphisms w.r.t. a monad T on Set, on
one hand, and (T, V )-categories and (T, V )-functors w.r.t. its lax extension ˆT to V -Rel, on the other hand.
Definition 36. Monoidal topology is a branch of categorical topology, which studies the properties of the
categories of the form (T, V )-Cat.
2.2. Examples of the categories (T, V )-Cat
2.2.1. Preordered sets and quasi-pseudo-metric spaces as V -categories
Remark 37. A V -category (X, a) consists of a set X and a V -relation X a
// X such that
(1)
X
1X
⩽
1X
// X
_ a

X,
which is equivalent to 1X ⩽ a, which is equivalent to k ⩽ a(x, x) for every x ∈ X;
(2)
X
1X
//
_a

⩽
X
_ a

X

a
// X,
which is equivalent to a·a ⩽ a, which is equivalent to y∈X a(x, y)⊗a(y, z) ⩽ a(x, z) for every x, z ∈ X,
which is equivalent to a(x, y) ⊗ a(y, z) ⩽ a(x, z) for every x, y, z ∈ X.
A V -functor (X, a)
f
−→ (Y, b) has the property that
X
f
//
_a

⩽
Y
_ b

X
f
// Y,
which is equivalent to f · a ⩽ b · f, which is equivalent to f(z)=y a(x, z) ⩽ b(f(x), y) for every x ∈ X and
every y ∈ Y , which is equivalent to a(x, z) ⩽ b(f(x), f(z)) for every x, z ∈ X.
Example 38. A 2-category is a pair (X, ⩽) such that x ⩽ x for every x ∈ X; and x ⩽ y, y ⩽ z imply x ⩽ z
for every x, y, z ∈ X. A 2-functor (X, ⩽)
f
−→ (Y, ⩽) is a map X
f
−→ Y such that x, z ∈ X and x ⩽ z imply
f(x) ⩽ f(z). As a result, 2-Cat is the category Prost of preordered sets and monotone maps.
Example 39. A P+-category is a pair (X, ρ) such that ρ(x, x) = 0 for every x ∈ X; and ρ(x, z) ⩽ ρ(x, y) +
ρ(y, z) for every x, y, z ∈ X. A P+-functor (X, ρ)
f
−→ (Y, ϱ) is a map X
f
−→ Y such that ϱ(f(x), f(z)) ⩽ ρ(x, z)
for every x, z ∈ X. As a result, 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.
6
2.2.2. Topological spaces as (T, V )-categories
Remark 40. It is well-known that the Eilenberg-Moore category of the ultrafilter monad on Set is the
category of compact Hausdorff topological spaces. One cannot extend this results to the whole category
Top, since the latter category is not of algebraic nature (e.g., bijective morphisms in Top are not necessarily
homeomorphisms). To get the whole category Top, one employs a lax extension of the ultrafilter monad.
Definition 41. Given a set X, a filter on X is a family x of subsets of X such that
(1) X ∈ x;
(2) A ∈ x and A ⊆ B imply B ∈ x;
(3) A, B ∈ x implies A B ∈ x.
A filter x is called proper provided that ∅ ̸∈ x. An ultrafilter x on a set X is a maximal element in the set
of proper filters on X, ordered by inclusion.
Example 42. Given a set X, every x ∈ X provides the principal ultrafilter ˙x = {A ⊆ X | x ∈ A} on X.
Remark 43. A proper filter x on X is an ultrafilter on X iff for every A ⊆ X, either A ∈ x or X\A ∈ x.
Definition 44. The ultrafilter monad β = (β, m, e) on Set is given by
(1) a functor Set
β
−→ Set, where βX = {x | x is an ultrafilter on X} for every set X, and βX
βf
−−→ βY is
defined by βf(x) = {B ⊆ Y | f−1
(B) ∈ x} for every map X
f
−→ Y ;
(2) a natural transformation 1Set
e
−→ β, where X
eX
−−→ βX is defined by eX(x) = ˙x;
(3) a natural transformation ββ
m
−→ β, where ββX
mX
−−→ βX is defined by mX(X) = Σ X (filtered sum or
Kowalsky sum), where A ∈ Σ X iff {x ∈ βX | A ∈ x} ∈ X.
Theorem 45. Given a relation X
r
// Y, define x ˆβr y iff for every A ∈ x and every B ∈ y, there exist
x ∈ A and y ∈ B such that x r y. Then ˆβ = (ˆβ, m, e) is a lax extension to Rel of the ultrafilter monad β, in
which, additionally, Rel
ˆβ
−→ Rel is a functor, and ˆβ ˆβ
m
−→ ˆβ is a natural transformation.
Remark 46. Every (β, 2)-category (X, a) has the following two properties:
(1) 1X ⩽ a · eX, which is equivalent to ˙x a x for every x ∈ X;
(2) a · ˆβa ⩽ a · mX, which is equivalent to X (ˆβa) x and x a x imply (Σ X) a x for every X ∈ ββX, every
x ∈ βX, and every x ∈ X.
Every (β, 2)-functor (X, a)
f
−→ (Y, b) satisfies the condition f · a ⩽ b · βf, which is equivalent to x a x
implies βf(x) b f(x) for every x ∈ βX and every x ∈ X.
Definition 47. Given a set X, a closure operation on X is a monotone map PX
c
−→ PX (w.r.t. the inclusion
order) such that 1P X ⩽ c and c·c ⩽ c (pointwise evaluation as maps). A closure operation c on X is finitely
additive provided that c( i∈I Ai) = i∈I c(Ai) for every finite family {Ai | i ∈ I} ⊆ PX (equivalently,
c(∅) = ∅ and c(A B) = c(A) c(B) for every A, B ∈ PX).
Proposition 48. A closure operation c on a set X is finitely additive iff the family τ = {X\A | c(A) = A}
is a topology on X, i.e., there exists a one-to-one correspondence between finitely additive closure operations
on X and topologies on X.
Proof. As an illustration, one could verify that τ is closed under finite intersections provided that c is
finitely additive. Given a finite family {X\Ai | i ∈ I} ⊆ τ, it follows that i∈I(X\Ai) = X\( i∈I Ai) =
X\( i∈I c(Ai))
(†)
= X\c( i∈I Ai)
(††)
∈ τ, where (†) relies on finite additivity of c, and (††) follows from the
property c · c = c of every closure operation c on X (observe that 1P X ⩽ c implies c ⩽ c · c).
7
Observe that τ is closed under arbitrary unions for every closure operation c on X, since given a family
{X\Ai | i ∈ I} ⊆ τ, it follows that i∈I(X\Ai) = X\( i∈I Ai) = X\( i∈I c(Ai))
(†)
= X\c( i∈I Ai) ∈ τ,
where (†) relies on the fact that i∈I c(Ai) = c( i∈I Ai), since, on the one hand, i∈I Ai ⊆ Aj for every
j ∈ I implies c( i∈I Ai) ⊆ c(Aj) for every j ∈ I implies c( i∈I Ai) ⊆ i∈I c(Ai); and, on the other hand,
i∈I c(Ai)
(††)
= i∈I Ai ⊆ c( i∈I Ai), where (††) relies on the fact that c(Ai) = Ai for every i ∈ I. □
Definition 49. Given a topological space (X, τ), a filter x on X converges to an element x ∈ X provided
that U ∈ x for every U ∈ τ such that x ∈ U. If x converges to x, then x is called a limit of x. The set of
limits of a filter x is denoted lim x.
Proposition 50. Given a finitely additive closure operation c on a set X, the following hold:
(1) for every A ⊆ X and every x ∈ X, x ∈ c(A) iff there exists x ∈ βX such that A ∈ x and x ∈ lim x;
(2) for every x ∈ βX and every x ∈ X, x ∈ lim x iff x ∈ c(A) for every A ∈ x.
Proof. As an illustration, one can show “⇒” of (2). Given A ∈ x, for every U ∈ τ (cf. Proposition 48)
such that x ∈ U, it follows that U ∈ x (since x ∈ lim x) and, therefore, U A ∈ x (since x is a filter), which
implies U A ̸= ∅ (since x is an ultrafilter). Thus, x ∈ c(A). In a similar way, one can show “⇐” of (1). □
Theorem 51. The category (β, 2)-Cat is isomorphic to the category Top.
Proof. The isomorphism between (β, 2)-categories and topological spaces is based in the idea that given a
set X, a (β, 2)-category structure βX a
// X on X represents a convergence relation between ultrafilters
on X and elements of X (i.e., a specifies which ultrafilter converges to which element). One then associates
with a a finitely additive closure operation c on X, and also shows that every finitely additive closure
operation c on X determines a convergence relation βX a
// X, which is a (β, 2)-category structure on X.
Given a (β, 2)-category (X, a), one defines a closure operation PX
clos(a)
−−−−→ PX on X by (clos(a))(A) =
{x ∈ X | there exists x ∈ βX such that A ∈ x and x a x} (cf. Proposition 50 (1)). Given a finitely additive
closure operation c on X, one defines a (β, 2)-category structure βX conv(c)
// X on X by x conv(c) x iff
x ∈ c(A) for every A ∈ x (cf. Proposition 50 (2)).
To show that, e.g., 1P X ⩽ clos(a), notice that given A ⊆ X, for every x ∈ A, it follows that A ∈ ˙x and
˙x a x, i.e., x ∈ (clos(a))(A), which implies A ⊆ (clos(a))(A). To show that, e.g., 1X ⩽ conv(c) · eX, notice
that given x ∈ X, it follows that ˙x conv(c) x, since given A ⊆ X, A ∈ ˙x implies x∈A⊆c(A), i.e., x∈c(A). □
2.2.3. Approach spaces as (T, V )-categories
Definition 52. An approach space is a pair (X, δ), where X is a set, and X × PX
δ
−→ [0, ∞] is a map
(approach distance) such that
(1) δ(x, {x}) = 0 for every x ∈ X;
(2) δ(x, ∅) = ∞ for every x ∈ X;
(3) δ(x, A B) = min{δ(x, A), δ(x, B)} for every x ∈ X and every A, B ⊆ X;
(4) δ(x, A) ⩽ δ(x, A(u)
) + u, where A(u)
= {y ∈ X | δ(y, A) ⩽ u} for every x ∈ X, A ⊆ X, u ∈ [0, ∞].
A morphism (X, δ)
f
−→ (Y, σ) of approach spaces is a non-expansive map X
f
−→ Y , i.e., σ(f(x), f(A)) ⩽ δ(x, A)
for every x ∈ X and every A ⊆ X. App is the category of approach spaces and non-expansive maps.
Remark 53. Approach spaces provide a unifying framework for topological, metric, and uniform spaces.
Remark 54. Every topological space (X, τ) gives an approach space (X, δ), in which
δ(x, A) =
0, x ∈ cl(A) (the closure of the set A w.r.t. τ)
∞, otherwise
for every x ∈ X and every A ∈ PX. One gets thus a full embedding Top → App.
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Remark 55. Every quasi-pseudo-metric space (X, ρ) gives an approach space (X, δ), in which δ(x, A) =
inf{ρ(y, x) | y ∈ A} for every x ∈ X and every A ∈ PX. One gets thus a full embedding QPMet → App.
Theorem 56. Given a P+-relation X r
// Y, define a map βX × βY
¯βr
−→ P+ by
¯βr(x, y) =
A∈x,B∈y x∈A,y∈B
r(x, y).
Then ¯β = (¯β, m, e) is a lax extension to P+-Rel of the ultrafilter monad β, in which, additionally, V -Rel
¯β
−→
V -Rel is a functor, and ¯β ¯β
m
−→ ¯β is a natural transformation.
Remark 57. Every (β, P+)-category (X, a) has the following two properties:
(1) 1X ⩽ a · eX, which is equivalent to a( ˙x, x) = 0 for every x ∈ X;
(2) a · ¯βa ⩽ a · mX, which is equivalent to a(Σ X, x) ⩽ ¯βa(X, x) + a(x, x) for every X ∈ ββX, x ∈ βX, x ∈ X.
Every (β, P+)-functor (X, a)
f
−→ (Y, b) satisfies the condition f · a ⩽ b · βf, which is equivalent to
b(βf(x), f(x)) ⩽ a(x, x) for every x ∈ βX and every x ∈ X.
Theorem 58. The category (β, P+)-Cat is isomorphic to the category App.
Proof. Following the analogy of Theorem 51, given a (β, P+)-category (X, a), one defines an approach
distance X × PX
clos(a)
−−−−→ [0, ∞] by (clos(a))(x, A) = inf{a(x, x) | x ∈ βA}. Given an approach space (X, δ),
one defines a (β, P+)-category structure βX conv(δ)
// X by (conv(δ))(x, x) = sup{δ(x, A) | A ∈ x}. □
Remark 59. Theorem 58 actually says that approach spaces provide “numerified topological spaces”, since
a classical convergence relation is replaced with a numerified “degree of convergence”.
2.2.4. Closure spaces as (T, V )-categories
Definition 60. A closure space is a pair (X, c), where X is a set, and PX
c
−→ PX is a closure operation on
X. A map (X, c)
f
−→ (Y, d) between closure spaces is continuous provided that f(c(A)) ⊆ d(f(A)) for every
A ⊆ X. Cls is the category of closure spaces and continuous maps.
Theorem 61. The lax extension ˆP of the powerset monad P provides the category (P, 2)-Cat, which is
isomorphic to the category Cls.
Proof. Every (P, 2)-category (X, a) has the following two properties:
(1) 1X ⩽ a · eX, which is equivalent to {x} a x for every x ∈ X;
(2) a · ˆPa ⩽ a · mX, which is equivalent to A ( ˆPa) B (i.e., for every y ∈ B, there exists A ∈ A such that
A a y) and B a x imply ( A) a x for every A ∈ PPX, every B ∈ PX, and every x ∈ X.
Every (P, 2)-functor (X, a)
f
−→ (Y, b) satisfies the condition f · a ⩽ b · Pf, which is equivalent to A a x
implies Pf(A) b f(x) (i.e., f(A) b f(x)) for every A ∈ PX and every x ∈ X.
Given a set X, there exists a bijective correspondence between (P, 2)-category structures PX a
// X
and closure operations PX
c
−→ PX, which is given by A a x iff x ∈ c(A) for every A ∈ PX, x ∈ X.
Additionally, a (P, 2)-functor (X, a)
f
−→ (Y, b) provides a continuous map (X, c)
f
−→ (Y, d) and vice versa.
To show that, e.g., 1P X ⩽ c (for a given (P, 2)-category structure PX a
// X on X), observe that for
every A ∈ PX, x ∈ A implies {x} a x (by item (1) above) implies {{y} | y ∈ A} ˆPa {x} and {x} a x implies
( {{y} | y ∈ A}) a x (by item (2) above) implies A a x implies x ∈ c(A), which results in A ⊆ c(A).
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To show that, e.g., 1X ⩽ a · eX (for a given closure operation PX
c
−→ PX on X), observe that for every
x ∈ X, x ∈ {x} ⊆ c{x} implies x ∈ c{x} implies {x} a x, which is exactly the condition of item (1) above.
To verify that a (P, 2)-functor (X, a)
f
−→ (Y, b) provides a continuous map (X, c)
f
−→ (Y, d), observe that for
every A ∈ PX, x ∈ c(A) implies A a x implies f(A) b f(x) (since f is a (P, 2)-functor) implies f(x) ∈ d(f(A)).
As a consequence, one obtains that f(c(A)) ⊆ d(f(A)).
To check that a continuous map (X, c)
f
−→ (Y, d) provides a (P, 2)-functor (X, a)
f
−→ (Y, b), observe that
for every A ∈ PX and every x ∈ X, A a x implies x ∈ c(A) implies f(x) ∈ d(f(A)) (since f is continuous)
implies f(A) b f(x), which is exactly the above-mentioned condition of (P, 2)-functors. □
Definition 62. There exists the finite-powerset monad Pfin = (Pfin, m, e) on Set, in which the functor
Set
Pfin
−−→ Set is given on objects by PfinX = {A ⊆ X | A is finite}. The natural transformations m and e
are the restrictions of the respective natural transformations of the powerset monad P on Set.
Definition 63. A closure space (X, c) is called finitary (or algebraic) provided that c(A) = B∈PfinA c(B)
for every A ∈ PX. Clsfin is the full subcategory of Cls of finitary closure spaces.
Theorem 64. The lax extension ˆPfin of the finite-powerset monad Pfin provides the category (Pfin, 2)-Cat,
which is isomorphic to the category Clsfin.
Proof. One uses the following modification of the proof of Theorem 61.
Given a set X, there exists a bijective correspondence between (Pfin, 2)-category structures PfinX
a
//X
and closure operations PfinX
cfin
−−→ PfinX, which is given by A a x iff x ∈ cfin(A) for every A ∈ PfinX, x ∈ X.
Moreover, a (Pfin, 2)-functor (X, a)
f
−→ (Y, b) provides a continuous map (X, cfin)
f
−→ (Y, dfin) and vice versa.
Given a set X, there also exists a bijective correspondence between closure operations PfinX
cfin
−−→ PfinX
and algebraic closure operations PX
c
−→ PX, which is given in one direction by x ∈ c(A) iff there exists
B ∈ PfinA such that x ∈ cfin(B) for every A ∈ PX, x ∈ X, and the opposite direction is the restriction of c to
PfinX. This correspondence respects continuity of maps, e.g., given a continuous map (X, cfin)
f
−→ (Y, dfin),
for every x ∈ X and every A ∈ PX, x ∈ c(A) implies the existence of B ∈ PfinA such that x ∈ cfin(B)
implies f(x) ∈ dfin(f(B)) (since f is continuous) implies f(x) ∈ d(f(B)) (since f(B) ∈ Pfin(f(A))). □
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