Elements of monoidal topology
Lecture 4: properties of the category V -Cat
Sergejs Solovjovs
Department of Mathematics, Faculty of Engineering, Czech University of Life Sciences Prague (CZU)
Kam´yck´a 129, 16500 Prague - Suchdol, Czech Republic
Abstract
This lecture shows that given a commutative unital quantale V , the category V -Cat, first, is symmetric
monoidal closed, confirming that the category Prost of preordered sets is cartesian closed, and the category
QPMet of quasi-pseudo-metric spaces is monoidal closed; and, second, has a functor taking a V -category
to its dual. This lecture also shows that every unital quantale V can be seen as a V -category, introduces the
category V -Mod of V -categories and V -modules, and describes its relationships with the category V -Cat.
1. Symmetric monoidal closed categories
1.1. Symmetric monoidal categories
Definition 1. A category C is called monoidal provided that it is equipped with a functor C × C
⊗
−→ C, a
distinguished C-object E, and natural isomorphisms A ⊗ (B ⊗ C)
αA,B,C
−−−−−→ (A ⊗ B) ⊗ C, E ⊗ A
λA
−−→ A, and
A ⊗ E
ρA
−−→ A for every C-objects A, B, C such that the following two diagrams commute
A ⊗ (B ⊗ (C ⊗ D))
1A⊗αB,C,D

αA,B,C⊗D
// (A ⊗ B) ⊗ (C ⊗ D)
αA⊗B,C,D
// ((A ⊗ B) ⊗ C) ⊗ D
A ⊗ ((B ⊗ C) ⊗ D) αA,B⊗C,D
// (A ⊗ (B ⊗ C)) ⊗ D
αA,B,C ⊗1D
OO
A ⊗ (E ⊗ B)
1A⊗λB
&&
αA,E,B
// (A ⊗ E) ⊗ B
ρA⊗1B
xx
A ⊗ B
for every C-objects A, B, C, D, and, moreover, E ⊗ E
λE
−−→ E = E ⊗ E
ρE
−−→ E.
Remark 2. One usually refers to the structure of a monoidal category C as follows: the functor ⊗ is the
tensor product, the object E is the unit, the natural isomorphism α is the associativity isomorphism, and
the natural isomorphism λ (resp. ρ) is the left unit (resp. right unit) isomorphism. Moreover, the two
commutative diagrams of Definition 1 are called the coherence conditions.
Remark 3. A monoidal category C is said to be strict provided that the natural isomorphisms α, λ, ρ in
Definition 1 are the identity natural isomorphisms, namely, A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C, E ⊗ A = A, and
A ⊗ E = A for every C-objects A, B, C. The coherence conditions are then trivially satisfied.
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) July 1, 2022
Definition 4. A monoidal category C is called symmetric provided that it is additionally equipped with
natural isomorphisms A⊗B
σA,B
−−−→ B⊗A for every C-objects A, B such that the following diagrams commute
A ⊗ (B ⊗ C)
αA,B,C
//
1A⊗σB,C

(A ⊗ B) ⊗ C
σA⊗B,C
// C ⊗ (A ⊗ B)
αC,A,B

A ⊗ (C ⊗ B) αA,C,B
// (A ⊗ C) ⊗ B
σA⊗C ⊗1B
// (C ⊗ A) ⊗ B
E ⊗ A
λA
""
σE,A
// A ⊗ E
ρA
||
A
A ⊗ B
σA,B
//
1A⊗B
%%
B ⊗ A
σB,A
yy
A ⊗ B
(1.1)
for every C-objects A, B, C. Since λE = ρE, the left-hand side of diagram (1.1) gives σE,E = 1E.
Remark 5. The commutative diagrams of Definition 4 are called the coherence conditions, i.e., associativity
coherence, unit coherence, and symmetry axiom, respectively. One should also observe that strict monoidal
categories generally fail to be symmetric.
Example 6.
(1) The category Set of sets and maps is symmetric monoidal w.r.t. cartesian product of sets. More
generally, every category with finite products is symmetric monoidal.
(2) The category of functors on a small category with the tensor product given by the composition of
functors is strict monoidal.
(3) The category Ab of abelian groups and group homomorphisms with the usual tensor product is symmetric
monoidal. More generally, given a commutative unital ring R, the category R-Mod of left R-modules
and left R-module homomorphisms is symmetric monoidal w.r.t. the tensor product of R-modules.
(4) The category Sup of -semilattices and -preserving maps equipped with the usual tensor product is
symmetric monoidal.
(5) Every unital quantale V is a strict monoidal category w.r.t. its multiplication. Commutative quantales
are additionally symmetric. More generally, a preordered set S, considered as a category, is strict
monoidal precisely when it is equipped with a monoid structure (⊗, k), whose multiplication S ×S
⊗
−→ S
is monotone. Commutative monoid structures provide additionally symmetric categories.
(6) Given a (symmetric) monoidal category C, Cop
is a (symmetric) monoidal category as well.
1.2. Monoidal functors
Definition 7. A morphism of monoidal categories C and D is a functor C
F
−→ D equipped with Dmorphisms
FA ⊗D FB
δA,B
−−−→ F(A ⊗C B) natural in C-objects A, B and a D-morphism ED
ε
−→ FEC
natural in ED, EC such that the following three diagrams commute
FA ⊗ (FB ⊗ FC)
αF A,F B,F C
//
1F A⊗δB,C

(FA ⊗ FB) ⊗ FC
δA,B⊗1F C

FA ⊗ F(B ⊗ C)
δA,B⊗C

F(A ⊗ B) ⊗ FC
δA⊗B,C

F(A ⊗ (B ⊗ C))
F αA,B,C
// F((A ⊗ B) ⊗ C)
2
E ⊗ FA
λF A
//
ε⊗1F A

FA
FE ⊗ FA
δE,A
// F(E ⊗ A)
F λA
OO FA ⊗ E
ρF A
//
1F A⊗ε

FA
FA ⊗ FE
δA,E
// F(A ⊗ E).
F ρA
OO
Notice that since δ is natural, the following diagram
FA ⊗ FB
F f⊗F g
//
δA,B

FC ⊗ FD
δC,D

F(A ⊗ B)
F (f⊗g)
// F(C ⊗ D)
commutes for every C-morphisms A
f
−→ C, B
g
−→ D.
Remark 8. The first three commutative diagrams of Definition 7 are called the coherence conditions, i.e.,
the associativity condition, and the two unit conditions, respectively.
Definition 9. A morphism of symmetric monoidal categories C and D is a functor C
F
−→ D, which is a
morphism of monoidal categories C and D, making additionally the following diagram commute
FA ⊗ FB
σF A,F B
//
δA,B

FB ⊗ FA
δB,A

F(A ⊗ B)
F σA,B
// F(B ⊗ A)
for every C-objects A, B.
Remark 10.
(1) The commutative diagram of Definition 9 is called the symmetry condition.
(2) Morphisms of (symmetric) monoidal categories are sometimes called monoidal functors.
Definition 11. A morphism of (symmetric) monoidal categories is called strong provided that the morphisms
δ and ε of Definition 7 are isomorphisms. It is called strict provided that δ and ε are identities. It
should be clear that strictness implies strongness, but not vice versa.
Example 12.
(1) The forgetful functor (Ab, ⊗, Z)
U
−→ (Set, ×, {∗}) is monoidal. Given two abelian groups A, B, one
defines UA × UB
δA,B
−−−→ U(A ⊗ B) by δA,B(a, b) = a ⊗ b, and {∗}
ε
−→ UZ by ε(∗) = 1. The monoidal
functor U is not strong.
(2) Every homomorphism of (commutative) unital quantales is a strict monoidal functor between (symmetric)
monoidal categories.
(3) Given two categories with finite products, every functor between them, which preserves finite products,
is strong monoidal.
3
1.3. Monoidal closed categories
Definition 13. An object A of a monoidal category C is called ⊗-exponentiable provided that the functors
C
A⊗(−)
−−−−→ C and C
(−)⊗A
−−−−→ C have right adjoints C
A −• (−)
−−−−−→ C and C
(−) •− A
−−−−−→ C, respectively.
Remark 14.
(1) The functors C
A −• (−)
−−−−−→ C and C
(−) •− A
−−−−−→ C of Definition 13 are called the right internal hom-functor
of A and the left internal hom-functor of A, respectively. One also sometimes uses the notation [A, −]
instead of (−) •− A, and [[A, −]] instead of A −• (−).
(2) The existence of the internal hom-functors implies, in particular, that every C-object B has an A⊗(−)co-universal
arrow A ⊗ (A −• B)
ev −• B
−−−−→ B and a (−) ⊗ A-co-universal arrow (B •− A) ⊗ A
evB •−
−−−−→ B,
i.e., for every C-morphisms A ⊗ C
f
−→ B and C ⊗ A
g
−→ B, there exist unique C-morphisms C
ˆf
−→ A −• B
and C
ˆg
−→ B •− A such that the following two diagrams commute
A ⊗ C
f
((
1A⊗ ˆf

A ⊗ (A −• B) ev −• B
// B
C ⊗ A
g
((
ˆg⊗1A

(B •− A) ⊗ A evB •−
// B.
(1.2)
(3) Since in a symmetric monoidal category C both functors A⊗(−) and (−)⊗A are naturally isomorphic,
their right adjoint functors A −• (−) and (−) •− A are naturally isomorphic as well, i.e., there is no need
to distinguish between right- and left-hom functors, and one can use any of the notations −• and •− .
The respective co-universal arrows are then called evaluation morphisms.
(4) For every ⊗-exponentiable C-object A, since the functor A ⊗ (−) (resp. (−) ⊗ A) has a right adjoint
A −• (−) (resp. (−) •− A), then A ⊗ (−) (resp. (−) ⊗ A) preserves the existing colimits, and A −• (−)
(resp. (−) •− A) preserves the existing limits.
Definition 15. A monoidal category C is called closed provided that every its object is ⊗-exponentiable.
Remark 16.
(1) A monoidal category C is sometimes called right closed (resp. left closed, biclosed) provided that for
every its object A, the functor A ⊗ (−) has a right adjoint (resp. the functor (−) ⊗ A has right adjoint,
both functors A ⊗ (−) and (−) ⊗ A have a right adjoint).
(2) Given a category C with the monoidal structure defined by finite products, one says exponentiable
instead of ×-exponentiable and generally writes BA
for the internal hom-object A −• B ∼= B •− A, which
is also called an exponential or power object. The morphism ˆf of diagram (1.2) is then called exponential
morphism for f. The category C itself is then said to be cartesian closed provided that it is closed.
Example 17.
(1) The category Set is cartesian closed. Given two sets A, B, the respective power object BA
is the set of
all maps B
α
−→ A, and the respective evaluation morphism A × BA evB
−−→ B is the usual evaluation map
given by ev(a, α) = α(a). Given a morphism A × C
f
−→ B, the exponential morphism C
ˆf
−→ BA
for f is
defined by ˆf(c) = f(−, c). It is easy to see that ˆf is a unique map making the next diagram commute
A × C
f
((
1A× ˆf

A × BA
evB
// B.
4
(2) The category Prost of preordered sets and monotone maps is cartesian closed. Given two preordered
sets A, B, the respective power object BA
is the set of all monotone maps B
α
−→ A, and the respective
evaluation morphism A × BA evB
−−→ B is the usual evaluation map given by ev(a, α) = α(a).
(3) The category Cat of small categories (the class of objects of a category is a set) and functors is cartesian
closed. Given two categories A, B, the respective power object is the functor category BA
, and the
respective evaluation morphism A × BA evB
−−→ B is defined on objects by evB(A, F) = FA and on
morphisms by evB(h, τ) = τA′ · Fh, where A
h
−→ A′
is an A-morphism and F
τ
−→ F′
is a natural
transformation. Given a functor A × C
F
−→ B, the exponential morphism C
ˆF
−→ BA
is defined by
A
ˆF C
−−→ B, ˆFC(A
h
−→ A′
) = F(A, C)
F (h,1C )
−−−−−→ F(A′
, C).
(4) The category Top of topological spaces and continuous maps is not cartesian closed, since the functor
Top
Q×(−)
−−−−→ Top (where Q is the space of rational numbers, with the topology induced by that of
the real line R) does not preserve quotients, and, thus, does not preserve coequalizers (notice that left
adjoint functors preserve the existing colimits).
(5) The category Ab of abelian groups is monoidal closed, where A −• B = Ab(A, B) with the pointwise
structure of an abelian group. More generally, the category R-Mod of left R-modules is monoidal closed,
where A −• B = R-Mod(A, B) with the pointwise structure of an R-module (recall from Example 6 (3)
that both categories Ab and R-Mod are symmetric).
(6) The category Sup of -semilattices is monoidal closed, where A −• B = Sup(A, B) with the pointwise
structure of a -semilattice (recall from Example 6 (4) that the category Sup is symmetric).
(7) A unital quantale V is monoidal closed. Given a ∈ V , the maps V
a −• (−)
−−−−−→ V and V
(−) •− a
−−−−−→ V are
defined by
a ⊗ c ⩽ b iff c ⩽ a −• b c ⊗ a ⩽ b iff c ⩽ b •− a
for every c, b ∈ V . In particular, a −• b = {c ∈ V | a ⊗ c ⩽ b} and b •− a = {c ∈ V | c ⊗ a ⩽ b}. If the
quantale V is commutative, then the maps a −• (−) and (−) •− a coincide (recall from Example 6 (5)
that V is then a symmetric monoidal category). Additionally, by Remark 14 (4), it follows that both
maps a −• (−) and (−) •− a are -preserving. In particular, they preserve the largest element ⊤V .
Proposition 18. Let C be a monoidal closed category, and let A, B, C be C-objects. There exist the
following natural isomorphisms:
(1) (A ⊗ B) −• C ∼= B −• (A −• C);
(2) C •− (A ⊗ B) ∼= (C •− B) •− A;
(3) (A −• C) •− B ∼= A −• (C •− B).
If C is cartesian closed, then CA×B ∼= (CA
)B ∼= (CB
)A
.
Proof. As an illustration, we show the proof of items (1) – (3) in case of C being a unital quantale V .
(1) Given v ∈ V , in view of Example 17 (7), it follows that v ⩽ (a⊗b) −• c iff (a⊗b)⊗v ⩽ c iff a⊗(b⊗v) ⩽ c
iff b ⊗ v ⩽ a −• c iff v ⩽ b −• (a −• c). As a result, one gets (a ⊗ b) −• c = b −• (a −• c).
(2) Given v ∈ V , in view of Example 17 (7), it follows that v ⩽ c •− (a⊗b) iff v⊗(a⊗b) ⩽ c iff (v⊗a)⊗b ⩽ c
iff v ⊗ a ⩽ c •− b iff v ⩽ (c •− b) •− a. As a result, one gets c •− (a ⊗ b) = (c •− b) •− a.
(3) Given v ∈ V , in view of Example 17 (7), it follows that v ⩽ (a −• c) •− b iff v⊗b ⩽ a −• c iff a⊗(v⊗b) ⩽ c
iff (a ⊗ v) ⊗ b ⩽ c iff a ⊗ v ⩽ c •− b iff v ⩽ a −• (c •− b). Thus, one gets (a −• c) •− b = a −• (c •− b). □
Remark 19. Given a monoidal closed category C and a C-object C, there exists a functor Cop (−) −• C
−−−−−→ C
defined for a C-morphism A
f
−→ B by (−) −• C(A
f
−→ B) = B −• C
f −• C
−−−−→ A −• C, where B −• C
f −• C
−−−−→
5
A −• C is the unique C-morphism making the diagram
A ⊗ (B −• C)
f⊗1B −• C
//
1A⊗(f −• C)

B ⊗ (B −• C)
evB
−• C

A ⊗ (A −• C)
evA
−• C
// C
commute.
Proposition 20. Given a symmetric monoidal closed category C, for every C-object C, it follows that the
functor Cop (−) −• C
−−−−−→ C has a left adjoint, which is given by the dual functor C
((−) −• C)op
−−−−−−−−→ Cop
.
Remark 21.
(1) For every object C of a symmetric monoidal closed category C, since the functor Cop (−) −• C
−−−−−→ C has
a left adjoint, it preserves the existing limits, i.e., it takes colimits to limits. In particular, for every
element c of a commutative unital quantale V , the map (−) −• c takes to .
(2) Given a (not necessarily unital) quantale V , the map (−) −• c (defined as in Example 17 (7)) takes to
, which can be seen as follows. Given a subset S ⊆ V , for every v ∈ V , v ⩽ ( S) −• c iff ( S)⊗v ⩽ c
iff s∈S(s ⊗ v) ⩽ c iff s ⊗ v ⩽ c for every s ∈ S iff v ⩽ s −• c for every s ∈ S iff v ⩽ s∈S(s −• c). As a
consequence, one gets that ( S) −• c = s∈S(s −• c). Similar result holds for the map c •− (−).
Example 22. If C is the cartesian closed category Set and C is the two-element set 2, then, for every set A,
the internal hom-object 2A
= A −• 2 is the powerset PA of A. The functor 2(−)
= (−) −• 2 is then precisely
the contravariant powerset functor Setop Q
−→ Set defined on a map B
f
−→ A by Q(A
fop
−−→ B) = PA
f−1
−−→ PB,
where f−1
(S) is the preimage of a subset S ⊆ A, i.e., f−1
(S) = {b ∈ B | f(b) ∈ S}.
2. Properties of the category V -Cat
2.1. Symmetric monoidal closed structure on the category V -Cat
Definition 23. Let V be a unital quantale, and let (X, a), (Y, b) be V -categories.
(1) Define a V -relation [−, −] on the set V -Cat((X, a), (Y, b)) = {X
f
−→ Y | f is a V -functor } by
[f, g] =
x∈X
b(f(x), g(x)),
and let [(X, a), (Y, b)] stand for the pair (V -Cat((X, a), (Y, b)), [−, −]).
(2) Define a V -relation a ⊗ b on the set X × Y (the cartesian product of the sets X and Y ) by
(a ⊗ b)((x1, y1), (x2, y2)) = a(x1, x2) ⊗ b(y1, y2)
for every x1, x2 ∈ X and every y1, y2 ∈ Y , and let (X, a) ⊗ (Y, b) stand for the pair (X × Y, a ⊗ b).
Proposition 24. Give a unital quantale V and V -categories (X, a), (Y, b), the following holds:
(1) [(X, a), (Y, b)] is a V -category;
(2) If the quantale V is commutative, then (X, a) ⊗ (Y, b) is a V -category.
Proof. In both cases, one shows the two required properties of a V -category (see Lecture 1).
6
(1) First, given f ∈ V -Cat((X, a), (Y, b)), it follows that [f, f] = x∈X b(f(x), f(x)) ⩾ x∈X k ⩾ k,
namely, k ⩽ [f, f]. Second, given f, g, h ∈ V -Cat((X, a), (Y, b)), it follows that [f, g] ⊗ [g, h] =
( x∈X b(f(x), g(x))) ⊗ ( x∈X b(g(x), h(x))) ⩽ x∈X(b(f(x), g(x)) ⊗ b(g(x), h(x))) ⩽ (since (Y, b) is
a V -category) ⩽ x∈X b(f(x), h(x)) = [f, h], namely, [f, g] ⊗ [g, h] ⩽ [f, h].
(2) First, given x ∈ X and y ∈ Y , it follows that (a ⊗ b)((x, y), (x, y)) = a(x, x) ⊗ b(y, y) ⩾ k ⊗ k = k,
namely, k ⩽ (a ⊗ b)((x, y), (x, y)). Second, given x1, x2, x3 ∈ X and y1, y2, y3 ∈ Y , it follows that
(a ⊗ b)((x1, y1), (x2, y2)) ⊗ (a ⊗ b)((x2, y2), (x3, y3)) = (a(x1, x2) ⊗ b(y1, y2)) ⊗ (a(x2, x3) ⊗ b(y2, y3)) =
(since the quantale V is commutative) = (a(x1, x2) ⊗ a(x2, x3)) ⊗ (b(y1, y2) ⊗ b(y2, y3)) ⩽ (since both
(X, a) and (Y, b) are V -categories) ⩽ a(x1, x3) ⊗ b(y1, y3) = (a ⊗ b)((x1, y1), (x3, y3)), namely, (a ⊗
b)((x1, y1), (x2, y2)) ⊗ (a ⊗ b)((x2, y2), (x3, y3)) ⩽ (a ⊗ b)((x1, y1), (x3, y3)). □
Remark 25. Notice that given V -categories (X, a), (Y, b) over a commutative unital quantale V , (X, a) ⊗
(Y, b) is not the product V -category (X, a) × (Y, b). Indeed, following the results of Lecture 2, V -Cat is
a topological category over Set, and, therefore, the limits in V -Cat are lifted from those in Set by the
forgetful functor. In particular, the product V -category of (X, a) and (Y, b) is V -category (X × Y, c), where
V -relation c is given by c((x1, y1), (x2, y2)) = a(x1, x2)∧b(y1, y2) for every x1, x2 ∈ X and every y1, y2 ∈ Y .
Example 26. If V = P+, then P+-categories (X, a), (Y, b) are quasi-pseudo-metric spaces (generalized
metric spaces in the sense of F. W. Lawvere). It then follows that [f, g] = supx∈X b(f(x), g(x)) is the usual
“sup-metric” on the function space [X, Y ], and, moreover, (a ⊗ b)((x1, y1), (x2, y2)) = a(x1, x2) + b(y1, y2)
equips X × Y with the usual “+-metric”.
Proposition 27. Given a V -category [(X, a), (Y, b)], [f, g] = x1,x2∈X a(x1, x2) −• b(f(x1), g(x2)) for every
f, g ∈ V -Cat((X, a), (Y, b)).
Proof. Given x1, x2 ∈ X, it follows that a(x1, x2)⊗b(f(x2), g(x2)) ⩽ (f is a V -functor) ⩽ b(f(x1), f(x2))⊗
b(f(x2), g(x2)) ⩽ ((Y, b) is a V -category) ⩽ b(f(x1), g(x2)), i.e., b(f(x2), g(x2))⩽a(x1, x2) −• b(f(x1), g(x2)).
Thus, [f, g] = x∈X b(f(x), g(x)) = x1,x2∈X b(f(x2), g(x2)) ⩽ x1,x2∈X a(x1, x2) −• b(f(x1), g(x2)).
Given x ∈ X, k⩽a(x, x) and Remark 21 (2) together imply a(x, x) −• b(f(x), g(x)) ⩽ k −• b(f(x), g(x)) ⩽
b(f(x), g(x)). Thus, it follows that x1,x2∈X a(x1, x2) −• b(f(x1), g(x2)) ⩽ x∈X a(x, x) −• b(f(x), g(x)) ⩽
x∈X b(f(x), g(x)) = [f, g], which finishes the proof. □
Remark 28. By analogy with Proposition 27, one can show that given a V -category [(X, a), (Y, b)], [f, g] =
x1,x2∈X b(f(x1), g(x2)) •− a(x1, x2) for every f, g ∈ V -Cat((X, a), (Y, b)).
Theorem 29. Given a commutative unital quantale V , the category V -Cat is symmetric monoidal closed.
Proof. First, define the tensor product functor V -Cat × V -Cat
⊗
−→ V -Cat by ⊗(((X1, a1), (Y1, b1))
(f,g)
−−−→
((X2, a2), (Y2, b2))) = (X1 × Y1, a1 ⊗ b1)
f×g
−−−→ (X2 × Y2, a2 ⊗ b2). To show that the functor is correct on
morphisms, notice that given x1, x′
1 ∈ X and y1, y′
1 ∈ Y1, it follows that (a1 ⊗ b1)((x1, y1), (x′
1, y′
1)) =
a1(x1, x′
1) ⊗ b1(y1, y′
1) ⩽ (both f and g are V -functors) ⩽ a2(f(x1), f(x′
1)) ⊗ b2(g(y1), g(y′
1)) = (a2 ⊗
b2)((f(x1), g(y1)), (f(x′
1), g(y′
1))) = (a2 ⊗ b2)((f × g)(x1, y1), (f × g)(x′
1, y′
1)).
Second, define the unit E = ({∗}, k), where k(∗, ∗) = k.
Third, given V -categories (X, a), (Y, b), and (Z, c), define the natural isomorphism (X, a) ⊗ ((Y, b) ⊗
(Z, c))
α(X,a),(Y,b),(Z,c)
−−−−−−−−−−→ ((X, a) ⊗ (Y, b)) ⊗ (Z, c) by α(X,a),(Y,b),(Z,c)(x, (y, z)) = ((x, y), z). Further, define the
natural isomorphism E ⊗ (X, a)
λ(X,a)
−−−−→ (X, a) by λ(X,a)(∗, x) = x (the projection map) and the natural
transformation (X, a) ⊗ E
ρ(X,a)
−−−−→ (X, a) by ρ(X,a)(x, ∗) = x (the projection map again). To show that, e.g.,
the map λ(X,a) is a V -functor, notice that for every x1, x2 ∈ X, it follows that (k ⊗ a)((∗, x1), (∗, x2)) =
k(∗, ∗) ⊗ a(x1, x2) = k ⊗ a(x1, x2) = a(x1, x2) = a(λ(X,a)(∗, x1), λ(X,a)(∗, x2)). Moreover, commutativity
of the diagrams of Definition 1 is immediate. For example, for the triangle, notice that (ρ(X,a) ⊗ 1(Y,b)) ·
α(X,a),E,(Y,b)(x, (∗, y)) = ρ(X,a) ×1(Y,b)((x, ∗), y) = (x, y) = 1(X,a) ×λ(Y,b)(x, (∗, y)) = 1(X,a) ⊗λ(Y,b)(x, (∗, y)).
7
Fourth, given V -categories (X, a), (Y, b), define the natural isomorphism (X, a)⊗(Y, b)
σ(X,a),(Y,b)
−−−−−−−→ (Y, b)⊗
(X, a) by σ(X,a),(Y,b)(x, y) = (y, x). The above structure then makes V -Cat a symmetric monoidal category.
Fifth, to show that the category V -Cat is closed, by Remark 14 (3), it is enough to show that given
V -categories (X, a) and (Y, b), the map (X, a)⊗[(X, a), (Y, B)]
ev(Y,b)
−−−−→ (Y, b), defined by ev(Y,b)(x, f) = f(x),
provides an (X, a) ⊗ (−)-co-universal arrow for (Y, b).
To check that the map ev(Y,b) provides a V -functor, notice that given x1, x2 ∈ X and f, g ∈ V -Cat((X, a),
(Y, b)), it follows that (a ⊗ [−, −])((x1, f), (x2, g)) = a(x1, x2) ⊗ [f, g] = (Proposition 27) = a(x1, x2) ⊗
( x′
1,x′
2∈X a(x′
1, x′
2) −• b(f(x′
1), g(x′
2))) ⩽ a(x1, x2) ⊗ (a(x1, x2) −• b(f(x1), g(x2))) ⩽ (Example 17 (7)) ⩽
b(f(x1), g(x2)) = b(ev(Y,b)(x1, f), ev(Y,b)(x2, g)).
Given a V -functor (X, a) ⊗ (Z, c)
f
−→ (Y, b), define a map Z
ˆf
−→ V -Cat((X, a), (Y, b)) by ˆf(z) = f(−, z).
To show that the map ˆf is correct, i.e., ˆf(z) is a V -functor for every z ∈ Z, notice that given x1, x2 ∈ X, it
follows that b(( ˆf(z))(x1), ( ˆf(z))(x2)) = b(f(x1, z), f(x2, z)) ⩾ (f is a V -functor) ⩾ (a ⊗ c)((x1, z), (x2, z)) =
a(x1, x2) ⊗ c(z, z) ⩾ ((Z, c) is a V -category) ⩾ a(x1, x2) ⊗ k = a(x1, x2).
To show that the map ˆf is a V -functor, one could observe that given z1, z2 ∈ Z, it follows that
[ ˆf(z1), ˆf(z2)] = (Definition 23 (1)) = x∈X b(( ˆf(z1))(x), ( ˆf(z2))(x)) = x∈X b(f(x, z1), f(x, z2)) ⩾ (f is
a V -functor) ⩾ x∈X(a ⊗ c)((x, x), (z1, z2)) ⩾ x∈X(a(x, x) ⊗ c(z1, z2)) ⩾ ((X, a) is a V -category) ⩾
x∈X(k ⊗ c(z1, z2)) = x∈X c(z1, z2) ⩾ c(z1, z2).
Lastly, it is easy to see that ˆf is the unique V -functor making the following triangle commute
(X, a) ⊗ (Z, c)
f
**
1(X,a)⊗ ˆf

(X, a) ⊗ [(X, a), (Y, b)] ev(Y,b)
// (Y, b).
□
Example 30.
(1) If V is the two-element unital quantale 2 = ({⊥, ⊤}, ∧, ⊤) (recall Lecture 1), then Theorem 29 confirms
that the category Prost of preordered sets is cartesian closed (cf. Example 17 (2)).
(2) If V is the extended real half-line P+ = ([0, ∞]op
, +, 0) (recall Lecture 1), then Theorem 29 confirms
that the category QPMet of quasi-pseudo-metric spaces (generalized metric spaces in the sense of
F. W. Lawvere) is monoidal closed. Following Remark 25, notice that given P+-categories (X, a), (Y, b),
the monoidal product (X, a) ⊗ (Y, b) is not the product P+-category (X, a) × (Y, b), since V -category
structure on (X, a) ⊗ (Y, b) is given by the “+-metric” (cf. Example 26), whereas V -category structure
on (X, a) × (Y, b) is given by c((x1, y1), (x2, y2)) = max{a(x1, x2), b(y1, y2)}.
2.2. Dual V -categories and their induced functor
Definition 31. Given a V -category (X, a), define a V -relation a◦
on X by a◦
(x, y) = a(y, x).
Proposition 32. Given a V -category (X, a) over a commutative unital quantale V , (X, a◦
) is a V -category.
Proof. First, given x ∈ X, it follows that a◦
(x, x) = a(x, x) ⩾ ((X, a) is a V -category) ⩾ k. Second, given
x, y, z ∈ X, it follows that a◦
(x, y) ⊗ a◦
(y, z) = a(y, x) ⊗ a(z, y) = (V is commutative) = a(z, y) ⊗ a(y, x) ⩽
((X, a) is a V -category) ⩽ a(z, x) = a◦
(x, z), i.e., a◦
(x, y) ⊗ a◦
(y, z) ⩽ a◦
(x, z). □
Remark 33. Given a V -category (X, a) over a commutative unital quantale V , (X, a)
op
= (X, a◦
) is called
the dual V -category of (X, a).
Proposition 34. Given a commutative unital quantale V , there exists a functor V -Cat
(−)op
−−−→ V -Cat
defined by ((X, a)
f
−→ (Y, b)) = (X, a◦
)
f
−→ (Y, b◦
).
Proof. Given a V -functor (X, a)
f
−→ (Y, b), it is enough to show that (X, a◦
)
f
−→ (Y, b◦
) is a V -functor.
Given x1, x2 ∈ X, a◦
(x1, x2) = a(x2, x1) ⩽ (f is a V -functor) ⩽ b(f(x2), f(x1)) = b◦
(f(x1), f(x2)). □
8
2.3. Unital quantale V as a V -category
Proposition 35. Given a unital quantale V , the pair (V, −• ) is a V -category.
Proof. Recall that Example 17 (7) defined the map V
(−) −• (−)
−−−−−−→ V by a −• b = {c ∈ V | a⊗c ⩽ b}, which
is a V -relation V (−) −• (−)
// V. We show the two required properties of a V -category (see Lecture 1).
(1) Given v ∈ V , v ⊗ k = v ⩽ v implies k ⩽ v −• v.
(2) Given u, v, w ∈ V , u −• v ⩽ u −• v and v −• w ⩽ v −• w imply u ⊗ (u −• v) ⩽ v and v ⊗ (v −• w) ⩽ w
imply u ⊗ (u −• v) ⊗ (v −• w) ⩽ v ⊗ (v −• w) ⩽ w implies (u −• v) ⊗ (v −• w) ⩽ u −• w. □
Example 36.
(1) If V is the two-element unital quantale 2 = ({⊥, ⊤}, ∧, ⊤) (recall Lecture 1), then (−) −• (−) is the
partial order of 2, i.e., for every u, v ∈ 2, it follows that u −• v = ⊤ iff u ⩽ v.
(2) If V is the extended real half-line P+ = ([0, ∞]op
, +, 0) (recall Lecture 1), then (−) −• (−) is the truncated
difference, i.e., for every u, v ∈ P+,
u −• v = inf{w ∈ [0, ∞] | v ⩽ u + w} =



v − u, u ⩽ v < ∞
0, v ⩽ u
∞, u < v = ∞.
2.4. The category V -Mod
2.4.1. Ordered categories and quantaloids
Definition 37. A category C is called preordered provided that every its hom-set C(A, B) is a preordered
set, and the composition of morphisms is monotone in both variables, i.e., given C-morphisms
A
f
−→ B
g1
//
g2
// C
h
−→ D, if g1 ⩽ g2, then h · g1 ⩽ h · g2 and g1 · f ⩽ g2 · f. Moreover, if the preorder on the
hom-sets of C is a partial order, then the category C is called partially ordered.
Remark 38. The condition on composition of morphism in a preordered category C of Definition 37 is
equivalent to the following: given C-morphisms A
f
−→ B
g1
//
g2
// C
h
−→ D, if g1 ⩽ g2, then h·g1 ·f ⩽ h·g2 ·f.
Example 39.
(1) Prost is a preordered category, where Prost((X, ⩽X), (Y, ⩽Y )) is equipped with a pointwise order.
(2) Sup is a partially ordered category.
(3) The category V -Rel of sets (as objects) and V -relations (as morphisms) is partially ordered by pointwise
evaluation of V -relations (for V -relations X
r
//
s
// Y, r ⩽ s iff r(x, y) ⩽ s(x, y) for every x ∈ X, y ∈ Y ).
(4) V -Cat is a partially ordered category with the partial order inherited from the category V -Rel.
(5) Every category is partially ordered if equipped with the partial order given by equality.
Remark 40.
(1) Given a preordered category C, its dual category Cop
is also preordered.
(2) For every preordered category C, there exists the conjugate preordered category Cco
, which has the same
morphisms but employs the dual preorder on hom-sets, i.e., Cco
(A, B) = (C(A, B), ⩾) = (C(A, B))
op
.
Moreover, it is easy to verify that Cop co
= Cco op
.
9
Definition 41. A morphism A
f
−→ B of a preordered category C is said to be a map provided that there
exists a C-morphism B
g
−→ A such that 1A ⩽ g · f and f · g ⩽ 1B. One uses the notation f ⊣ g, where f is
the left adjoint and g is the right adjoint of the adjunction.
Remark 42.
(1) The terminology of Definition 41 is motivated by the category Rel, where a relation X r
// Y is a
map in Rel exactly when it is the graph of a morphism X
r
−→ Y in Set. The existence of a relation
Y s
// X such that 1X ⩽ s · r means that for every x ∈ X, there exists y ∈ Y such that x r y and
y s x; and r · s ⩽ 1Y means that for every x ∈ X and every y1, y2 ∈ Y , if y1 s x and x r y2, then y1 = y2.
One can thus define a unique map X
r
−→ Y in Set by r(x) = y iff x r y. One can also check that s = r◦
.
(2) In every preordered category C, a right adjoint g of a map f as in Definition 41 is uniquely determined
up to “∼=”, i.e., if both g1 and g2 are right adjoints of f, then g1 ⩽ g2 and g2 ⩽ g1. Moreover, in a
partially ordered category C, a right adjoint g to a map f is determined uniquely.
Definition 43. A category C is a said to be a quantaloid provided that every its hom-set C(A, B) is a
-semilattice, and the composition of morphisms is -preserving in both variables, i.e., given C-morphisms
A
f
−→ B
gi
−→ C
h
−→ D with i ∈ I, it follows that h · ( i∈I gi) = i∈I(h · gi) and ( i∈I gi) · f = i∈I(gi · f).
Remark 44. Every quantaloid is a partially ordered category.
Example 45.
(1) The categories Sup and V -Rel are quantaloids.
(2) A category C partially ordered by equality is a quantaloid iff its hom-sets have at most one element.
(3) Unital quantales are precisely the quantaloids with one object.
(4) Given a quantaloid C, the dual category Cop
is a quantaloid, but the conjugate category Cco
is generally
not a quantaloid, since composition of morphisms will generally preserve and not .
Definition 46. A homomorphism of quantaloids C
F
−→ D is a functor which preserves on hom-sets, i.e.,
given C-morphisms A
fi
−→ B for i ∈ I, it follows that F( i∈I fi) = i∈I Ffi.
Example 47. Every homomorphism of unital quantales provides a homomorphism of quantaloids.
Definition 48. A quantaloid C is said to be involutive provided that it comes equipped with a quantaloid
homomorphism Cop (−)◦
−−−→ C (called involution) such that C◦
= C for every C-object C and (f◦
)◦
= f for
every C-morphism A
f
−→ B. In particular, given C-morphisms A
fi
−→ B for i ∈ I, ( i∈I fi)◦
= i∈I f◦
i .
2.4.2. V -modules
Definition 49. Given V -categories (X, a), (Y, b) over a unital quantale V , a V -relation X r
// Y is
called V -module (also V -bimodule, V -profunctor, or V -distributor) provided that r ·a ⩽ r and b·r ⩽ r. One
denotes a V -module between V -categories (X, a) and (Y, b) by (X, a) ◦
r
// (Y, b).
Proposition 50. For a V -relation X
r
// Y between V -categories (X, a) and (Y, b) equivalent are:
(1) r is a V -module;
(2) r · a = r and b · r = r;
(3) b · r · a = r.
10
Proof. For (1) ⇒ (2), it will be enough to verify that given a V -module (X, a) ◦
r
// (Y, b), it follows
that r ⩽ r · a and r ⩽ b · r. For the former, notice that given x ∈ X and y ∈ Y , one gets that (r · a)(x, y) =
x′∈X a(x, x′
) ⊗ r(x′
, y) ⩾ a(x, x) ⊗ r(x, y) ⩾ ((X, a) is a V -category) ⩾ k ⊗ r(x, y) = r(x, y); and for the
latter, observe that given x ∈ X and y ∈ Y , one obtains that (b · r)(x, y) = y′∈Y r(x, y′
) ⊗ b(y′
, y) ⩾
r(x, y) ⊗ b(y, y) ⩾ ((Y, b) is a V -category) ⩾ r(x, y) ⊗ k = r(x, y).
For (2) ⇒ (3), notice that b · r · a = b · (r · a) = b · r = r.
For (3) ⇒ (1), observe that, first, r · a = 1Y · r · a ⩽ ((Y, b) is a V -category) ⩽ b · r · a = r and, second,
b · r = b · r · 1X ⩽ ((X, a) is a V -category) ⩽ b · r · a = r. □
Example 51. If V = 2, then 2-modules are precisely the classical modules between preordered sets, i.e.,
relations X
r
// Y (where (X, ⩽X), (Y, ⩽Y ) are preordered sets) such that (⩽Y )·r·(⩽X) ⩽ r. The latter
condition means that for every x1, x2 ∈ X, y1, y2 ∈ Y , x2 ⩽X x1 and x1 r y1 and y1 ⩽Y y2 together imply
x2 r y2, i.e., the map Xop
× Y
r
−→ 2 is monotone, where Xop
× Y is given the component-wise preorder.
Proposition 52.
(1) If (X, a) ◦
r
// (Y, b), (Y, b) ◦
s
// (Z, c) are V -modules, then X s·r
// Z is a V -module.
(2) Given a V -category (X, a), X a
// X is a V -module.
Proof.
(1) In view of Proposition 50, c · s · r · b = (c · s) · (r · b) = (both r and s are V -modules) = s · r.
(2) First, a · a ⩽ a ((X, a) is a V -category), and, second, a = a · 1X ⩽ ((X, a) is a V -category) ⩽ a · a. As a
consequence, it follows that a · a = a. □
Remark 53.
(1) In view of Proposition 50 (2) and Proposition 52, there exists the category V -Mod of V -categories (as
objects) and V -modules (as morphisms), where given a V -category (X, a), it follows that V -relation
X a
// X provides the identity morphism on (X, a).
(2) The category V -Mod is a partially ordered category with the partial order on hom-sets inherited from
the partially ordered category V -Rel. Moreover, the category V -Mod is a quantaloid with in hom-sets
formed by pointwise evaluation precisely as in the category V -Rel.
Remark 54. Recall from Lecture 1 that for a unital quantale V , there is a functor Set
(−)◦
−−−→ V -Rel, which
is a non-full embedding if V has at least two elements. There also exists a functor Setop (−)◦
−−−→ V -Rel.
Lemma 55. Given a V -functor (X, a)
f
−→ (Y, b) over a unital quantale V , it follows that a · f◦
⩽ f◦
· b.
Proof. Since f is a V -functor, it follows that f · a ⩽ b · f. Recall from Lecture 1 that 1X ⩽ f◦
· f and
f · f◦
⩽ 1Y for every map X
f
−→ Y . Thus, a ⩽ f◦
· f · a ⩽ f◦
· b · f implies a · f◦
⩽ f◦
· b · f · f◦
⩽ f◦
· b. □
Proposition 56.
(1) There exists a functor V -Cat
(−)∗
−−−→ V -Mod defined by ((X, a)
f
−→ (Y, b))∗ = (X, a) ◦
f∗
// (Y, b), where
f∗ = b · f, i.e., f∗(x, y) = b(f(x), y) for every x ∈ X and every y ∈ Y .
(2) There exists a functor (V -Cat)
op (−)∗
−−−→ V -Mod defined by ((X, a)
f
−→ (Y, b))∗
= (Y, b) ◦
f∗
// (X, a),
where f∗
= f◦
· b, i.e., f∗
(y, x) = b(y, f(x)) for every x ∈ X and every y ∈ Y .
Proof.
11
(1) To show that f∗ is a V -module, consider Definition 49: f∗ · a = b · f · a ⩽ (f is a V -functor) ⩽ b · b · f ⩽
((Y, b) is a V -category) ⩽ b · f = f∗ and b · f∗ = b · b · f ⩽ ((Y, b) is a V -category) ⩽ b · f = f∗.
To show that (−)∗ preserves composition of morphisms, notice that given V -functors (X, a)
f
−→ (Y, b),
(Y, b)
g
−→ (Z, c), it follows that (g · f)∗ = c·g·f = c·g·1Y ·f ⩽ ((Y, b) is a V -category) ⩽ c·g·b·f = g∗ ·f∗.
Moreover, g∗ ·f∗ = c·g ·b·f ⩽ (g is V -functor) ⩽ c·c·g ·f ⩽ ((Z, c) is a V -category) ⩽ c·g ·f = (g · f)∗.
To show that (−)∗ preserves identities, notice that given a V -category (X, a), (1X)∗ = a · 1X = a.
(2) To show that f∗
is a V -module, consider Definition 49: f∗
· b = f◦
· b · b ⩽ ((Y, b) is a V -category)
⩽ f◦
·b = f∗
and a·f∗
= a·f◦
·b ⩽ (Lemma 55 for f) ⩽ f◦
·b·b ⩽ ((Y, b) is a V -category) ⩽ f◦
·b = f∗
.
To show that (−)
∗
preserves composition of morphisms, notice that given V -functors (X, a)
f
−→ (Y, b),
(Y, b)
g
−→ (Z, c), it follows that (g · f)
∗
= (g · f)
◦
· c = f◦
· g◦
· c = f◦
· 1Y · g◦
· c ⩽ ((Y, b) is a V -category)
⩽ f◦
· b · g◦
· c = f∗
· g∗
. Moreover, f∗
· g∗
= f◦
· b · g◦
· c ⩽ (Lemma 55 for g) ⩽ f◦
· g◦
· c · c ⩽ ((Z, c) is
a V -category) ⩽ f◦
· g◦
· c = (g · f)
◦
· c = (g · f)
∗
.
To show that (−)
∗
preserves identities, one should observe that given a V -category (X, a), it follows
that (1X)
∗
= (1X)
◦
· a = 1X · a = a. □
Remark 57.
(1) The functors of Proposition 56 provide a structured version of the functors of Remark 54, i.e., Set
(−)◦
−−−→
V -Rel
(−)◦
←−−− Setop
is replaced with V -Cat
(−)∗
−−−→ V -Mod
(−)∗
←−−− (V -Cat)
op
.
(2) In case the quantale V has at least two elements, unlike the functor Set
(−)◦
−−−→ V -Rel, the functor
V -Cat
(−)∗
−−−→ V -Mod is not faithful. Consider, e.g., a V -category (R, k), where k(x, y) = k for every
real numbers x, y. Then every map R
f
−→ R provides a V -functor (R, k)
f
−→ (R, k). However, f∗ = k · f
implies f∗(x, y) = k(f(x), y) = k for every x, y ∈ R, which then gives f∗ = k = 1(R,k).
Proposition 58. Given a V -functor (X, a)
f
−→ (Y, b), it follows that f∗ · f∗
⩽ (1Y )
∗
and (1X)
∗
⩽ f∗
· f∗.
Proof. First, f∗ · f∗
= b · f · f◦
· b ⩽ (since f · f◦
⩽ 1Y ) ⩽ b · b ⩽ ((Y, b) is a V -category) ⩽ b = 1Y · b =
(1Y )
◦
· b = (1Y )
∗
. Second, f∗
· f∗ = f◦
· b · b · f ⩾ (f is a V -functor) ⩾ a · f◦
· f · a ⩾ (since f◦
· f ⩾ 1X)
⩾ a · a = (Proposition 52 (2)) = a = 1X · a = (1X)
◦
· a = (1X)
∗
. □
Remark 59.
(1) Proposition 58 provides a structured version of the adjunction f◦ ⊣ f◦
valid in V -Rel for every map
X
f
−→ Y , in the form of f∗ ⊣ f∗
valid in V -Mod for every V -functor (X, a)
f
−→ (Y, b).
(2) (1Y )
∗
and (1X)
∗
in Proposition 58 can be replaced with (1Y )∗ and (1X)∗.
Proposition 60. Given the dual V -category functor V -Cat
(−)op
−−−→ V -Cat, it follows that (fop
)∗ = (f∗
)
◦
and (fop
)
∗
= (f∗)
◦
for every V -functor (X, a)
f
−→ (Y, b).
Proof. Given a V -functor (X, a)
f
−→ (Y, b), since ((X, a)
f
−→ (Y, b))op
= (X, a◦
)
f
−→ (Y, b◦
) by Proposition 34,
it follows that (fop
)∗ = b◦
· f = (f◦
· b)
◦
= (f∗
)
◦
and (fop
)
∗
= f◦
· b◦
= (b · f)
◦
= (f∗)
◦
. □
References
[1] F. Borceux, Handbook of Categorical Algebra. Volume 2: Categories and Structures, Cambridge University Press, 1994.
[2] M. M. Clementino and D. Hofmann, Exponentiation in V-categories, Topology Appl. 153 (2006), no. 16, 3113–3128.
[3] P. Eklund, J. Guti´errez Garc´ıa, U. H¨ohle, and J. Kortelainen, Semigroups in Complete Lattices. Quantales, Modules and
Related Topics, vol. 54, Cham: Springer, 2018.
[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] G. M. Kelly, Basic Concepts of Enriched Category Theory, Repr. Theory Appl. Categ. 2005 (2005), no. 10, 1–136.
[6] F. W. Lawvere, Metric spaces, generalized logic and closed categories, Repr. Theory Appl. Categ. 1 (2002), 1–37.
[7] S. Mac Lane, Categories for the Working Mathematician, 2nd ed., Springer-Verlag, 1998.
[8] K. I. Rosenthal, Quantales and Their Applications, Addison Wesley Longman, 1990.
[9] K. I. Rosenthal, The Theory of Quantaloids, Addison Wesley Longman, 1996.
12