Topology and its Applications 153 (2006) 3113–3128
www.elsevier.com/locate/topol
Exponentiation in V-categories
Maria Manuel Clementino a,∗, Dirk Hofmann b
a Departamento de Matemática, Universidade de Coimbra, 3001-454 Coimbra, Portugal
b Departamento de Matemática, Universidade de Aveiro, 3810-193 Aveiro, Portugal
Received 20 February 2004; accepted 30 November 2004
Abstract
For a Heyting algebra V which, as a category, is monoidal closed, we obtain characterizations
of exponentiable objects and morphisms in the category of V-categories and apply them to some
well-known examples. In the case V = R+ these characterizations of exponentiable morphisms and
objects in the categories (P)Met of (pre)metric spaces and non-expansive maps show in particular
that exponentiable metric spaces are exactly the almost convex metric spaces, while exponentiable
complete metric spaces are the complete totally convex ones.
© 2005 Elsevier B.V. All rights reserved.
MSC: 18D15; 18D20; 18B35; 18A25
Keywords: Exponentiable object; Exponentiable morphism; V-category; Premetric space; Almost convex metric
space; Totally convex metric space
0. Introduction
In 1973, Lawvere [10] observed that premetric spaces can be thought of as enriched
categories over [0,∞]: a premetric d :X × X → [0,∞] can be interpreted as the hom-
functor of a category so that the inequalities
0 d(x, x),
* Corresponding author.
E-mail addresses: mmc@mat.uc.pt (M.M. Clementino), dirk@mat.ua.pt (D. Hofmann).0166-8641/$ – see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.topol.2005.01.038
3114 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128d(x, y) + d(y, z) d(x, z)
play the role of the unit and of the composition laws of a category.
Indeed, a V-category in the sense of Eilenberg and Kelly [8] is a set X endowed with a
map X×X → V, (x, y) → X(x,y), and specifications of identity morphisms and compo-
sition law
I → X(x,x),
X(x, y) ⊗ X(y, z) → X(x, z)
satisfying unity and associativity axioms, that can be expressed by commutative diagrams
in the category V. (Here ⊗ is a tensor product in V with neutral element I .)
In our crucial example of premetric spaces, the category V is the complete real half-line
[0,∞], with categorical structure u → v if u v, with tensor product + and unit 0. Since
our category V is given by a lattice, commutativity of diagrams, and hence also the unity
and associativity axioms, come for free.
Another interesting example arises for V the two-element chain 2 = {false true}, with
the tensor product ∧, that coincides with the categorical one, and neutral element true. For
a map X × X → 2, or equivalently for a binary relation in X, the unit and composition
laws read as
true x x,
(x y) ∧ (y z) (x z),
so that (X,) is exactly a preordered set.
If V = Set, and if the tensor product is the categorical one, then a V-category is exactly
a small category.
Moreover, in the former examples the notion of V-functor gives natural morphisms: an
R+-functor is a non-expansive map, a 2-functor is a monotone map, while a Set-functor is
just a functor.
The tensor product in V induces naturally a tensor product in the category V-Cat of
V-categories and V-functors. Lawvere proved that with V also V-Cat is a monoidal closed
category. Since in the examples V = 2 and V = Set the tensor and the categorical products
coincide, this gives in particular that 2-Cat and Cat(= Set-Cat) are Cartesian closed cat-
egories. Cartesian closedness of R+-Cat does not follow from Lawvere’s result, since the
tensor product + does not coincide with the categorical product max. In fact this category
is not Cartesian closed, although R+ is. Moreover, none of the categories listed is locally
Cartesian closed. It is possible to avoid this problem working instead in the larger category
V-RGph of reflexive V-graphs, which is a quasi-topos whenever V is locally Cartesian
closed (see [5]).
The goal of this paper is to identify exponentiable morphisms and objects in V-Cat
when V is a lattice, that is, a small category with at most one morphism between each pair
of objects. Then the embedding V-Cat ↪→ V-RGph is full, a property that plays an essen-
tial role in this work. Our main result is Theorem 3.4 which characterizes exponentiable
morphisms (hence in particular exponentiable objects) in V-Cat whenever the lattice V is
complete and locally Cartesian closed, which means exactly that V is a complete Heyt-
ing algebra. This result gives in particular the characterization of exponentiable monotone
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 3115maps in POrd obtained by Tholen [12] and new characterizations of exponentiable mor-
phisms and objects in the category Met of metric spaces and non-expansive maps.
Finally we observe that some of the proofs presented in this paper follow the guidelines
of the proof of the characterization of exponentiable continuous maps in Top obtained
in [4], which raises the problem of whether these techniques can be applied to obtain
such a characterization in the more general context of reflexive and transitive lax algebras
described in [3] or [6].
In Section 1 we present the basic results on V-categories we need throughout. Sec-
tion 2 gives an account on exponentiability, including exponentiation in the category of
reflexive V-graphs. Section 3 contains our main results, namely the characterizations of
exponentiable morphisms and objects in V-Cat. The last section deals with examples,
giving a special attention to exponentiability in the category Met of metric spaces and
non-expansive maps.
1. The category of V-categories
1.1. Throughout V is a complete lattice equipped with a symmetric tensor product ⊗,
with unit k, and with right adjoint hom; that is, for each u,v,w ∈ V,
u ⊗ v w ⇐⇒ v hom(u,w).
As a category, V is said to be a closed symmetric monoidal category.
A V-enriched category (or simply V-category) is a pair (X,a) with X a set and a :X ×
X → V a map such that:
(R) for each x ∈ X, k a(x, x);
(T) for each x, x′, x′′ ∈ X, a(x, x′) ⊗ a(x′, x′′) a(x, x′′).
Given V-categories (X,a) and (Y, b), a V-functor f : (X,a) → (Y, b) is a map f :X → Y
such that, for each x, x′ ∈ X, a(x, x′) b(f (x), f (x′)). They form the category V-Cat of
V-categories and V-functors.
1.2. We list now two guiding examples of such categories, obtained when V is the two-
element chain 2 = {false true}, with the monoidal structure given by ∧ and “true”, and
when V is the (complete) real half-line [0,∞], with the categorical structure induced by
the relation (i.e., a → b means a b) and with tensor product +, which we will denote
by R+.
For V = 2, with the usual notation x x′ :⇐⇒ a(x, x′) = true, axioms (R) and (T) read
as:
∀x ∈ X x x;
∀x, x′, x′′ ∈ X x x′ & x′ x′′ ⇒ x x′′
that is, (X,) is a preordered set. A 2-functor is then a map f : (X,) → (Y,) between
preordered sets such that
∀x, x′ ∈ X x x′ ⇒ f (x) f (x′);
3116 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128that is, f is a monotone map. Hence 2-Cat is exactly the category POrd of preordered sets
and monotone maps.
An R+-category is a set X endowed with a map a :X × X → [0,∞] such that
∀x ∈ X 0 a(x, x);
∀x, x′, x′′ ∈ X a(x, x′) + a(x′, x′′) a(x, x′′);
that is, a :X×X → [0,∞] is a premetric in X. AnR+-functor is a map f : (X,a) → (Y, b)
between premetric spaces satisfying the following inequality:
∀x, x′ ∈ X a(x, x′) b(f (x), f (x′)),
which means that f is a non-expansive map. Therefore the category R+-Cat coincides
with the category PMet of premetric spaces and non-expansive maps. (For more details,
see [10,6].)
1.3. The V-functor hom : V × V → V provides V with a structure of V-category. In-
deed, for each u,v,w ∈ V:
k hom(u,u),
hom(u, v) ⊗ hom(v,w) hom(u,w).
The first inequality follows from k ⊗ u = u u, while the second is a consequence of
u ⊗ hom(u, v) ⊗ hom(v,w) v ⊗ hom(v,w)w.
1.4. The tensor product ⊗ of V induces a tensor product in V-Cat, which we will
denote also by ⊗: for each pair (X,a), (Y, b) of V-categories, (X,a) ⊗ (Y, b) := (X ×
Y,a ⊗ b), where (a ⊗ b)((x, y), (x′, y′)) = a(x, x′)⊗ b(y, y′); it is clear that, for each pair
of V-functors f : (X,a) → (Y, b) and g : (X′, a′) → (Y ′, b′), the map f × g :X × X′ →
Y × Y ′ becomes a V-functor between the corresponding V-categories.
This tensor product has a unit element, K = ({∗}, k), a singleton set with k(∗,∗) = k.
Moreover, it was shown in [10]:
1.5. Theorem. The category V-Cat equipped with the tensor product ⊗ is a closed cate-
gory.
The description of its right adjoint hom, constructed in [10], becomes very simple in
this context: for each pair (X,a) and (Y, b) of V-categories, hom((X,a), (Y, b)) has as
underlying set XY = {f : (X,a) → (Y, b); f is a V-functor}, with structure d defined by:
d(f,g) =
∧
x∈X
b
(
f (x), g(x)
)
.
1.6. In case ⊗ coincides with the categorical product, as it is the case when V = 2,
the category of V-categories is automatically Cartesian closed by the previous theorem.
Whenever ⊗ is not the categorical product, this result is no longer valid. For instance,
PMet, studied in detail in the last section (see 4.2), is not Cartesian closed.
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 3117Even in the case ⊗ is the categorical product, Cartesian closedness is not inherited
by the slice categories: for instance, POrd is not locally Cartesian closed (see [12] and
Theorem 4.1).
2. The category of reflexive V-graphs
2.1. If we drop the axiom (T) in the definition of V-category, we end up with the
category V-RGph of reflexive V-graphs and V-functors. That is, objects of V-RGph are
pairs (X,a), where X is a set and a :X × X → V is a map such that k a(x, x) for
every x ∈ X; morphisms f : (X,a) → (Y, b) in V-RGph are maps f :X → Y such that
a(x, x′) b(f (x), f (x′)), for all x, x′ ∈ X.
This category contains V-Cat as a full subcategory, and, moreover, as it was shown
in [3]:
2.2. Theorem. In the commutative diagram
V-Cat V-RGph
Set
the (full) embedding is reflective, with identity maps as reflections, and the forgetful func-
tors are topological.
In contrast with the situation in V-Cat, exponentiation in V-RGph is easily described,
based on exponentiation in Set and V, as developed in [5].
2.3. For the forthcoming study of exponentiation in V-Cat, the description of expo-
nentials in V-RGph is crucial. Our first observation in this direction is that the use of
partial products as introduced in [7] turns out to simplify this study. Next we summarize
the results needed for that.
2.4. Theorem. [11,7] For a morphism f :X → Y in a category C, the following conditions
are equivalent:
(i) f is exponentiable, i.e. f × − : C/Y → C/Y has a right adjoint;
(ii) the ‘change-of-base’ functor f ×Y − : C/Y → C/X has a right adjoint;
(iii) the ‘pullback’ functor X ×Y − : C/Y → C has a right adjoint;
(iv) C has partial products over f .
3118 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128We recall that C has partial products over f :X → Y if, for each object Z in C, there
is a diagram
Z X ×Y Pev π1
π2
X
f
P
p
Y
such that every diagram
Z X ×Y Qev′ π
′
1
π ′2
X
f
Q
q
Y
factors as p · t = q and ev · (1X × t) = ev′ by a unique morphism t :Q → P .
In [5] it is shown that, when V is a locally Cartesian category—which, in our situation
just means V is a Heyting algebra—V-RGph has partial products over every V-functor
f : (X,a) → (Y, b). In fact:
2.5. Theorem. [5] If V is a Heyting algebra, then V-RGph is a quasitopos.
Here we will sketch only the construction of partial products, since they will play a cru-
cial role in the subsequent study.
2.6. For each V-functor f : (X,a) → (Y, b) between reflexive V-graphs and for each
reflexive V-graph (Z, c), the partial product (P, d) of (Z, c) over f is defined as follows:
P = {(s, y); y ∈ Y and s : (f−1(y), a)→ (Z, c) is a V-functor}
(here a :f −1(y) × f−1(y) → V is just the restriction of a :X × X → V); to obtain the
structure d on P , for each (s, y), (s′, y′) ∈ P , x ∈ f−1(y) and x′ ∈ f −1(y′), we first form
in V the partial product v((s, x), (s′, x′)) of c(s(x), s′(x′)) over a(x, x′) b(y, y′); then
d((s, y), (s′, y′)) is the multiple pullback of the (v((s, x), (s′, x′)))x∈f −1(y),x′∈f−1(y′) in the
lattice ↓b(y, y′); that is
d
(
(s, y), (s′, y′)
)=
⎧⎪⎪⎨
⎪⎪⎩
∧
x∈f−1(y), x′∈f−1(y′)
v((s, x), (s′, x′)),
if f −1(y) = ∅ = f −1(y′),
b(y, y′), otherwise.
Since
v
(
(s, x), (s′, x′)
)=
∧
x∈f−1(y), x′∈f−1(y′)
∨{
v ∈ V; v b(y, y′) and
a(x, x′) ∧ v c(s(x), s′(x′))},
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 3119using the distributivity of binary meets over arbitrary joins, we obtain the following de-
scription of d((s, y), (s′, y′)), which is easier to handle is this context:
d
(
(s, y), (s′, y′)
)= max{v ∈ V; v b(y, y′) & ∀x ∈ f−1(y)∀x′ ∈ f −1(y′)
a(x, x′) ∧ v c(s(x), s′(x′))}.
3. Exponentiation in V-Cat
3.1. In order to characterize exponentiable morphisms, and consequently exponen-
tiable objects, in V-Cat, we first state some auxiliary results.
3.2. Lemma. The following conditions are equivalent:
(i) k is the top element of V;
(ii) for every u,v,w ∈ V,
(u ∧ v) ⊗ w (u ⊗ w) ∧ v.
3.3. Proposition. If V is a Heyting algebra, k is the top element of V, f : (X,a) → (Y, b) is
an exponentiable morphism in V-Cat and (Z, c) is a V-category, then the partial products
of (Z, c) over f in V-Cat and in V-RGph coincide.
Proof. Let
(Z, c) (X ×Y P, d˜)ev π1
π2
(X,a)
f
(P, d)
p
(Y, b)
and
(Z, c) (X ×Y P ′, d˜ ′)
π ′1ev′
π ′2
(X,a)
f
(P ′, d ′) p
′
(Y, b)
be the partial products of (Z, c) over f in V-RGph and V-Cat, respectively. Since V-Cat
is closed under pullbacks and the latter diagram lies in particular in V-RGph, there exists
a unique V-functor t : (P ′, d ′) → (P, d) such that p · t = p′ and ev · (1× t) = ev′. It is easy
to check that t :P ′ → P is a bijection, using the universal properties of both diagrams. We
therefore assume, for simplicity, that t is an identity map. To show it is an isomorphism, that
is, d = d ′, let (s0, y0), (s1, y1) ∈ P . The pair ({0,1}, e) with e(0,1) = d((s0, y0), (s1, y1))
and e(0,0) = e(1,1) = k is a V-category. Hence, the diagram in V-Cat
(Z, c) (X ×Y {0,1}, e˜)ev·ι (X,a)
f
({0,1}, e) p·ι (Y, b)
3120 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128where ι : ({0,1}, e) → (P, d) is the inclusion map, with ι(0) = (s0, y0) and ι(1) = (s1, y1),
induces a V-functor ({0,1}, e) → (P ′, d ′) by the universal property of (P ′, d ′). Hence
d((s0, y0), (s1, y1)) = e(0,1) d ′((s0, y0), (s1, y1)) and the conclusion follows.
3.4. Theorem. If V is a Heyting algebra and k is the top element of V, then a V-functor
f : (X,a) → (Y, b) is exponentiable in V-Cat if and only if, for each x0, x2 ∈ X, y1 ∈ Y
and for each v0, v1 ∈ V such that v0 b(f (x0), y1) and v1 b(y1, f (x2)),∨
x∈f−1(y1)
(
a(x0, x) ∧ v0
)⊗ (a(x, x2) ∧ v1
)
a(x0, x2) ∧ (v0 ⊗ v1). (∗)
Proof. We first show that condition (∗) above guarantees the exponentiability of f . More
precisely, we show that, for any V-category (Z, c), its partial product (P, d) over f , formed
in V-RGph, is a V-category; that is, it satisfies axiom (T):
∀(s0, y0), (s1, y1), (s2, y2) ∈ P
d
(
(s0, y0), (s1, y1)
)⊗ d((s1, y1), (s2, y2)
)
d
(
(s0, y0), (s2, y2)
)
.
Let u0 := d((s0, y0), (s1, y1)), u1 := d((s1, y1), (s2, y2)) and u := d((s0, y0), (s2, y2)). For
every xi ∈ f−1(yi), i = 0,1,2, we have
u0 b(y0, y1) and a(x0, x1) ∧ u0 c
(
s0(y0), s1(y1)
)
,
u1 b(y1, y2) and a(x1, x2) ∧ u1 c
(
s1(y1), s2(y2)
)
.
Hence, for every x0 ∈ f −1(y0) and x2 ∈ f −1(y2),
u0 ⊗ u1 b(y0, y1) ⊗ b(y1, y2) b(y0, y2), and
a(x0, x2) ∧ (u0 ⊗ u1)
∨
x∈f−1(y1)
(
a(x0, x) ∧ u0
)⊗ (a(x, x2) ∧ u1
)
∨
x∈f−1(y1)
c
(
s0(x0), s1(x)
)⊗ c(s1(x), s2(x2)
)
c
(
s0(x0), s2(x2)
)
.
Therefore, u0 ⊗ u1 u as claimed.
To show the necessity of the condition, we consider x0, x2 ∈ X, y0 = f (x0), y2 = f (x2),
y1 ∈ Y , v0, v1 ∈ V as in (∗), and we define a triple of maps
s0 :f
−1(y0) → V
x → a(x0, x),
s1 :f
−1(y1) → V
x → a(x0, x) ∧ v0,
s2 :f
−1(y2) → V
x →
∨
−1
(
a(x0, x1) ∧ v0
)⊗ (a(x1, x) ∧ v1
)
.x1∈f (y1)
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 3121The map s0 is a V-functor, since, for each x, x′ ∈ f −1(y0),
a(x0, x) ⊗ a(x, x′) a(x0, x′)
⇒ a(x, x′) hom(a(x0, x), a(x0, x′)
)= hom(s0(x), s0(x′)
)
.
In order to check that s1 is a V-functor, let x, x′ ∈ f−1(y1). Then
s1(x) ⊗ a(x, x′) =
(
a(x0, x) ∧ v0
)⊗ a(x, x′)
(
a(x0, x) ⊗ a(x, x′)
)∧ v0 a(x0, x′) ∧ v0 = s1(x′)
⇒ a(x, x′) hom(s1(x), s1(x′)
)
.
Finally we have to check that s2 is also a V-functor: for x, x′ ∈ f −1(y2),
s2(x) ⊗ a(x, x′) =
∨
x1∈f−1(y1)
(
a(x0, x1) ∧ v0
)⊗ (a(x1, x) ∧ v1
)⊗ a(x, x′)
∨
x1∈f−1(y1)
(
a(x0, x1) ∧ v0
)⊗ ((a(x1, x) ⊗ a(x, x′)
)∧ v1
)
∨
x1∈f−1(y1)
(
a(x0, x1) ∧ v0
)⊗ (a(x1, x′) ∧ v1
)
= s2(x′);
hence, a(x, x′) hom(s2(x), s2(x′)).
Finally, it is easy to check that
v0 d
(
(s0, y0), (s1, y1)
)
and v1 d
(
(s1, y1), (s2, y2)
)
,
since: v0 b(y0, y1) by hypothesis, and, for every x′0 ∈ f −1(y0), x′1 ∈ f −1(y1),
a(x′0, x′1) ∧ v0 hom
(
s0(x
′
0), s1(x
′
1)
)
follows from
s0(x
′
0) ⊗
(
a(x′0, x′1) ∧ v0
)= a(x0, x′0) ⊗
(
a(x′0, x′1) ∧ v0
)
(
a(x0, x
′
0) ⊗ a(x′0, x′1)
)∧ v0
a(x0, x′1) ∧ v0 = s1(x′1);
analogously, v1 b(y1, y2) by hypothesis, and, for each x′1 ∈ f −1(y1), x′2 ∈ f −1(y2),
a(x′1, x′2) ∧ v1 hom
(
s1(x
′
1), s2(x
′
2)
)
follows easily from the inequalities
s1(x
′
1) ⊗
(
a(x′1, x′2) ∧ v1
)= (a(x0, x′1) ∧ v0
)⊗ (a(x′1, x′2) ∧ v1
)
∨
x1∈f−1(y1)
(
a(x0, x1) ∧ v0
)⊗ (a(x1, x′2) ∧ v1
)= s2(x′2).
Therefore, since we are assuming that (P, d) is a V-category, we may conclude that
d
(
(s0, y0), (s2, y2)
)
v0 ⊗ v1,
3122 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128which means in particular that
a(x0, x2) ∧ (v0 ⊗ v1) hom
(
s0(x0), s2(x2)
)= hom(k, s2(x2)
)= s2(x2);
that is,
a(x0, x2) ∧ (v0 ⊗ v1)
∨
x1∈f−1(y1)
(
a(x0, x1) ∧ v0
)⊗ (a(x1, x2) ∧ v1
)
,
which completes the proof.
3.5. Corollary. If V is a Heyting algebra with k its top element, then a V-category (X,a)
is exponentiable in V-Cat if and only if
∀x0, x2 ∈ X ∀v0, v1 ∈ V∨
x∈X
(
a(x0, x) ∧ v0
)⊗ (a(x, x2) ∧ v1
)
a(x0, x2) ∧ (v0 ⊗ v1).
3.6. Corollary. If the tensor product ⊗ and the categorical product coincide in V, then a
V-functor f : (X,a) → (Y, b) is exponentiable in V-Cat if and only if
∀x0, x2 ∈ X ∀y1 ∈ Y∨
x∈f−1(y1)
(
a(x0, x) ∧ a(x, x2)
)
a(x0, x2) ∧ b
(
f (x0), y1
)∧ b(y1, f (x2)
)
.
Proof. If ⊗ = ∧, one has∨
x∈f−1(y1)
(
a(x0, x) ∧ v0
)⊗ (a(x, x2) ∧ v1
)
= v0 ∧ v1 ∧
∨
x∈f−1(y1)
(
a(x0, x) ∧ a(x, x2)
)
,
and
v0 ∧ v1 ∧
∨
x∈f−1(y1)
(
a(x0, x) ∧ a(x, x2)
)
a(x0, x2) ∧ v0 ∧ v1
⇐⇒
∨
x∈f−1(y1)
(
a(x0, x) ∧ a(x, x2)
)
a(x0, x2) ∧ v0 ∧ v1.
Finally it is clear that it is enough to check the latter inequality for v0 = b(f (x0), y1) and
v1 = b(y1, f (x2)).
3.7. Remark. The description of quotients obtained in [9] gives an alternative way of
showing that (∗) is sufficient for exponentiability. First, one can prove that a pullback
of a quotient map g :Z → Y along f is again a quotient map. Roughly speaking, (∗)
guarantees that, for x0, x2 in X, each zigzag in Z mapped to f (x0), f (x2) can be lifted in
an appropriate way to a zigzag in X ×Y Z mapped to x0, x2. Finally, it is easy to check
that property (∗) is pullback-stable, hence the change of base functor f ×Y − preserves
quotients, and therefore, by Freyd’s Adjoint Functor Theorem, it has a right adjoint.
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 31234. Examples
4.1. Exponentiation in POrd
We consider first exponentiation in V-Cat for V = 2, that is, in the category POrd
of preordered sets and monotone maps. In this category the condition of Corollary 3.6
trivializes except for b(f (x0), y1) = b(y1, f (x2)) = a(x0, x2) = true. Then it means that:
∨
x∈f−1(y1)
a(x0, x) ∧ a(x, x2) = true;
that is, there exists x ∈ X such that f (x) = y1 and x0 x x2. Therefore we have:
Theorem. [12] A monotone map f : (X,) → (Y,) between preordered sets is exponen-
tiable in POrd if and only if
∀x0, x2 ∈ X ∀y1 ∈ Y x0 x2 & f (x0) y1 f (x2)
⇒ ∃x1 ∈ X: f (x1) = y1 & x0 x1 x2:
f (x0) f (x2) y1
x0 x2
x1
We observe that, if f : (X,) → (Y,) is an embedding, this condition can be restated
as:
Corollary. An embedding f : (X,) → (Y,) is exponentiable in POrd if and only if
↓f (X) ∩ ↑f (X) = f (X).
Using the closure defined by ↑, this just means that f (X) is the intersection of an open
and a closed subset of Y , which resembles the characterization of exponentiable embed-
dings in Top as the locally closed embeddings (see [11, Corollary 2.7]).
4.2. Exponentiation in PMet
Now we consider V = R+ = ([0,∞],) the (complete) real half-line. As we have
already observed, the category V-Cat is the category PMet of premetric spaces and non-
expansive maps. Here condition (∗) of Theorem 3.4 may be also simplified, as we explain
next. In the sequel we use the natural order in the real numbers, which is the opposite of the
3124 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128categorical one, so that (∗) reads as: for each x0, x2 ∈ X, y1 ∈ Y , and for each v0, v1 ∈R+
such that v0 b(f (x0), y1) and v1 b(y1, f (x2)),
inf
x∈f−1(y1)
(
a(x0, x) ∨ v0
)+ (a(x, x2) ∨ v1
)
a(x0, x2) ∨ (v0 + v1).
We recall from Section 2 that, given a non-expansive map f : (X,a) → (Y, b) and a pre-
metric space (Z, c), the structure d in the partial product P is given by
d
(
(s, y), (s′, y′)
)= min{v ∈R+; v b(y, y′) & ∀x ∈ f−1(y) ∀x′ ∈ f −1(y′)
a(x, x′) ∨ v c(s(x), s′(x′))},
for every (s, y), (s′, y′) ∈ P .
Theorem. A non-expansive map f : (X,a) → (Y, b) between premetric spaces is expo-
nentiable in PMet if and only if, for each x0, x2 ∈ X, y1 ∈ Y and u0, u1 ∈ R+ such
that u0 b(f (x0), y1), u1 b(y1, f (x2)) and u0 + u1 = max{a(x0, x2), b(f (x0), y1) +
b(y1, f (x2))} < ∞,
∀ε > 0 ∃x1 ∈ f−1(y1): a(x0, x1) < u0 + ε and a(x1, x2) < u1 + ε. (∗∗)
Proof. First we show that condition (∗∗) is necessary for exponentiability. For u0, u1 as
above, Theorem 3.4 together with u0 + u1 a(x0, x2) assure that
inf
x∈f−1(y1)
(
a(x0, x) ∨ u0
)+ (a(x, x2) ∨ u1
)
u0 + u1,
and the conclusion follows.
Conversely, assume that condition (∗∗) is satisfied.
We first observe that, when max{a(x0, x2), b(f (x0), y1) + b(y1, f (x2))} = ∞, (∗) is
immediately verified.
Let v0, v1 ∈R+ be such that v0 b(f (x0), y1) and v1 b(y1, f (x2)).
If a(x0, x2) b(f (x0), y1)+b(y1, f (x2)), then necessarily u0 = b(f (x0), y1) and u1 =
b(y1, f (x2)) in (∗∗). We therefore have
∀ε > 0 ∃x1 ∈ f−1(y1): a(x0, x1) < u0 + ε v0 + ε and
a(x1, x2) < u1 + ε v1 + ε,
which implies that
inf
x∈f−1(y1)
(
a(x0, x) ∨ v0
)+ (a(x, x2) ∨ v1
)= v0 + v1 = a(x0, x2) ∨ (v0 + v1).
Consider now the case a(x0, x2) b(f (x0), y1)+ b(y1, f (x2)). If v0 + v1 a(x0, x2),
using (∗∗) for u0 v0 and u1 v1 such that u0 + u1 = a(x0, x2), we conclude that
inf
x∈f−1(y1)
(
a(x0, x) ∨ v0
)+ (a(x, x2) ∨ v1
)= v0 + v1 = a(x0, x2) ∨ (v0 + v1).
If v0 + v1 < a(x0, x2), let u0 = v0 and u1 = a(x0, x2) − v0 in (∗∗). Then
inf
x∈f−1(y1)
(
a(x0, x) ∨ v0
)+ (a(x, x2) ∨ v1
)
inf
x∈f−1(y1)
(
a(x0, x) ∨ u0
)+ (a(x, x2) ∨ u1
)
a(x0, x2) = a(x0, x2) ∨ (v0 + v1).
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 3125We observe that, defining, for a premetric space (X,a) and X′ ⊆ X,
↓S := {x ∈ X; ∃x′ ∈ X: d(x′, x) < ∞} and
↑S := {x ∈ X; ∃x′ ∈ X: d(x, x′) < ∞},
one has:
Corollary A. An embedding f : (X,a) → (Y, b) in PMet is exponentiable if and only if
↓f (X) ∩ ↑f (X) = f (X).
From the Theorem one derives also the following characterization of exponentiable
objects in PMet:
Corollary B. A premetric space (X,a) is exponentiable in PMet if and only if, for each
x0, x2 ∈ X and u0, u1 ∈R+ such that u0 + u1 = a(x0, x2),
∀ε > 0 ∃x1 ∈ X: a(x0, x1) < u0 + ε and a(x1, x2) < u1 + ε.
Using this characterization it is obvious that:
• Q+ with the hom structure (inherited from R+), and Q and Q+ with the usual Euclid-
ean metric, are exponentiable in PMet;
• finite premetric spaces with non-trivial premetric (i.e. having points whose distance
differs from 0 and ∞) are not exponentiable in PMet.
These examples give an interesting contrast with the situation in Top: with their induced
topology the former ones are not exponentiable in Top while the latter ones are.
4.3. Exponentiation in Met
We finally study the situation in the category Met of (symmetric, separated, with non-
infinite distances) metric spaces and non-expansive maps.
Proposition. Let (X,a), (Y, b) and (Z, c) be metric spaces and let f : (X,a) → (Y, b) be
a non-expansive map.
(a) If there exists the partial product (P, d) of (Z, c) over f in PMet, then the premetric
d is symmetric and separated.
(b) If the partial products of (Z, c) over f exist in both PMet and Met, then they coincide.
Proof. (a) From the description of d above, it is obvious that symmetry of d is inherited
by the symmetry of a, b and c. Moreover, d((s, y), (s′, y′)) = 0 means that b(y, y′) = 0,
hence y = y′, and, for every x, x′ ∈ f −1(y), a(x, x′) c(s(x), s′(x′)). In particular, for
x = x′ it follows that c(s(x), s′(x)) = 0, hence s = s′.
(b) Let (P, d) and (P ′, d ′) be the partial products of (Z, c) over f in PMet and
Met, respectively. To show that they coincide, it is enough to show that d cannot take
3126 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128the value ∞. By the universal property of (P, d), there exists a non-expansive bijection
t : (P ′, d ′) → (P, d). Hence, d ′ d and the result follows.
Theorem. A non-expansive map f : (X,a) → (Y, b) between metric spaces is exponen-
tiable in Met if and only if it satisfies condition (∗∗) and has bounded fibres.
Proof. We first show that boundedness of the fibres of f , together with (∗∗), is sufficient
for exponentiability.
By the proposition above, it remains to be shown that d does not take the value ∞. For
each (s, y), (s′, y′) ∈ P , boundedness of f −1(y) and f −1(y′) guarantees that s(f −1(y))
and s′(f −1(y′)) are bounded subsets of (Z, c), hence {c(s(x), s′(x′)); x ∈ f −1(y), x′ ∈
f−1(y′)} is bounded as well; let u(s, s′) be its supremum. Then d((s, y), (s′, y′))
b(y, y′) ∨ u(s, s′), hence d is a metric.
To show the reverse implication, we check first that boundedness of the fibres is neces-
sary for exponentiability of f . Let f be exponentiable. For any y ∈ Y , consider (Z, c) =
(f −1(y), a), the identity s : (Z, c) → (Z, c) and a constant map s′ : (Z, c) → (Z, c), taking
every point x into a fixed x0 ∈ f −1(y). Then
∞ > d((s, y), (s′, y))min{v; v b(y, y) & ∀x ∈ f −1(y)
a(x, x) ∨ v c(s(x), s′(x))}
= min{v; ∀x ∈ f−1(y) v a(x, x0)
}
,
hence f −1(y) is a bounded subset of X.
It remains to be shown that condition (∗∗) is necessary for exponentiability. First we
observe that (∗∗) is trivially verified when X = ∅. For X = ∅, we first check that, if f is
exponentiable in Met, then f is surjective. Assume that y1 ∈ Y \ f (X). Consider x0 ∈ X,
y0 = f (x0), and
v = sup{a(x, x′); x, x′ ∈ f−1(y0)
}
.
Then v < ∞ because f has bounded fibres. Let Z := {0,1}, with c(0,1) = max{v, b(y0,
y1)+ b(y1, y0)}+ 1. Define the constant maps s :f −1(y0) → Z, s′′ :f−1(y0) → Z, which
send every x to 0 and 1, respectively, and let s′ :f−1(y1) = ∅ → Z. Then it is easy to check
that
d
(
(s, y0), (s
′, y1)
)= b(y0, y1),
d
(
(s′, y1), (s′′, y0)
)= b(y1, y0),
d
(
(s, y0), (s
′′, y0)
)= c(0,1),
which shows that d is not transitive, since
d
(
(s, y0), (s
′′, y0)
)= c(0,1) > b(y0, y1) + b(y1, y0)
= d((s, y0), (s′, y1)
)+ d((s′, y1), (s′′, y0)
)
.
Finally the result follows from an easy adaptation of the argumentation of the proof
of Theorem 3.4, using the maps s0, s1, s2, replacing the hom structure in R+ by the usual
metric structure in [0,∞[. Continuity of these maps follows from the continuity of s0, s1, s2
M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128 3127of Theorem 3.4 since the metric structure is just the symmetrization of the hom structure
of R+. The only detail to check is that, in the definition of s2,
s2 :f
−1(y2) → [0,∞[
x → inf
x1∈f−1(y1)
(
a(x0, x1) ∨ v0
)+ (a(x1, x) ∨ v1
)
,
the meet is non-empty, but this follows immediately from the surjectivity of f .
In the particular case of metric spaces, we conclude directly the following
Corollary A. A metric space (X,a) is exponentiable in Met if and only if it is bounded
and, for each x0, x2 ∈ X and u0, u1 ∈R+ such that u0 + u1 = a(x2, x0):
∀ε > 0 ∃x1 ∈ X: a(x0, x1) < u0 + ε and a(x1, x2) < u1 + ε:
Metric spaces satisfying the condition above have been studied in different contexts: in
[1], as almost 3-hyperconvex metric spaces; in [2], under the name almost convex spaces,
where this condition is used in the study of topologies in the hyperspace of closed subsets
of X. In fact, exponentiability of f is equivalent to the “composition of” (closed) balls (see
[2, Proposition 4.1.4]), as stated below. We denote by Su(x0) (Su(x0)) the open (closed)
ball with center x0 and radius u.
Corollary B. For a bounded metric space (X,a), the following conditions are equivalent:
(i) (X,a) is exponentiable;
(ii) ∀x0 ∈ X ∀u,v ∈ ]0,∞[ Su(Sv(x0)) = Su+v(x0);
(iii) ∀x0 ∈ X ∀u,v ∈ ]0,∞[ Su(Sv(x0)) = Su+v(x0).
Finally, from the Theorem we obtain an interesting characterization of exponentiable
complete metric spaces.
Corollary C. For a complete metric space (X,a), the following conditions are equivalent:
(i) (X,a) is exponentiable;
(ii) (X,a) is bounded and totally convex, i.e.
∀x0, x2 ∈ X ∀u0, u1 ∈ [0,∞[ u0 + u1 = a(x0, x2)
⇒ ∃x1 ∈ X: a(x0, x1) = u0 & a(x1, x2) = u1.
3128 M.M. Clementino, D. Hofmann / Topology and its Applications 153 (2006) 3113–3128Acknowledgements
The authors acknowledge partial financial assistance by Centro de Matemática da Uni-
versidade de Coimbra/FCT and Unidade de Investigação e Desenvolvimento Matemática
e Aplicações da Universidade de Aveiro/FCT.
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