ClaesAGT.dvi


@
Applied General Topology

c© Universidad Politécnica de Valencia

Volume 9, No. 1, 2008

pp. 21-32

Exponentiality for the construct of affine sets

V. Claes

Abstract. The topological construct SSET of affine sets over the

two-point set S contains many interesting topological subconstructs

such as TOP, the construct of topological spaces, and CL, the con-

struct of closure spaces. For this category and its subconstructs carte-

sian closedness is studied. We first give a classification of the subcon-

structs of SSET according to their behaviour with respect to exponen-

tiality. We formulate sufficient conditions implying that a subconstruct

behaves similar to CL. On the other hand, we characterize a conglom-

erate of subconstructs with behaviour similar to TOP. Finally, we

construct the cartesian closed topological hull of SSET.

2000 AMS Classification: 54A05, 54C35, 18D15

Keywords: topological construct, affine space, cartesian closed category,
cartesian closed topological hull, exponential object

1. Introduction

The lack of natural function spaces in a topological construct that is not
cartesian closed, has long been recognized as an akward situation for various
applications in homotopy theory and topological algebra and for use in infi-
nite dimensional differential calculus. For references to the original sources one
might consult [22, 11, 12, 18, 21]. Topological constructs like TOP or the larger
construct CL and several others that are commonly used by topologists, how-
ever are not cartesian closed. For the construct TOP of topological spaces this
problem is extensively studied in the literature, see for example [17, 8, 4, 5, 19].
For the subconstruct CL of closure spaces, the author studied cartesian closed-
ness in [6]. To remedy these facts, topologists have applied various methods.
Either they dealt with the local problem, the description of exponential objects
and with the construction of cartesian closed subconstructs, or they looked for
larger cartesian closed constructs. A topological construct is cartesian closed if
and only if every object X is exponential in the sense that the functor X × −
preserves coproducts and quotients. The reason for TOP not being cartesian



22 V. Claes

closed is that X ×− does not always preserve quotients, except for corecompact
X. For CL it is just the other way around, X × − generally does not preserve
coproducts, except for X being indiscrete.

TOP as well as CL are fully embedded in the construct SSET of affine
spaces and affine maps which is a host for many other subconstructs that are
important to topologists [13, 14] (see next section for the exact definitions). In
this paper we investigate the problem of cartesian closedness for SSET and
we describe the exponential objects and deduce results for its subconstructs.
We prove that for a non-indiscrete affine space X, the functor X × − does not
preserve coproducts in SSET and we describe a conglomerate of subconstructs
of SSET (to which CL belongs), in which this negative result goes through. On
the other hand, we determine a large subconstruct of SSET in all topological
subconstructs of which (like for instance TOP) the functor X ×− does preserve
coproducts. In the final section of the paper we describe the cartesian closed
topological hull of SSET. Remark that, as was observed by E. Giuli [13], our
definition of affine spaces and maps, as we recall it in the next section, only
differs slightly from the normal Boolean Chu spaces and continuous maps, as
introduced by V. Pratt to model concurrent computation. Objects in SSET
have a structure containing constants. This assumption makes SSET into a
well-fibred topological construct in the sense of [1], which has the property that
cartesian closedness is equivalent to the existence of ”nice” function spaces.

2. Exponential objects in SSET and in its subconstructs.

An affine space X over the two point set S = {0, 1} is a structured set, where
the structure on the underlying set X is a collection of subsets of X. The sets
belonging to the structure are called the “open” sets of X. An affine map
from X −→ Y is a function f such that inverse images of open sets are open.
An affine space can be isomorphically described in a functional way: An affine
space (over S) (X, A) consists of a set X and a subset A of the powerset SX . An
affine map f : (X, A) → (Y, B) is a function f such that β ◦ f ∈ A for all β ∈ B.
In this paper we will use the functional description. We will restrict ourselves
to the affine spaces whose affine structure contains the constant functions 0 and
1. As in [13], the corresponding construct of affine spaces and affine maps will
be denoted by SSET. In that paper it was proved that SSET is a well-fibred
topological construct.

An object X in a category with finite products is exponential if the functor
X ×− has a right adjoint. In a well-fibred topological construct X, this notion
can be characterized as follows: X is exponential in X iff for each X-object
Y the set HomX(X, Y ) can be supplied with the structure of a X-object - a
function space or a power object Y X - such that

(1) the evaluation map ev: X × Y X → Y is a X-morphism
(2) for each X-object Z and each X-morphism f : X × Z → Y , the map

f ∗ : Z → Y X defined by f ∗(z)(x) = f (x, z) is a X-morphism.



Exponentiality for the affine sets 23

It is well known that in the setting of a topological construct X, an object X is
exponential in X iff X × − preserves final episinks [15], [16]. Moreover, small
fibredness of X ensures that this is equivalent to the condition that X ×− pre-
serves quotients and coproducts. A well-fibred topological construct X is said
to be cartesian closed (or to have function spaces) if every object is exponential.

Before characterizing the exponential objects in the subcategories of SSET,
we first prove some useful results for subconstructs of SSET.

We first recall the following result from [20].

Proposition 2.1. [20] Every topological subcategory X of SSET is a bicore-
flective subcategory of some full bireflective subcategory Y of SSET.

The following propositions can be proved using techniques similar to those
developed for TOP in [17].

Proposition 2.2. Every topological subcategory of SSET is closed under the

formation of retracts in SSET.

Proposition 2.3. Every non-trivial topological subcategory X of SSET con-

tains all complemented topological spaces.

In order to investigate the interaction of products and coproducts in SSET
and in its subconstructs, we first look at the construction of coproducts in
SSET and in its subconstructs.

Let (Xi, Ai)i∈I be a family of affine sets, then one can easily verify that the
coproduct in SSET of these objects is given by (∐Xi, A = { ⊔

i∈I
αi : αi ∈ Ai})

with

⊔
i∈I

αi : ∐
i∈I

Xi → S : (xi, i) → αi(xi)

Proposition 2.4. Let X be a non-trivial topological subconstruct of SSET

and (Xi, Ai)i∈I a family of X-objects. For every i ∈ I and every αi ∈ Ai,
the function ⊔

j∈I
βj belongs to the affine structure A∐Xi on the coproduct ∐

i∈I
Xi

whenever βi = αi and βj = 0 for all j 6= i or βj = 1 for all j 6= i.

Proof. Let Y be the bireflective subcategory of SSET such as in proposition
2.1 and let (Xi, Ai)i∈I be a family of X-objects. For r = (ri)i∈I ∈ Π

i∈I
Xi, we

define the function

fr : ∐
i∈I

Xi → Π
i∈I

Xi × I : (xi, i) → ((yj )j∈I , i)

with yi = xi and yj = rj for j 6= i. Let A be the initial affine structure for the
source (fr : ∐

i∈I
Xi → Π

i∈I
(Xi, Ai) × (I, S

I ))r∈ Π
i∈I

Xi . Then, A = A1 ∪ A2 with



24 V. Claes

A1 = {1J ◦ prI ◦ fr | J ⊂ I, r ∈ Π
i∈I

Xi}

= ∪
J⊂I

{ ⊔
i∈I

βi | βi = 1 if i ∈ J and βi = 0 if i /∈ J}

A2 = {αi ◦ prXi ◦ fr | i ∈ I, αi ∈ Ai, r ∈ Π
i∈I

Xi}

= ∪
i∈I

{ ⊔
j∈I

βj | βi = αi ∈ Ai and βj = 1 if j 6= i} ∪

∪
i∈I

{ ⊔
j∈I

βj | βi = αi ∈ Ai and βj = 0 if j 6= i}

From the previous proposition follows that the categories X and Y contain
the discrete affine sets, and in particular (I, SI ). Since Y is closed under the
formation of initial structures in SSET, it follows that ( ∐

i∈I
Xi, A) belongs to

Y. For all i ∈ I, the map ji : (Xi, Ai) → ( ∐
i∈I

Xi, A) is affine. Hence, ( ∐
i∈I

Xi, A)

is coarser than the coproduct ( ∐
i∈I

Xi, A∐YXi ) in the category Y. Since X is a

bicoreflective subcategory of Y, this implies that A ⊂ A∐XXi = A∐YXi . �

Hence, every non-trivial subcategory of SSET for which the affine structures
are closed under arbitrary suprema is closed under the formation of coproducts
in SSET.

We will now recall a general method to construct hereditary bicoreflective
subcategories of SSET [9], [10] ,[13].
In order to define a subconstruct of SSET we put an algebra structure on
S = {0, 1}. Recall that an algebra structure on the set S is a class of operations

Ω = {ωi : S
ni → S | i ∈ I}

of arbitrary arities. Hence the ni are arbitrary cardinal numbers, and there is
no condition on the size of the indexing system I. We assume that Ω contains
the constant operations. For every set X, by point-wise extension, the powerset
SX carries an algebra structure. We denote by SSET(Ω) the subconstruct of
SSET consisting of those affine sets (X, A) for which A is a Ω-subalgebra of the
function algebra SX . The objects in SSET(Ω) are called affine sets over the
algebra (S, Ω). For example the category CL of closure spaces and the category
TOP of topological spaces can be obtained this way. The construct obtained
this way, by considering for Ω the class containing the constant operations and
the complementation¯: S → S defined by (̄0) = 1,̄ (1) = 0 will be denoted by
SSET(C).

Lemma 2.5. SSET(Ω) is a subcategory of SSET(C) whenever Ω contains an
operation ωT : S

T → S that satisfies the following condition:
∃ (xt)t∈T , (yt)t∈T ∈ S

T such that ωT ((xt)t∈T ) = 0, ωT ((yt)t∈T ) = 1 and
xt = 0 implies yt = 0 for all t ∈ T .



Exponentiality for the affine sets 25

Proof. Let (X, A) be an SSET(Ω)-object. For α ∈ A, define the serie functions
(ft : X → S)t∈T as follows.

ft =







0 if xt = 0
1 if yt = 1
α if xt = 1, yt = 0

Since A contains all constant functions, we have that ft ∈ A for all t ∈ T .
Then, A contains the function ωT ◦ Π

t∈T
ft : X → S : x → ωT (ft(x))t∈T .

If α(x) = 1, then ωT ◦ Π
t∈T

ft(x) = ωT ((xt)t∈T ) = 0 =¯◦ α(x).

If α(x) = 0, then ωT ◦ Π
t∈T

ft(x) = ωT ((yt)t∈T ) = 1 =¯◦ α(x). We can conclude

that¯◦ α = ωT ◦ Π
t∈T

ft ∈ A and thus (X, A) is an SSET(C)-object. �

Lemma 2.6. If X is a non-trivial topological subcategory of SSET and D2
is the two point discrete space, then for every non-indiscrete object (X, A) the
following holds:

(X, A) × D2 ∈ X ⇒ (X, A) is not an exponential object in X

Proof. For a non-constant function α ∈ A, 0 ⊔ α is an element of AX⊔X , while
it is not contained in AX×D2 . �

The previous negative result has important consequences with respect to
exponential objects in SSET and to cartesian closedness of topological sub-
constructs.

Corollary 2.7. If X is a topological subconstruct of SSET which is finitely

productive in SSET, then the class of exponential objects in X coincides with

the class of indiscrete spaces.

We now characterize a conglomerate of subconstructs of SSET, which are
not finitely productive in SSET, in which the class of exponential objects also
coincides with the class of indiscrete objects.

Proposition 2.8. For every category SSET(Ω) that is not a subcategory of
SSET(min : S2 → S, max : S2 → S) the exponential objects are exactly the
indiscrete affine sets.

Proof. Suppose that SSET(Ω) has a non-indiscrete exponential object (X, A).
Then we have that (X, A)⊔(X, A) is isomorphic to (X, A)×D2. By proposition
2.4, it follows that 0 ⊔ α belongs to AX×D2 for every α ∈ A. The product
AX×D2 is the smallest Ω-subalgebra of S

X×D2 containing B = { α ◦ prX | α ∈
A} ∪ { prD2 , prD2 : (x, a) → a}.
Hence, 0⊔α = ωα((γ ◦prX )γ∈A, prD2 , prD2 ) with ωα : S

A∪S → S a composition
of algebraic operations of Ω. For a non-constant function α ∈ A, there exists
x, y ∈ X such that α(x) = 1 and α(y) = 0. Define (fγ : S × S → S)γ∈A as
follows:

• If γ(x) = γ(y), put fγ the constant function with value γ(x).
• If γ(x) = 1 and γ(y) = 0, put fγ = pr1 : S × S → S : (a, b) → a



26 V. Claes

• If γ(x) = 0 and γ(y) = 1, put fγ =¯ ◦ pr1 : S × S → S : (a, b) → a

Then, for b ∈ S we have:

• ωα ◦ ( Π
γ∈A

fγ × pr2 × pr2)(0, b) = ωα((γ(y))γ∈A, b, b̄) =

ωα((γ ◦ prX )γ∈A, prD2 , prD2 )(y, b) = 0 ⊔ α (y, b) = 0 = min(0, b).
• ωα ◦ ( Π

γ∈A
fγ × pr2 × pr2)(1, b) = ωα((γ(x))γ∈A, b, b̄) =

ωα((γ ◦ prX )γ∈A, prD2 , prD2 )(x, b) = 0 ⊔ α (x, b) = b = min(1, b).

This implies that min(a, b) = ωα ◦ ( Π
γ∈A

fγ × pr2 × pr2)(a, b), which means that

the operation min : S ×S → S can be written in terms of the operation ωα, the
complementation ¯ and the constant functions. If SSET(Ω) is a subcategory
of SSET(C), it now follows that SSET(Ω) is a subcategory of SSET(min :
S2 → S).
For the categories SSET(Ω) that are not embedded in SSET(C), it follows
from lemma 2.5 that:

(1) ωα((γ(x))γ∈A, 0, 0) = 0,
because ωα((γ(x))γ∈A, 0, 1) = 0 ⊔ α(x, 0) = 0

(2) ωα((0)γ∈A, 1, 0) = 0,
because ωα((γ(y))γ∈A, 1, 0) = 0 ⊔ α(y, 1) = 0

(3) ωα((0)γ∈A, 0, 0) = 0

By defining fγ = pr1 : S × S → S : (a, b) → a if γ(x) = 1 and otherwise fγ = 0,
we have that ωα ◦ ( Π

γ∈A
fγ × pr2 × 0)(a, b) = min(a, b). Indeed, for b ∈ S we

have

• ωα ◦ ( Π
γ∈A

fγ × pr2 × 0)(0, b) = ωα((0)γ∈A, b, 0) = 0

• ωα ◦ ( Π
γ∈A

fγ × pr2 × 0)(1, b) = ωα((γ(x))γ∈A, b, 0) = b

Where the last equation follows from previous observation (1) and the fact that
ωα((γ(x))γ∈A, 1, 0) = 0 ⊔ α(x, 1) = 1

So in each category SSET(Ω) which has an non-indiscrete exponential ob-
ject, A is closed under finite minima for every object (X, A). It can be proved
in a similar way that A is closed under finite maxima. One can easily prove
that the indiscrete affine sets are exponential. So we can conclude that the
exponential objects of SSET(Ω) are exactly the indiscrete affine sets. �

Thus for the categories SSET, CL and SSET(C) the exponential objects are
exactly the indiscrete objects. From the proof of previous proposition follows
that in all the categories SSET(Ω) which are not embedded in SSET(min :
S2 → S, max : S2 → S), the functor X × − does not preserve coproducts for
non-indiscrete objects X. In SSET(min : S2 → S, max : S2 → S) itself, the
functor X × − preserves coproducts for some non-indiscrete objects X. In the
following proposition these objects are characterized.



Exponentiality for the affine sets 27

Proposition 2.9. In the construct SSET(min : S2 → S, max : S2 → S), we
have that the functor (X, A)×− preserves coproducts if and only if A is a finite
set.

Proof. It can be easily verified that for (X, A), with A a finite set, (X, A) ×
∐

i∈I
(Yi, Bi) is isomorphic to ∐

i∈I
(X, A)×(Yi, Bi) for every collection SSET(min :

S2 → S, max : S2 → S)-objects (Yi, Bi)i∈I .
Suppose now that the functor (X, A) × − preserves coproducts, then we have
that (X, A) × (A, SA) is isomorphic to ∐

α∈A
(X, A). We consider the function

⊔
α∈A

α : ∐
α∈A

(X, A) → S : (x, α) → α(x)

Since SSET(min : S2 → S, max : S2 → S) is a bicoreflective subcategory
of SSET, this function ⊔

α∈A
α belongs to A ∐

α∈A
(X,A) and thus ⊔

α∈A
α belongs

to AX×(A,SA). The product AX×(A,SA) is the smallest subalgebra containing

B = { α ◦ prX | α ∈ A} ∪ { f ◦ prA | f ∈ S
A}. Hence, there exists a finite

set I and for every i ∈ I there exist αi ∈ A, fi ∈ S
A such that ⊔

α∈A
α =

max
i∈I

min(αi ◦ prX , fi ◦ prA). For β ∈ A, set Iβ = {i ∈ I | fi(β) = 1}. For every

x ∈ X, we have: β(x) = ⊔
α∈A

α (x, β) = max
i∈I

min(αi(x), fi(β)) = max
i∈Iβ

αi(x).

Since the set I is finite, this implies that A also shall be finite. �

3. Subconstructs in which the functor X × − preserves
coproducts

The categories considered in the previous section fail to be cartesian closed
because the functor X ×− does not preserve coproducts. From the last propo-
sition, it follows that the condtion, SSET(Ω) is a subcategory of SSET(min :
S2 → S, max : S2 → S), is not a sufficient condition such that the functor
X × − preserves coproducts for all objects X. In this section we formulate a
sufficient condition. It is known [17] that in TOP and in all its topological
subconstructs the functor X × − preserves coproducts for all objects X. We
generalize these results to a larger subconstruct of SSET.

Definition 3.1. Let D be the full subcategory of SSET with objects all affine

sets (X, A) that satisfy the following two conditions.

(D1) α ∈ A, β ∈ A ⇒ min(α, β) ∈ A
(D2) {αi | i ∈ I} ⊂ A and min(αi, αj ) = 0 for each i 6= j ⇒ max

i∈I
αi ∈ A

It is clear that TOP is a subcategory of this category D. For a collection
B ⊂ SX we can define a D-structure A on X as follows. Let C consist of
all finite minima of elements of B ∪ {0, 1}. By adding to C the maxima of
collections functions (αi)i∈I of C that satisfy the condition min(αi, αj ) = 0 for
each i 6= j, we get a D-structure A. Moreover, A is the smallest D-structure
on X containing B. B is called the subbase of A and C the base of A.



28 V. Claes

Proposition 3.2. D is a topological category.

Proof. For a family of functions (fi : X → (Xi, Ai))i∈I with (Xi, Ai) ∈ D the
D-structure A generated by the subbase B = {αi ◦ fi | αi ∈ Ai, i ∈ I} is the
unique initial structure on X for the given source. �

Proposition 3.3. D is a bicoreflective subcategory of SSET

Proof. For an affine set (X, A) the bicoreflection is given by

1X : (X, A
′) → (X, A)

with A′ the D-structure generated by the subbase A. �

Remark that D is not a hereditary subcategory of SSET as follows from
the next example.

Example 3.4. Let X = {0, 1, 2, 3} and A = {0, 1, 1{1,3}, 1{2,3}, 1{3}}, then
(X, A) is a D-object. Then (Y, A|Y ) = ({0, 1, 2}, {0, 1, 1{1}, 1{2}}) is the
SSET-subspace of (X, A) with underlying set {0, 1, 2}. min(1{1}, 1{2}) = 0
and max(1{1}, 1{2}) = 1{1,2} /∈ A|Y , so (Y, A|Y ) does not belong to the category
D.

Hence, there is no algebraic structure Ω on S such that D has the form
SSET(Ω).

Proposition 3.5. If SSET(Ω) is a subcategory of D, then it is a subcategory
of TOP or a subcategory of SSET(C).

Proof. For an arbitrary set I, take ∞ /∈ I and define for every i ∈ I the function
αi : I ∪ {∞} → S with αi(i) = 1 and αi(x) = 0 for x 6= i.
Let A be the smallest Ω-subalgebra of SI∪{∞} containing {αi | i ∈ I}. Then,
(I ∪ {∞}, A) is an SSET(Ω)-object.
Since SSET(Ω) is a subcategory of D, we have that max

i∈I
αi belongs to A.

Hence, max
i∈I

αi = ωI (αi)i∈I , with ωI : S
I → S a composition of algebraic

operations of Ω. This gives the following information about ωI :

• ∀j ∈ I : ωI (αi(j))i∈I = max(αi(j))i∈I = 1
• ωI (0)i∈I = ωI (αi(∞)i∈I ) = max(αi(∞)i∈I ) = max(0)i∈I = 0

Now two cases can arise:

(1) ∀x 6= (0)i∈I ∈ S
I : ωI (x) = 1.

In this case ωI = max
i∈I

(2) There exists a x 6= (0)i∈I ∈ S
I such that ωI (x) = 0. Choose j ∈ I

such that prj (x) 6= 0 and define (yi)i∈I ∈ S
I with yj = 1 and yi = 0

for i 6= j. Then we have ωI (x) = 0, ωI (yi)i∈I = ωI (αi(j))i∈I = 1
and pri(x) = 0 implies yi = 0. It then follows from lemma 2.5 that
SSET(Ω) is a subcategory of SSET(C).

We can conclude that either for every set I, ωI = max
i∈I

and thus SSET(Ω) is a

subcategory of TOP or SSET(Ω) is a subcategory of SSET(C). �



Exponentiality for the affine sets 29

It can be verified that coproducts are universal in D, i.e. coproducts are
preserved under pullbacks along arbitrary morphisms. From proposition 2.4
and the condition (D2) follows that D and its subconstructs are closed under
the formation of coproducts in SSET. Combining this with 2.2, the following
theorem can be proved.

Theorem 3.6. In every topological subcategory of D coproducts are preserved

by the functor X × −.

Corollary 3.7. The exponential objects of a subcategory of D are the objects

X for which the functor X × − preserves quotients.

4. Cartesian closed topological hull of SSET

In section 2, we proved that for finitely productive subcategories of SSET,
products do not distribute over coproducts. If we want to work in a cartesian
closed construct in which products are formed similarly to the ones in SSET,
we have to consider a larger scope. We will look for cartesian closed topological
constructs that are larger than SSET and in which SSET is finally densely
embedded. We know from [7] that quotients in SSET are productive. In fact we
have the same situation as for the category CL [6]. For CL a cartesian closed
extension was constructed in [6] using the method presented by J. Adámek
and J. Reiterman in [2]. We first look if this method is also applicable to
SSET.

Definition 4.1. For affine sets (X, A) and (Y, B) we consider the collection of
functions N = {Γβ|β ∈ B} on Hom(X, Y ) with Γβ : Hom(X, Y ) → S defined
by Γβ(f ) = 1 iff β ◦ f = 1.

Analogous to CL, we can prove the following result for this structure on the
Hom-sets of SSET.

Proposition 4.2. If M ⊆ Hom(X, Y ) is a subset endowed with an affine
structure M such that the evaluation map ev: (X, A) × (M, M) → (Y, B) is
an affine map, then the following conditions hold:

(1) N|M ⊆ M
(2) ev: X × (M, N|M ) → Y is affine.

This shows that SSET is a type of category as considered in 4.3 of [2].
Consider the following superconstruct K of SSET. Objects of K are triples
(X, A, A) where X is a set, A is a cover of X such that

U ′ ⊆ U, U ∈ A ⇒ U ′ ∈ A

and A is an affine structure on X which is A-final in the sense that (i :
(U, A|U ) → (X, A))U∈A is final in SSET.
The members of A are called generating sets. A morphism in K,

f : (X, A, A) → (Y, B, B)

is a function that is affine (with respect to (X, A) and (Y, B)) and preserves
the generating sets: U ∈ A ⇒ f (U ) ∈ B.



30 V. Claes

SSET is fully embedded in K by identifying (X, A) with (X, P(X), A).
By the general theorem in [2] it follows that K is a cartesian closed topological
category in which SSET is finally densely embedded.
A cartesian closed well-fibred topological construct Y is called a cartesian closed
topological hull (CCT hull) of a construct X if Y is a finally dense extension
of X with the property that any finally dense embedding of X into a cartesian
closed topological construct can be uniquely extended to Y. Starting from the
cartesian closed extension K of SSET, we will now construct the cartesian
closed topological hull of SSET. Here again we will work as we did for CL. In
particular, we apply the construction of the cartesian closed hull, using power-
closed collections, described by J. Adámek, J. Reiterman and G.E. Strecker in
II.2 and II.3 in [3]. We first recall from [3] some definitions and the construction
of the CCT-hull applied to categories with productive quotients.

Definition 4.3. Let X be a construct and let H, K be X-objects and X a set.
A function h : X × H → K is called a multimorphism if for each x ∈ X,
h(x, −) : H → K defined by h(x, −)(y) = h(x, y) is a morphism.

Definition 4.4. Let X be a well-fibred topological construct. A collection C of
objects (A, U) with A ⊆ X is said to be power-closed in a set X provided that
C contains each object (A0, U0) with A0 ⊆ X with the following property:
Given a multimorphism h : X × H → K such that for each (A, U) ∈ C
the restriction h|A : (A, U) × H → K is a morphism, then the restriction
h|A0 : (A0, U0) × H → K is also a morphism.
We denote by PC(K) the category of power-closed collections. Objects are pairs
(X, C), where X is a set and C is a power-closed collection in X. Morphisms
f : (X, C) → (Y, D) are functions from X to Y such that for each (A, U) ∈ C
the final object of the restriction fA : (A, U) → f (A) is in D.

Theorem 4.5. [3] Any well-fibred topological construct with productive quo-
tients has a CCT hull. Moreover, this hull is precisely the category of power-

closed collections.

Next we define a suitable subconstruct of the cartesian closed extension K
of SSET.

Definition 4.6. Let K∗ be the full subconstruct of K whose objects are the

K-objects (X, A, A) that satisfy the following condition:
If V ⊂ X /∈ A, then there exists a set Z ⊆ X with V ∩ Z 6= ∅, V 6⊂ Z and such
that: ∀U ∈ A : U ∩ Z = ∅ or U ⊆ Z.

In order to prove that K∗ is the CCT hull of SSET we establish an isomor-
phism between K∗ and the category of power-closed collections of SSET.

Proposition 4.7. For each object (X, A, A) of K∗ the collection of affine
spaces CX = {(U, B) | U ∈ A, A|U ⊆ B} is power-closed.



Exponentiality for the affine sets 31

Proof. If (X0, A0) is an affine set with X0 ⊆ X and (X0, A0) /∈ CX , then
either X0 /∈ A or A|X0 6⊆ A0. If A0 is not finer than A|X0 , there exists an
α ∈ A such that α|X0 /∈ A0. For an arbitrary affine space H, the function
α ◦ prX : X × H → S with S the Sierpinski space is a multimorphism. For
(U, B) ∈ CX the restriction h|U = α|U ◦ prU : (U, B) × H → S is affine and the
restriction h|X0 = α|X0 ◦ prX0 : (X0, A0) × H → S is not affine.
If X0 /∈ A, then there exists a subset Z of X, not containing X0 and intersecting
X0 such that for all U ∈ A we have that U ∩ Z = ∅ or U ⊆ Z.
Take an affine set H that has a non-constant function γ ∈ AH . The function
h = min(1Z ◦ prX , γ ◦ prH ) : X × H → S is a multimorphism. For (U, B) ∈ CX
the restriction h|U is either the constant function 0 or γ ◦ prH . Hence, the
restrictions h|U : (U, B)×H → S are affine for all (U, B) ∈ CX . Since X0∩Z 6= ∅
and X0 6⊆ Z, we have that h|X0 6= α ◦ prH and h|X0 6= β ◦ prX0 for all α ∈ AH
and β ∈ AX0 . Therefore the restriction h|X0 : (X0, A0) × H → S is not
affine. �

Proposition 4.8. If C is a power-closed collection of SSET-objects in a set
X then there exists a unique K∗-object (X, A, A) such that C = CX .

Proof. For a power-closed collection C of SSET-objects in X, we can prove
analogously to CL [6] that (X, A, A) where A = {U ⊆ X | (U, B) ∈ C for someB}
and A the final structure determined by the sink of inclusion maps (i : (U, B) →
X)(U,B)∈C is a K

∗-object such that C = CX . �

Theorem 4.9. K∗ is the CCT hull of SSET.

Proof. It follows from the two previous propositions that the functor F : K∗ →

PC(SSET) defined by F (X
f
→ X′) = (X, CX )

f
→ (X′, CX′ ) is bijective on ob-

jects. In a similar way as for CL in [6], one can prove that F is an isomorphism.
From 4.5 it then follows that K∗ is the CCT-hull of SSET. �

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Received July 2006

Accepted December 2006

Veerle Claes (vclaes@vub.ac.be)
WISK-IR, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussel, Belgium