Theory and Applications of Categories, Vol. 8, No. 2, pp. 16–32.

EXPONENTIABLE MORPHISMS: POSETS, SPACES, LOCALES, AND GROTHENDIECK TOPOSES

SUSAN NIEFIELD

ABSTRACT. In this paper, we consider those morphisms p: P ⟶ B of posets for which the induced geometric morphism of presheaf toposes is exponentiable in the category of Grothendieck toposes. In particular, we show that a necessary condition is that the induced map p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in the category of topological spaces, where P^{↓} is the space whose points are elements of P and open sets are downward closed subsets of P. Along the way, we show that p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable if and only if p: P ⟶ B is exponentiable in the category of posets and satisfies an additional compactness condition. The criteria for exponentiability of morphisms of posets is related to (but weaker than) the factorization-lifting property for exponentiability of morphisms in the category of small categories (considered independently by Giraud [G] and Conduché [C]).

1. Introduction

Let A be a category with finite limits. Recall that an object X of A is called exponentiable if the functor − × X: A ⟶ A has a right adjoint (denoted by ( )^{X}). The category A is called cartesian closed if every object is exponentiable, and it is called locally cartesian closed if the slice categories A/T are cartesian closed, for all objects T of A, where A/T is the category whose objects are morphisms p: X ⟶ T of A and morphisms are commutative triangles in A. We shall say that a morphism p: X ⟶ T of A is exponentiable if p is exponentiable in A/T, or equivalently, if the pullback functor p^{∗}: A/T ⟶ A/T has a right adjoint. Note that we shall follow the customary abuse of notation and write − ×_{T} X for − × p, when the morphism p is unambiguous.

Exponentiability has been studied in many categories including

Top topological spaces and continuous maps

Cat small categories and functors

Pos partially-ordered sets and order-preserving maps

Loc locales and morphisms of locales

GTop Grothendieck toposes and geometric morphisms

Unif uniform spaces and uniformly continuous maps

Received by the editors 2000 January 13 and, in revised form, 2001 January 11. Transmitted by Peter Johnstone. Published on 2001 January 29. 2000 Mathematics Subject Classification: 06D22, 06B35, 18B25, 18B30, 18D15, 54C35, 54D45. Key words and phrases: presheaf topos, poset, locale, exponentiable, factorization lifting, metastably

locally compact, discrete opfibration.

\begin{matrix}c \\ ◯ \end{matrix} Susan Niefield, 2001. Permission to copy for private use granted.

Aff affine schemes and morphism of affine schemes

This paper concerns the first five categories listed above. For the latter two, we refer the reader to [N3].

We began by asking to what extent the presheaf functor

PSh: Cat ⟶ GTop

preserves and reflects exponentiability. This question arose when the author was working with Marta Bunge on [BN]. Since it is well-known that this functor preserves products and exponentiable objects, though not equalizers (see [J1]), we did not expect that exponentiable morphisms would be preserved, but we made little progress towards an answer. Subsequently, the author realized that this problem might be solved by first restricting to partially-ordered sets, solving the problem for the presheaf functor Pos ⟶ GTop, and later generalizing the results to Cat. The advantage of working with posets would be that one could exploit the equivalence between the category PSh(P) of presheaves on P and the category Sh(P^{↓}) of sheaves on the space P^{↓} whose points are elements of P and open sets are downward closed subsets, and thus make use of known results for topological spaces.

First, we consider some of the relevant history, partly to provide a context for the reader, but also to recall those results which will be used in this paper.

Interest in exponentiability of objects in non-cartesian closed categories is related to the study of suitable topologies on function spaces, for if X is exponentiable in Top, then taking Y = 1 in the natural bijection

Top(Y × X, Z) ≅ Top(Y, Z^{X})

it is easy to see that Z^{X} can be identified with the set Top(X, Z) of continuous maps from X to Z. Perhaps the first definitive result in this area appeared in the 1945 paper “On topologies for function spaces” [F] by R. H. Fox, where he clearly stated the problem as follows:

Given topological spaces X, T, and Y and a function h from X × T to Y which is continuous in x for each fixed t, there is associated a function h^{∗} from T to F = Y^{X}, the space whose elements are continuous functions from X to Y. … It would be desirable to so topologize F that the functions h^{∗} which are continuous are precisely those which correspond to continuous h. It has been known for a long time that this is possible if X satisfies certain conditions, chief among which is the condition of local compactness.

and showed that a separable metrizable space is exponentiable if and only if it is locally compact.

The characterization of exponentiable spaces was finally achieved by Day and Kelly in their 1970 paper “On topological quotients preserved by pullback or products” [DK], where they proved that, for a space X, the functor

− × X: Top ⟶ Top

preserves quotients if and only if the lattice O(X) of open sets of X is (what is now known as) a continuous lattice, in the sense of Scott [S], i.e., O(X) satisfies V = ⋁{U|U << V }, where U << V if every open cover of V has a finite subfamily that covers U. Since −×X preserves coproducts in any case, preservation of quotients (or equivalently, coequalizers) is necessary and sufficient for exponentiability in Top, by Freyd’s Special Adjoint Functor Theorem [F]. Day and Kelly also showed that continuity of O(X) coincides with local compactness of X for Hausdorff spaces. For a thorough treatment of exponentiability in Top and related function space problems see Isbell’s article [I] on the influence of the Day and Kelly paper.

For non-Hausdorff spaces, there were two nonequivalent definitions of local compactness under consideration in the 1970’s. One hypothesized the existence of a compact neighborhood of each point (see [K]) while the other required arbitrarily small such neighborhoods (see [M]). In 1978, Hofmann and Lawson [HL] showed that the sober spaces satisfying the latter definition are those for which O(X) is continuous, and hence, precisely the sober spaces which are exponentiable in Top. Thus, the second definition seemed to be the appropriate one for non-Hausdorff spaces. Then in 1982, Eilenberg [E] proposed that continuity of O(X) be taken to be the definition of local compactness for a general space X. For this to make sense, he suggested that we need only rename the relation << by saying “U is compact in V ” in place of “U is way-below in V.” Thus, we could say X is called locally compact if every open set V can be covered by open sets U such that U is compact in V, and we have generalized the definition of local compactness in such a way that it coincides precisely with exponentiability. It is this definition of local compactness (for spaces and locales) that we shall follow in this paper.

In the decade following the Day/Kelly paper, the relationship between exponentiability and local compactness was generalized to locales and toposes. In [H], Hyland showed that a locale is exponentiable in Loc if and only if it is locally compact. Then Johnstone and Joyal generalized this result to Grothendieck toposes. They began with the definition of continuous posets based on the Hofmann and Stralka [HS] characterization of continuous lattices as those for which the map ⋁: Idl(A) ⟶ A has a left adjoint. Using Grothendieck’s [Gr] notion of the ind-completion Ind(E) of a locally small category E, they defined E to be a continuous category if it has filtered colimits and the functor \underrightarrow{lim}: Ind(E) ⟶ E has a left adjoint. They then proved that E is exponentiable in GTop if and only if it is a continuous category. In addition, they showed that for a locale X, the topos Sh(X) of sheaves on X is exponentiable in GTop if and only if X is metastably locally compact. This is a property implying local compactness but using a strengthening of << in its place.

The results of [DK] were also relativized by Niefield in [N3], where the characterization of exponentiable spaces was generalized to Top/T. As corollaries, it was shown that if X is locally compact and T is Hausdorff, then any continuous map X ⟶ T is exponentiable in Top, and if X is a subspace of T, then the inclusion X ⟶ T is exponentiable in Top if and only if X is locally closed (i.e., the intersection of an open and a closed subspace of T). The latter was extended to Loc and GTop in [N2]. The results of [N3] were also used by the author in [N4] to show that for sober spaces X and T (in which points of T are locally closed), a continuous map p: X ⟶ T is exponentiable in Top if and only if O(p): O(X) ⟶ O(T) is exponentiable in Loc if and only if p_{∗}(Ω_{X}) is locally compact as an internal locale in Sh(T), where p_{∗}: Sh(X) ⟶ Sh(T) is the direct image of the geometric morphism induced by p.

There has also been considerable interest in exponentiability in Cat. It has long been known that Cat is cartesian closed but not locally cartesian closed. The exponentiable morphisms of Cat were characterized by Giraud [Gi] (and later rediscovered by Conduché [C]) as those satisfying a certain factorization lifting property. In [BN], Bunge and Niefield introduced the notion of a locally closed subcategory and showed that the inclusion of a full subcategory has this factorization lifting property if and only if it is locally closed if and only if the corresponding geometric morphism of presheaf toposes is locally closed.

Thus, we know that the presheaf functor Pos ⟶ GTop preserves exponentiable objects (since every poset P is exponentiable as is every presheaf topos PSh(P)) and exponentiable inclusions (since they are locally closed in both cases). The idea in the general case is to proceed as follows. Suppose p: P ⟶ B is an order-preserving map for which the induced geometric morphism p: PSh(P) ⟶ PSh(B) is exponentiable in GTop. Then using the equivalence PSh(P) ≃ Sh(P^{↓}), we know that p: Sh(P^{↓}) ⟶ Sh(B^{↓}) is exponentiable as well, and since the Johnstone/Joyal work [JJ] is constructive and hence applies to the relative case, it follows that p_{∗}(Ω_{P^{↓}}) is locally compact in the category Loc(Sh(B^{↓})) of internal locales in Sh(B^{↓}). But, Loc(Sh(B^{↓})) ≃ Loc/O(B^{↓}) (see [JT] or [J2]), and so using the results from [N4], it follows that p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top. Thus, we are led to considering the effect of the functor ( )^{↓}: Pos ⟶ Top on exponentiability.

We begin, in §2, by showing that a weakened version of the Giraud-Conduché characterization of exponentiable morphisms in Cat can be used for the category Pos. In §3, we review the exponentiability criteria from [N3] for morphisms of Top and show that it applies to T_{0}-spaces, as well. We then use these results to show, in §4, that p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top if and only if p: P ⟶ B is exponentiable in Pos and satisfies an additional compactness condition. In §5, we adapt these results to include locales, in the case where B^{↓} is a sober space. We conclude, in §6, by showing that these conditions on p: P ⟶ B are necessary (but not sufficient) for the exponentiability of the corresponding geometric morphism PSh(P) ⟶ PSh(B) in GTop. As a consequence, we give an example of a discrete opfibration p: C ⟶ B for which the corresponding geometric morphism of presheaf toposes is not exponentiable in GTop, thus providing a counterexample to a “theorem” appearing in a preliminary version of [BN]. We also show that the presheaf functor Cat ⟶ GTop neither preserves or nor reflects exponentiable morphisms.

2. Exponentiability in Pos

In this section, we characterize exponentiable morphism of the category of posets, and show that they include those which are exponentiable in Cat. Our proofs here are similar to those of Giraud [Gi] but we include them here for the sake of completeness, and since they are simpler than those for Cat.

We begin with some notation. For x ≤ y in a poset P, let [x, y] denote the interval {z ∈ P |x ≤ z ≤ y}. Given p: P ⟶ B in Pos and b ∈ B, the fiber of P over b is the subposet P_{b} = p^{−1}(b) of P. For A ⊆ P, we shall write A_{b} for A ∩ P_{b}.

2.1. Definition. A poset morphism p: P ⟶ B is called an interpolation-lifting map if [x, y]_{b} ≠ ∅, for all x ≤ y in P and b ∈ [px, py].

Note that this says that given x ≤ y in P and px ≤ b ≤ py in B, the following diagram can be completed:

P  x ≤ z ≤ y
↓ | | |
B  px ≤ b ≤ py

Thus, this is a consequence of the Giraud-Conduché factorization lifting property characterizing exponentiability in Cat, but there is no assumption of the connectedness of [x, y]_{b} here.

2.2. Theorem. A morphism p: P ⟶ B is exponentiable in Pos if and only if it is an interpolation-lifting map.

Proof. Suppose p: P ⟶ B is an interpolation-lifting map. Given q: Q ⟶ B, let Q^{P} denote the set of pairs (σ, b), where b ∈ B and σ: P_{b} ⟶ Q_{b} is a morphism of posets. Define (σ, b) ≤ (σ′, b′) if b ≤ b′ and σx ≤ σ′x′, for all (x, x′) ∈ P_{b} × P_{b′} such that x ≤ x′ in P. Then ≤ is clearly reflexive and antisymmetric. For transitivity, suppose (σ, b) ≤ (σ′, b′) and (σ′, b′) ≤ (σ′′, b′′). Then b ≤ b′′ since b ≤ b′ and b′ ≤ b′′. Given (x, x′′) ∈ P_{b} × P_{b′′} such that x ≤ x′′, since px ≤ b ≤ px′ and p is an interpolation-lifting map, there exists x′ ∈ P_{b′} such that x ≤ x′ ≤ x′′. Then σx ≤ σ′′x′′, since σx ≤ σ′x′ and σx′ ≤ σ′′x′′, and it follows that (σ, b) ≤ (σ′′, b′′). Therefore, Q^{P} is a poset and the projection map Q^{P} ⟶ B is clearly order-preserving. Moreover, one easily checks that ( )^{P} is a functor and the natural maps ϵ: Q^{P} ×_{B} P ⟶ Q, ϵ((σ, b), x) = σx, and η: Q ⟶ (Q ×_{B} P)^{P}, η(y) = ((y, −), qy), are order-preserving and satisfy the adjunction identities. Therefore, p: P ⟶ B is exponentiable in Pos.

Conversely, suppose that p: P ⟶ B is exponentiable in Pos. To show that p is an interpolation-lifting map, suppose px ≤ b ≤ py, where x ≤ y in P. Then the morphism 3 ⟶ B induced by px ≤ b ≤ py, gives rise to a pushout in Pos/B of the form

[figure: B; 2; 3; 1; 2]

where 1 = {0}, 2 = {0, 1}, and 3 = {0, 1, 2} are the linearly ordered one, two, and three element posets, respectively. Since − ×_{B} P preserves pushouts (being a left adjoint), it follows that the corresponding diagram

[figure: 2 ×_{B} P; 3 ×_{B} P; 1 ×_{B} P; 2 ×_{B} P]

is a pushout in Pos. It is not difficult to show that this pushout can be identified with the set

Q = P_{px≤b} ⋃_{P_{b}} P_{py≤b}

where P_{px≤b} and P_{py≤b} correspond to the two copies of 2 ×_{B} P in the pushout above, and the order on Q is the smallest partial order containing that of P_{px≤b} and P_{py≤b}. Thus, for u ∈ P_{px} and v ∈ P_{py}, we have u ≤ v if there exists w ∈ P_{b} such that u ≤ w and w ≤ v. Since Q ≅ 3 ×_{B} P and (0, x) ≤ (2, y) in 3 ×_{B} P, there exists (1, z) in 3 ×_{B} P such that (0, x) ≤ (1, z) ≤ (2, y), and it follows that p: P ⟶ B is an interpolation-lifting map, as desired.

3. Exponentiability in Top

In this section, we recall the characterization of exponentiable morphisms of Top presented in [N1, N3] and show that it also applies to the category Top_{0} of T_{0}-spaces. The latter will be of interest in §4 since P^{↓} is T_{0} for every poset P. The proof for Top_{0} is essentially the same as that of Top appearing in [N3]. We will include just enough here to show that the relevant spaces are T_{0}.

Recall that if X is a topological space, then H ⊆ O(X) is called Scott-open if it is upward closed, i.e., U ∈ H and U ⊆ V implies V ∈ H, and it satisfies the finite union property, {U_{α}}_{α∈A} ⊆ O(X) and ⋃_{α∈A} U_{α} ∈ H implies ⋃_{α∈F} U_{α} ∈ H, for some finite F ⊆ A.

For a continuous map p: X ⟶ T of spaces, families H ⊆ ⋃_{t∈T} O(X_{t}) of open subsets of fibers are of interest. To simplify notation, we write ⋂ H for the set whose fiber over t ∈ T is given by

(⋂ H)_{t} = ⋂ H_{t} = ⋂{V_{t}|V_{t} ∈ H_{t}}

3.1. Theorem. The following are equivalent for p: X ⟶ T in Top_{0}:

(a) p is exponentiable in Top.

(b) p is exponentiable in Top_{0}.

(c) Given x ∈ U_{px} ∈ O(X_{px}), there exists H ⊆ ⋃_{t∈T} O(X_{t}) such that (i) U_{px} ∈ H_{px}

(ii) H_{t} is Scott-open, for all t ∈ T

(iii) {t ∈ T |V_{t} ∈ H_{t}} is open in T, for all V ∈ O(X)

(iv) ⋂ H is a neighborhood of x in X.

Proof. The proof of (a) ⇔ (c) can be found in [N3] but is similar to that of (b) ⇔ (c) presented below.

Suppose p satisfies (c). For a continuous map q: Z ⟶ T of T_{0}-spaces, define Z^{X} as follows. The points of Z^{X} are pairs (σ, t), where σ: X_{t} ⟶ Z_{t} is a continuous map and t ∈ T. The topology on Z^{X} is generated by sets of the form

⟨H, W ⟩ = {(σ, t)|σ^{−1}(W_{t}) ∈ H_{t}}

where H ⊆ ⋃_{t∈T} O(X_{t}) satisfies (ii) and (iii) above and W is open in Z. Now, the projection π: Z^{X} ⟶ T is easily seen to be continuous. To see that Z^{X} is a T_{0}-space, suppose (σ, t) ≠ (σ′, t′) in Z^{X}. If t ≠ t′, then there is an open set G of T containing one but not both of t and t′, and so π^{−1}(G) is an open set of Z^{X} containing one but not both of (σ, t) and (σ′, t′). If t = t′, then σ ≠ σ′, and so σx ≠ σ′x for some x ∈ X_{t}. Since Z is T_{0}, there is an open set W of Z containing one but not both of σx and σ′x. Without loss of generality, assume σx ∈ W and σ′x ∉ W. Taking x ∈ U_{t} = σ^{−1}(W_{t}), we get H satisfying (i)-(iv) above. Then (σ, t) ∈ ⟨H, W ⟩, since U_{t} ∈ H_{t}, and (σ′, t′) ∉ ⟨H, W ⟩, for otherwise (σ′)^{−1}(W_{t}) ∈ H_{t} and so x ∈ ⋂ H_{t} ⊆ (σ′)^{−1}(W_{t}) contradicting that σ′x ∉ W. A straightforward calculation shows that ϵ: Z^{X} ×_{T} X ⟶ Z, ϵ ((σ, t), x) = σx and η: Z ⟶ (Z ×_{T} X)^{X}, η(z) = ((z, −), qz) are continuous, and that the adjunction identities hold. Note that the continuity of ϵ uses condition (c) but that of η holds in any case.

Conversely, suppose p is exponentiable in Top_{0}. Proceeding as in [N3], we consider the projection T ×2 ⟶ T, where 2 = {0, 1} is the Sierpinski space with {0} open but not {1}, and identify (T × 2)^{X} with ⋃_{t∈T} O(X_{t}). Then the counit ϵ: (T × 2)^{X} ×_{T} X ⟶ T × 2 is given by

ϵ(U_{t}, x) = \{ \begin{matrix}(t, 0) & if x ∈ U_{t} \\ (t, 1) & if x ∉ U_{t} \end{matrix}

under this identification. To prove (c), suppose x ∈ U_{px} ∈ O(X_{px}). Then by continuity of ϵ, there exists an open set H ⊆ ⋃_{t∈T} O(X_{t}) and an open set V ⊆ X such that (U_{px}, x) ∈ H ×_{T} V and ϵ(H ×_{T} V) ⊆ T × {0}. One easily sees that H satisfies (i) and (iv). The proofs of (ii) and (iii) in [N3] make use of continuous maps Z ⟶ (T × 2)^{X}, for certain continuous maps q: Z ⟶ T. To see that this proof works for Top_{0}, it suffices to show that the relevant spaces Z are T_{0}. But, they are T, T × 2, and the spaces Â defined as follows. Let A be any infinite set, and consider the space Â whose points are finite subsets of A together with A itself, and open sets are U ⊆ Â such that A ∉ U, or A ∈ U and ↑ F ⊆ U, for some finite F ⊆ A. Now, T and T × 2 are clearly T_{0} since T is. To see that Â is T_{0}, consider F ≠ G in Â. If both are finite subsets of A, then U = {F } is an open subset of Â containing F but not G. If only one is finite, say F, then G = A, and there exists a ∈ A such that a ∉ F. Then ↑ {a} is an open set of Â containing G but not F, and so Â is T_{0}, as desired.

Note that the proof of (b) ⇔ (c) can be adapted to the category Sob of sober spaces since T, T × 2, and the spaces Â can easily seen to be sober when T is sober. For Â, one shows that the irreducible closed sets are of the form {A} and {F }, where F ⊆ A is finite. Thus, we see that if p: X ⟶ T is exponentiable in Sob, then it is exponentiable in Top. With a further assumption on T, we can do even better [N4]. In particular, we need not assume that X is sober, only that p̃: X̃ ⟶ T is exponentiable in Sob to get that p: X ⟶ T is exponentiable in Top, where : Top ⟶ Sob is the reflection of Top in Sob, i.e., the left adjoint to the inclusion Sob ⟶ Top.

Recall that T is called a T_{D}-space if points of T are locally closed. Note that for such a space, the inclusion {t} ⟶ T is exponentiable in Top, for all t ∈ T, since locally closed inclusions are exponentiable [N3].

3.2. Corollary. If T is a sober T_{D}-space and p̃: X̃ ⟶ T is exponentiable in Sob, then p: X ⟶ T is exponentiable in Top.

Proof. By the above remarks, we know that p̃: X̃ ⟶ T is exponentiable in Top, and hence satisfies (c) of Theorem 3.1. To see that p is exponentiable, we will show that p: X ⟶ T also satisfies (c). Since the points of T are contained in those of X̃ and O(X) ≅ O(X̃), it suffices to show that O(X_{t}) ≅ O(X̃_{t}), for all t ∈ T. Note that does not preserve pullbacks so the canonical map O(X_{t}) ⟶ O(X̃_{t}) need not be an isomorphism, in general. To see that it is, in this case, it suffices to show that the exponentials (T × 2)^{t} in Top/T are sober spaces, for then we have natural isomorphisms

Top/T (X_{t}, T × 2) ≅ Top/T (X ×_{T} t, T × 2)
≅ Top/T (X, (T × 2)^{t})
≅ Top/T (X̃, (T × 2)^{t})
≅ Top/T (X̃ ×_{T} t, T × 2)
≅ Top/T (X̃_{t}, T × 2)

and the desired result follows.

To see that (T × 2)^{t} is sober, first note that it can be identified with T ∪ {⋆}, where ⋆ is an additional point over t. We will show that F is an irreducible closed subset of this space if (1) F is an irreducible closed subset of T and ⋆ ∉ F, or (2) F = Cl_{T} (t) ∪ {⋆}, where Cl_{T} denotes the closure in T. In the first case, F = Cl_{T ∪{⋆}}(u), where u is the unique point of T such that F = Cl_{T} (u). In the second case, F = Cl_{T ∪{⋆}}(⋆). Note that uniqueness is clear in both cases.

Using the description of the topology on (T × 2)^{t} from the proof of Theorem 3.1, we see that closed sets F of T ∪ {⋆} can be described as follows. If ⋆ ∈ F, then t ∈ F, since the fibers of the complement of F must be Scott-open by (ii), and F \ {⋆} must be closed in T by (iii). If ⋆ ∉ F, then F must be closed in T, by (iii). Thus, closed sets F of T ∪{⋆} satisfy (1) F is closed in T and ⋆ ∉ F, or (2) F = E ∪ {⋆}, where E is closed in T. In the first case, F is clearly irreducible in T ∪ {⋆} if and only if it is irreducible in T. In the second case, since E ∪ {⋆} = E ∪ (Cl_{T} (t) ∪ {⋆}), it will be irreducible if and only if E = Cl_{T} (t), and so F is of the desired form.

In [N4], it was shown that the converse of this corollary also holds but the method of proof involved internal locales in the topos of sheaves on T, and so will be delayed until we consider locales in §5.

4. Posets and Spaces

In this section, we consider the extent to which the functor ( )^{↓}: Pos ⟶ Top preserves and reflects exponentiability of morphisms. It is well-known that it preserves exponentiable objects. Indeed, every poset P is exponentiable, since Pos is cartesian closed, and P^{↓} is exponentiable in Top since it is easily seen to be locally compact.

4.1. Lemma. If p: P ⟶ B and q: Q ⟶ B are in Pos, then the induced map (P ×_{B} Q)^{↓} ⟶ P^{↓} ×_{B^{↓}} Q^{↓} is an isomorphism of spaces.

Proof. Since the map is clearly a bijection, it suffices to show that it is an open map. Suppose W is open in (P ×_{B} Q)^{↓}. Then W is downward closed, and it easily following that ↓x ×_{B^{↓}} ↓y ⊆ W, for all (x, y) ∈ P^{↓} ×_{B^{↓}} Q^{↓}, and so W is open in P^{↓} ×_{B^{↓}} Q^{↓}.

Let X be a topological space and x, y ∈ X. Define x ≤ y, if y ∈ U ⇒ x ∈ U, for all U open in X. Then ≤ is clearly reflexive and transitive, and it is antisymmetric if X is a T_{0}-space. Note that ≤ is the opposite of the specialization order on X (in the sense of [J3]). Thus, X becomes a poset, which we denote by X^{•}. Moreover, if f: X ⟶ Z, then f^{•}: X^{•} ⟶ Z^{•} is easily seen to be order-preserving.

4.2. Lemma. This construction defines a functor ( )^{•}: Top_{0} ⟶ Pos which is a right inverse right adjoint to ( )^{↓}: Pos ⟶ Top_{0}.

Proof. A straightforward calculation shows that (P^{↓})^{•} = P, for all posets P, and that the identity function ϵ: (X^{•})^{↓} ⟶ X is continuous, and so ( )^{•} is right inverse right adjoint to ( )^{↓} with counit ϵ and unit given by the identity function η: P ⟶ (P^{↓})^{•}.

We shall see below that exponentiability of p: P ⟶ B in Pos is necessary but not sufficient for the exponentiability of p^{↓}: P^{↓} ⟶ B^{↓} in Top. An additional assumption of p will be needed.

4.3. Definition. A poset morphism p: P ⟶ B is called hereditarily compact if ↓x ∩ p^{−1}(↓b) is compact in P^{↓}, for all x ∈ P and b ≤ px.

4.4. Theorem. The following are equivalent for p: P ⟶ B in Pos:

(a) p: P ⟶ B is an interpolation-lifting map which is hereditarily compact.

(b) p: P ⟶ B is exponentiable in Pos and hereditarily compact.

(c) p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top.

(d) p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top_{0}.

Proof. Since (a) ⇔ (b) by Theorem 2.2 and (c) ⇔ (d) by Theorem 3.1, it suffices to show that (d) ⇒ (b) and (a) ⇒ (c).

For (d) ⇒ (b), suppose p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top_{0}. Then

Pos/B (Q ×_{B} P, R) ≅ Top_{0}/B^{↓} ((Q ×_{B} P)^{↓}, R^{↓})
≅ Top_{0}/B^{↓} (Q^{↓} ×_{B^{↓}} P^{↓}, R^{↓})
≅ Top_{0}/B^{↓} (Q^{↓}, (R^{↓})^{P^{↓}})
≅ Pos/(B^{↓})^{•} (Q, ((R^{↓})^{P^{↓}})^{•})
≅ Pos/B (Q, ((R^{↓})^{P^{↓}})^{•})

where all the bijections ≅ are natural, and the first one holds since ( )^{↓} is full and faithful, the second by Lemma 4.1, the third by the exponentiability of P^{↓}, the fourth since ( )^{↓} is left adjoint to ( )^{•}, and the fifth since (B^{↓})^{•} = B. Therefore, p: P ⟶ B is exponentiable in Pos.

To see that p: P ⟶ B is hereditarily compact, suppose b ≤ px and consider

x ∈ (↓x)_{px} ∈ O(P_{px}^{↓})

By Theorem 3.1, there exists H ⊆ ⋃_{t∈B^{↓}} O(P_{t}^{↓}) such that

(i) (↓x)_{px} ∈ H_{px}

(ii) H_{t} is Scott-open, for all t ∈ B^{↓}

(iii) {t ∈ B|V_{t} ∈ H_{t}} is open in B^{↓}, for all V ∈ O(X)

(iv) ⋂ H is a neighborhood of x in P^{↓}

Since (↓x)_{px} ∈ H_{px} and b ≤ px, we know (↓x)_{b} ∈ H_{b} by (iii). Since (↓x)_{b} = ⋃_{y∈(↓x)_{b}}(↓y)_{b}, applying (ii) we see that

(↓y_{1})_{b} ∪ … ∪ (↓y_{n})_{b} ∈ H_{b}

for some y_{1}, …, y_{n} ∈ (↓x)_{b}. We claim that

↓x ∩ p^{−1}(↓b) = (↓y_{1} ∪ … ∪ ↓y_{n}) ∩ p^{−1}(↓b)

which is compact in P^{↓}, since y_{1}, …, y_{n} ∈ p^{−1}(↓b). Since y_{1}, …, y_{n} ≤ x, we know that

(↓y_{1} ∪ … ∪ ↓y_{n}) ∩ p^{−1}(↓b) ⊆ ↓x ∩ p^{−1}(↓b)

Since (↓y_{1} ∪ … ∪ ↓y_{n})_{b} ∈ H_{b}, and open sets are downward closed in B^{↓}, applying (iii) we see that (↓y_{1} ∪ … ∪ ↓y_{n})_{t} ∈ H_{t}, for all t ≤ b in B. Since ⋂ H is a neighborhood of x in P^{↓} by (iv), we know that ↓x ⊆ ⋂ H, and so (↓x)_{t} ⊆ (⋂ H)_{t} ⊆ (↓y_{1} ∪ … ∪ ↓y_{n})_{t}, for all t ≤ b. Therefore, ↓x ∩ p^{−1}(↓b) ⊆ (↓y_{1} ∪ … ∪ ↓y_{n}) ∩ p^{−1}(↓b), as desired.

For (a) ⇒ (c), suppose p: P ⟶ B is an interpolation-lifting map which is hereditarily compact. To show that the map p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top, we will show that it satisfies Theorem 3.1(c). Let x ∈ U_{px} ∈ O(P_{px}^{↓}), and consider

H = \{ \begin{matrix}\{W_{t} ∈ O(P_{t}^{↓})|(↓x)_{t} ⊆ W_{t}\} & if t ≤ px \\ ∅ & otherwise \end{matrix}

Then clearly H satisfies (i) as well as the upward closure part of (ii), and ⋂ H is a neighborhood of x in P^{↓}, since ↓x ∩ p^{−1}(↓px) ⊆ ⋂ H by definition of H.

To see that H satisfies the finite union part of (ii), suppose ⋃_{α∈A}(U_{α})_{t} ∈ H_{t}, for some t ∈ B. Then t ≤ px and (↓x)_{t} ⊆ ⋃_{α∈A}(U_{α})_{t}. We claim that ↓x ∩ p^{−1}(↓t) ⊆ ⋃_{α∈A} U_{α} ∩ p^{−1}(↓t). Suppose y ∈ ↓x ∩ p^{−1}(↓t). Then since p is an interpolation-lifting map, y ≤ x, and py ≤ t ≤ px, there exists z ∈ P_{t} such that y ≤ z ≤ x. Since z ∈ (↓x)_{t}, we know z ∈ ⋃_{α∈A}(U_{α})_{t}, and since ⋃_{α∈A} U_{α} is downward closed, it follows that y ∈ ⋃_{α∈A} U_{α} ∩ p^{−1}(↓t), as desired. Now, since ↓x ∩ p^{−1}(↓t) is compact, it follows that ↓x ∩ p^{−1}(↓t) ⊆ ⋃_{α∈F} U_{α} ∩ p^{−1}(↓t), for some finite F ⊆ A, and so ⋃_{α∈F} (U_{α})_{t} ∈ H_{t}, as desired.

It remains to show that (iii) holds. Consider G = {t ∈ B|V_{t} ∈ H_{t}}, where V is open in P^{↓}. To show that G is open, suppose t ∈ G and u ≤ t. Then V_{t} ∈ H_{t}, and so t ≤ px and (↓x)_{t} ⊆ V_{t}. We claim that (↓x)_{u} ⊆ V_{u}. Let y ∈ (↓x)_{u}. Then y ≤ x and py = u ≤ t ≤ px, and so, since p is an interpolation-lifting map, we know t = pz, were y ≤ z ≤ x. Then z ∈ V (since z ∈ (↓x)_{t} ⊆ V_{t}) and so y ∈ V (since V is open in P^{↓}). Therefore, (↓x)_{u} ⊆ V_{u}, and it follows that u ∈ G, making G an open set, to complete the proof.

We conclude this section with an example of an exponentiable morphism of Pos whose corresponding continuous map is not exponentiable in Top.

4.5. Example. Let B = 2 and P = {x, y_{1}, y_{2}, …}, where y_{i} ≤ x, for all i, and there are no other comparable elements, and define p: P ⟶ B by px = 1 and py_{i} = 0, for all i. Then p is exponentiable in Pos, but p^{↓}: P^{↓} ⟶ B^{↓} is not exponentiable in Top.

Proof. Clearly, p is an interpolation-lifting map (it is a discrete opfibration) but ↓x ∩ p^{−1}(↓0) = {y_{1}, y_{2}, …} is not compact. Therefore, p^{↓}: P^{↓} ⟶ B^{↓} is not exponentiable in Top, by Theorem 4.4.

5. Posets and Locales

The main purpose of this section is to consider the extent to which

Pos \underrightarrow{( )^{↓}} Top \underrightarrow{O} Loc

preserves and reflects exponentiable morphisms. As with ( )^{↓}, it preserves exponentiable objects, since P^{↓} is locally compact, for every poset P.

In [N4], it was shown that if T is a sober T_{D}-space and p: X ⟶ T is in Top, then the conditions of Theorem 3.1 are equivalent to the exponentiability of p̃: X̃ ⟶ T in Sob, as well as the corresponding morphism O(p): O(X) ⟶ O(T) in Loc. The latter is obtained using the equivalence (described in [JT]) of Loc/O(T) with the category Loc(Sh(T)) of internal locales in Sh(T).

Now, B^{↓} is a T_{D}-space, for every poset B, and so if we assume that it is sober, then we can combine the results of Theorem 4.4 with this extended version of Theorem 3.1 to determine the relationship between exponentiability in Pos and Loc. Although this combination requires no further justification, we shall recall part of the approach in [N4] and present a simpler proof of Theorem 5.2 (a) ⇒ (e) in the posetal case below.

The following two results appeared in [N4]. We omit the proof of the first as it would be identical to that of [N4] but we include the second as it is a slight variation.

5.1. Lemma. If p: X ⟶ T and q: Y ⟶ T are morphisms of sober spaces such that O(p): O(X) ⟶ O(T) is exponentiable in Loc, then the induced morphism O(Y ×_{T} X) ⟶ O(Y) ×_{O(T)} O(X) is an isomorphism in Loc.

5.2. Proposition. If p: X ⟶ T is a morphism of sober spaces such that O(p): O(X) ⟶ O(T) is exponentiable in Loc, then p: X ⟶ T is exponentiable in Sob.

Proof. Suppose that q: Y ⟶ T and r: Z ⟶ T are morphisms of sober spaces. Then we have natural bijections

Sob/T (Y ×_{T} X, Z) ≅ Loc/O(T)(O(Y ×_{T} X), O(Z))
≅ Loc/O(T)(O(Y) ×_{O(T)} O(X), O(Z))
≅ Loc/O(T)(O(Y), O(Z)^{O(X)})
≅ Sob/T (Y, pt(O(Z)^{O(X)}))

where pt is the right adjoint to O: Sob ⟶ Loc.

Recall that if p: E ⟶ B is a localic geometric morphism of toposes, then p_{∗}(Ω) is an internal locale in B and E is equivalent to the topos of sheaves on the locale p_{∗}(Ω). In particular, if p: X ⟶ T is a continuous map of spaces and p: Sh(X) ⟶ Sh(T) is the corresponding geometric morphism, then p_{∗}(Ω_{X}) is the internal locale in Sh(T) whose value at an open set G of T is the locale O(p^{−1}(G)), and for G′ ⊆ G, the restriction map O(p^{−1}(G)) ⟶ O(p^{−1}(G′)) is given by V |_{G′} = V ∩ p^{−1}(G′) (see [JT] or [J2]).

Following [J2] and [J4], local compactness in Sh(T) can by described as follows. Recall that for an ideal I of p_{∗}(Ω_{X}) defined over an open set G of T, we have

⋁ I = ⋃{U|U ∈ I(G′), for some G′ ⊆ G}

Thus, to show that p_{∗}(Ω_{X}) is locally compact it suffices to show that for all x ∈ V ∈ O(X), there exist open neighborhoods U and G of x and px, respectively, such that G ⊆ [[U << V]], where [[ϕ]] denotes the truth-value of the formula ϕ in the topos Sh(T). To show that G ⊆ [[U << V]], it suffices to show that for all ideals I defined over an open set G′ ⊆ G of T,

G′ ⊆ [[V ⊆ ⋁ I]] ⇒ G′ ⊆ [[U ∈ I]]

or equivalently

V ∩ p^{−1}(G′) ⊆ ⋁ I ⇒ U ∩ p^{−1}(G′) ∈ I(G′)

5.3. Theorem. The following are equivalent for p: P ⟶ B in Pos such that B^{↓} is a sober space:

(a) p: P ⟶ B is an interpolation-lifting map which is hereditarily compact.

(b) p: P ⟶ B is exponentiable in Pos and hereditarily compact.

(c) p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top.

(d) p^{↓}: P^{↓} ⟶ B^{↓} is exponentiable in Top_{0}.

(e) p_{∗}(Ω_{P^{↓}}) is locally compact in Loc(Sh(B^{↓})).

(f) p_{∗}(Ω_{P^{↓}}) is exponentiable in Loc(Sh(B^{↓})).

(g) O(p^{↓}): O(P^{↓}) ⟶ O(B^{↓}) is exponentiable in Loc.

(h) p̃^{↓}: P̃^{↓} ⟶ B^{↓} is exponentiable in Sob.

Proof. Note that (a) – (d) are equivalent by Theorem 4.4, (e) ⇒ (f) by Hyland’s theorem [H], (f) ⇒ (g) follows from Loc(Sh(B^{↓})) ≃ Loc/O(B^{↓}), (g) ⇒ (h) is a consequence of applying Proposition 5.2 to p̃^{↓}: P̃^{↓} ⟶ B^{↓}, and (h) ⇒ (c) holds by Corollary 3.2. Thus, it suffices to prove (a) ⇒ (e).

Suppose p: P ⟶ B satisfies (a). Then by the above description of local compactness, it suffices to show that if x ∈ V ∈ O(P^{↓}), then

↓px ⊆ [[↓x << V]]

So, suppose that V ∩ p^{−1}(G) ⊆ ⋁ I, for some ideal I defined over an open set G ⊆ ↓px. To see that ↓x∩p^{−1}(G) ∈ I(G), it suffices to show that ↓x∩p^{−1}(↓b) ∈ I(↓b), for all b ∈ G.

Since ↓x ∩ p^{−1}(↓b) is compact in P^{↓}, we know it is of the form ↓y_{1} ∪ … ∪ ↓y_{n}, for some y_{1}, …, y_{n} ∈ ↓x ∩ p^{−1}(↓b). We claim that we can assume py_{i} = b, for all i. Since p: P ⟶ B is an interpolation-lifting map and py_{i} ≤ b ≤ px, we know there exists z_{i} ∈ P such that pz_{i} = b and y_{i} ≤ z_{i} ≤ x, for all i, and so replacing x_{i} by z_{i} gives the desired result.

Since x ∈ V ⊆ ⋁ I = ⋃{U|U ∈ I(G′), for some G′ ⊆ ↓b} and y_{i} ≤ x, we know y_{i} ∈ U_{i}, for some G′_{i} ⊆ ↓b and U_{i} ∈ I(G′_{i}). But, then b ∈ G′_{i} since y_{i} ∈ U_{i} and py_{i} = b, and so G′_{i} = ↓b. Since I is an ideal, it follows that ↓y_{1} ∪ … ∪ ↓y_{n} ∈ I(↓b), and so ↓x ∩ p^{−1}(↓b) ∈ I(↓b), as desired. Therefore, ↓px ⊆ [[↓x << V]], and so p_{∗}(Ω_{P^{↓}}) is locally compact in Loc(Sh(B^{↓})).

6. Posets and Toposes

In this section, we consider the extent to which the presheaf functor

PSh: Pos ⟶ GTop

preserves and reflects exponentiable morphisms. Although we are able to give only a partial answer, we shall use these results to fully answer the analogous question for the presheaf functor defined on Cat.

If P is a poset, then it is not difficult to show that PSh(P) is a localic topos whose subobjects of 1 correspond to downward closed subsets of P, and so it follows that PSh(P) is equivalent to the topos Sh(P^{↓}) of sheaves on P^{↓}. Moreover, if p: P ⟶ B is a morphism of posets, then the geometric morphism PSh(P) ⟶ PSh(B) is identified with the geometric morphism Sh(P^{↓}) ⟶ Sh(B^{↓}) via this equivalence. Since the latter is a localic geometric morphisms, we are led to consider exponentiability of localic toposes.

In [JJ], Johnstone and Joyal showed that a localic topos Sh(A) is exponentiable if and only if A is metastably locally compact, a condition that implies (but is stronger than) the local compactness of A. They also showed that every stably locally compact locale (in the sense of [J3]) satisfies this property. Note that although these results were proved for Grothendieck toposes, the authors remark that they hold for bounded toposes over any topos with a natural number object, and hence, they apply to GTop/Sh(T), for any space T.

Using this description of exponentiable geometric morphisms, we will see, in the following example, that the presheaf functor Cat ⟶ GTop does not reflect exponentiable morphisms.

6.1. Example. Let P = 2 × 2, B = 3, and p: P ⟶ B be given by p(x, y) = x + y. Then the induced geometric morphism PSh(P) ⟶ PSh(B) is exponentiable in GTop but p is not exponentiable in Cat.

Proof. First, p is not exponentiable in Cat since it does not satisfy the connectedness condition in the Giraud-Conduché lifting property. From the proof of (h) ⇒ (a) in Theorem 5.3, wee see that ↓x << ↓x in p_{∗}(Ω_{P^{↓}}|_{↓px}), for all x ∈ P, and it follows that p_{∗}(Ω_{P^{↓}}) is stably locally compact in Sh(T), and so PSh(P) ⟶ PSh(T) is exponentiable in GTop.

6.2. Theorem. Suppose B^{↓} is a sober space and p: P ⟶ B is a morphism of posets such that the geometric morphism p: PSh(P) ⟶ PSh(B) is exponentiable in GTop. Then p: P ⟶ B satisfies the equivalent conditions of Theorem 5.3. In particular, p: P ⟶ B is exponentiable in Pos and hereditarily compact.

Proof. By the above remarks, we know that p_{∗}(Ω_{P^{↓}}) is locally compact in Loc(Sh(B^{↓})), and the desired result follows.

6.3. Corollary. The presheaf functor Pos ⟶ GTop reflects exponentiable morphisms.

6.4. Corollary. There is a discrete opfibration p: C ⟶ B such that the geometric morphism p: PSh(C) ⟶ PSh(B) is not exponentiable in GTop.

Proof. The poset morphism p: P ⟶ B of Example 4.5 is clearly a discrete opfibration when considered as a morphism in Cat. The corresponding geometric morphism of presheaf toposes is not exponentiable in GTop since p is not hereditarily compact, as ↓x ∩ p^{−1}(↓b) = {y_{1}, y_{2}, …}, when b = 0. 6.5. Corollary. The presheaf functors Pos ⟶ GTop and Cat ⟶ GTop do not preserve exponentiable morphisms. Proof. The poset morphism p: P ⟶ B of Example 4.5 is exponentiable in Pos and Cat but, as seen in the proof of Corollary 6.4, the corresponding geometric morphism of presheaf toposes is not exponentiable in GTop.

It is well-known that the presheaf functor does not preserve pullbacks, since as noted in the introduction, it does not preserve equalizers [J1]. However, in the posetal case, we can use the equivalence of PSh(P) with Sh(P^{↓}) to get the following version of Lemma 5.1 for presheaf toposes. 6.6. Corollary. Suppose p: P ⟶ B is a morphism of posets such that B^{↓} is a sober space. If the geometric morphism p: PSh(P) ⟶ PSh(B) is exponentiable in GTop, then the induced geometric morphism

PSh(Q ×_{B} P) ⟶ PSh(Q) ×_{PSh(B)} PSh(P)

is an equivalence, for all q: Q ⟶ B in Pos. Proof. It suffices to show that the geometric morphism

Sh(Q^{↓} ×_{B^{↓}} P^{↓}) ⟶ Sh(Q^{↓}) ×_{Sh(B^{↓})} Sh(P^{↓})

is an equivalence. Since the functor Sh: Loc ⟶ GTop preserves pullbacks [JT], it suffices to show that the induced morphism

O(Q^{↓} ×_{B^{↓}} P^{↓}) ⟶ O(Q^{↓}) ×_{O(B^{↓})} O(P^{↓})

is an isomorphism in Loc. But, we know that O(p^{↓}): O(P^{↓}) ⟶ O(B^{↓}) is exponentiable in Loc, by Theorem 6.2, and so the desired result follows from Lemma 5.1.

We conclude with an example showing that the additional compactness condition on an exponentiable morphism p: P ⟶ B of Pos is not sufficient to ensure the exponentiability in GTop of the corresponding geometric morphism of presheaf toposes. In particular, we give an example of a morphism p: P ⟶ B such that p_{∗}(Ω_{P^{↓}}) is locally compact but not metastably locally compact in Loc(Sh(B^{↓})). 6.7. Example. Let B = 2 and P = {x, z_{1}, z_{2}, y_{1}, y_{2}, …}, where y_{i} ≤ z_{1} ≤ x and y_{i} ≤ z_{2} ≤ x, for all i, and there are no other comparable pairs, and define p: P ⟶ B by px = 1, pz_{1} = pz_{2} = 0, and py_{i} = 0, for all i. Then p_{∗}(Ω_{P^{↓}}) is locally compact but not metastably locally compact in Loc(Sh(B^{↓})). Proof. It is not difficult to show that p_{∗}(Ω_{P^{↓}}) satisfies Theorem 5.3 (a), and so it is locally compact. To show that it is not metastably locally compact one shows that [[↓x <<< ↓x]] ≠ B using Proposition 5.11 of [JJ] and showing that the cover ↓z_{1} ∪ ↓z_{2} defined over {0} does not satisfy ↓z_{1} ∩ ↓z_{2} <<< ↓z_{1} ∩ ↓z_{2}.

Theory and Applications of Categories, Vol. 8, No. 2 31

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Department of Mathematics Union College Schenectady, NY 12308, USA

Email: niefiels@union.edu

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John Baez, University of California, Riverside: baez@math.ucr.edu Michael Barr, McGill University: barr@barrs.org Lawrence Breen, Université Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Jean-Luc Brylinski, Pennsylvania State University: jlb@math.psu.edu Aurelio Carboni, Università dell Insubria: carboni@fis.unico.it P. T. Johnstone, University of Cambridge: ptj@dpmms.cam.ac.uk G. Max Kelly, University of Sydney: maxk@maths.usyd.edu.au Anders Kock, University of Aarhus: kock@imf.au.dk F. William Lawvere, State University of New York at Buffalo: wlawvere@acsu.buffalo.edu Jean-Louis Loday, Université de Strasbourg: loday@math.u-strasbg.fr Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Andrew Pitts, University of Cambridge: Andrew.Pitts@cl.cam.ac.uk Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz James Stasheff, University of North Carolina: jds@math.unc.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: walters@fis.unico.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca