Theory and Applications of Categories, Vol. 16, No. 11, 2006, pp. 218–235.

EXPONENTIABILITY IN LAX SLICES OF TOP

SUSAN NIEFIELD

Abstract.

We consider exponentiable objects in lax slices of Top with respect to the specialization order (and its opposite) on a base space B. We begin by showing that the lax slice over B has binary products which are preserved by the forgetful functor to Top if and only if B is a meet (respective, join) semilattice in Top, and go on to characterize exponentiability over a complete Alexandrov space B.

1. Introduction

Let Top denote the category of topological space and continuous maps. Recall that b ≤ c in the specialization order on a T_{0} space B, if whenever U is open in B and b ∈ U, then c ∈ U.

A lax slice of Top is a category of the form Top↗B defined as follows for any fixed T_{0} space B. Objects of Top↗B are continuous maps p: X [figure]B and morphisms are triangles

[figure: X; B; p; X; Y; f; Y; B; q; ≤]

which commute up to the specialization order on B, i.e., px ≤ qfx, for all x ∈ X, or equivalently, f(p^{−1}U) ⊆ q^{−1}U, for all U open in B.

In [5], Funk used the lax slice over the Sierpinski space 2 to study “homotopy of marked spheres” by identifying Top↗2 with the category whose objects are pairs (X, X_{U}), where X_{U} is an open subset of a topological space X, and morphisms f: (X, X_{U}) [figure](Y, Y_{U}) are continuous maps f: X [figure] Y such that f(X_{U}) ⊆ Y_{U}. In the preliminaries, he shows that this category has products given by

(X, X_{U}) × (Y, Y_{U}) = (X × Y, X_{U} × Y_{U})

and he characterizes exponentiable objects as follows. A pair (Y, Y_{U}) is exponentiable if and only if Y is exponentiable in Top and the set

Z_{U}^{Y} = {σ: Y [figure] Z|σ(Y_{U}) ⊆ Z_{U}}

Received by the editors 2005-12-07 and, in revised form, 2006-06-05.

Transmitted by W. Tholen. Published on 2006-06-16.

2000 Mathematics Subject Classification: 18B30, 18A40, 18A25, 54C35, 54F05, 06F30.

Key words and phrases: exponentiable space, function space, lax slice, specialization order.

\begin{matrix}c \\ ◯ \end{matrix} Susan Niefield, 2006. Permission to copy for private use granted.

is open in the corresponding function space Z^{Y}, for all pairs (Z, Z_{U}).

Several natural questions arise. Is there an intrinsic characterization of these exponentiable pairs, i.e., one that depends only on (Y, Y_{U}) and not directly on the function spaces Z^{Y}? Are there other lax slices Top↗B in which the product is preserved by the forgetful functor to Top? If so, what are their exponentiable objects? These results would be of interest since lax slices can be used to study other configurations of open subsets of topological spaces. Finally, to study diagrams of closed sets, one can consider the analogous questions for the op-lax slices Top↙B, whose morphisms are diagrams that commute up to ≥. In particular, f: X [figure] Y is a morphism

[figure: X; B; p; X; Y; f; Y; B; q; ≥]

if and only if p^{−1}U ⊇ f^{−1}q^{−1}U, for all U open in B if and only if p^{−1}F ⊆ f^{−1}q^{−1}F, for all F closed in B if and only if f(p^{−1}F) ⊆ q^{−1}F, for all F closed in B. For example, Top↙2 can be identified with the category of pairs (X, X_{F}), where X_{F} is a closed subset of X, and morphisms are continuous maps f: X [figure] Y such that f(X_{F}) ⊆ Y_{F}.

The paper proceeds as follows. After a brief review of exponentiability in section two and the introduction of background material in sections three and four, these questions are answered for lax and op-lax slices in sections five and six, respectively.

2. Preliminaries

Let T be a category with binary products. Recall that an object Y is called exponentiable if the functor − × Y: T [figure] T has a right adjoint, often denoted by ( )^{Y}. The category T is called cartesian closed if every object is exponentiable.

When T = Top, taking X to be a one-point space in the natural bijection

Top(X × Y, Z) ≅ Top(X, Z^{Y})

one sees that Z^{Y} can be identified with the set Top(Y, Z) of continuous maps from Y to Z, and so the question of exponentiability becomes one of finding suitable topologies on the function spaces Z^{Y}.

The first exponentiability results in print appear to be the 1945 paper [3] of Fox, where it was shown that a separable metric space is exponentiable if and only if it is locally compact. A complete characterization was achieved by Day and Kelly in their 1970 paper [2], when they proved the functor − × Y: Top [figure] Top preserves quotients if and only if the lattice O(Y) of open sets of Y is a continuous lattice, in the sense of Scott [9], and that a Hausdorff space satisfies this property if and only if it is locally compact. Since − × Y preserves coproducts in any case, preservation of quotients is necessary and sufficient for exponentiability in Top. Note that although sufficiency follows from Freyd’s Special Adjoint Functor Theorem [4], one can construct the exponentials as follows, and use the continuity of O(Y) to establish the exponentiability adjunction.

Recall that a subset H of O(Y) 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, ⋃_{α∈A} U_{α} ∈ H ⇒ ⋃_{α∈F} U_{α} ∈ H, for some finite F ⊆ A. Then the sets of the form

⟨H, W ⟩ = {σ ∈ Z^{Y} |σ^{−1}(W) ∈ H}

where H is Scott-open in Y and W is open in Z, generate a topology on Z^{Y} which agrees with the compact-open topology when Y is locally compact. Moreover, ( )^{Y}: Top [figure] Top is always a functor, and it is the right adjoint to − × Y when O(Y) is a continuous lattice.

A characterization of exponentiability in the strict slices Top/B first appeared in [7]. Since the product is given by the fiber product there, function spaces are formed relative to the fibers of spaces. Here we need only consider the full function spaces Z^{Y}, since products in the lax slices we consider below are preserved by the forgetful functor. Note that this excludes all T_{1} spaces B for, since the specialization order on any T_{1} space B is discrete, the lax and strict slices over B coincide, and so products are clearly not preserved by the forgetful functor, unless B consists of a single point. However, exponentiability in lax slices over T_{1} spaces is completely understood as it is just that of the strict slices characterized in [7]. For more on exponentiability in Top/B and other strict slice categories, the reader is also referred to [8].

Finally, Johnstone’s book [6] is a good source for background on posets and lattices, including the specialization order and Scott topology.

3. Products in Lax Slices

In this section, we consider the existence of binary products in Top↗B and Top ↙B. This can be achieved simultaneously if we let ≤ denote an arbitrary partial order on B and work in the category Top/≤B whose objects are continuous map p: X [figure]B and morphisms are triangles which commute up to ≤.

Recall that for a partially-ordered set B, the upward closed subsets form a topology known as the Alexandrov topology on B. These spaces, known as Alexandrov spaces, are precisely those in which arbitrary intersections of open sets are open [1]. In particular, every finite T_{0} space is Alexandrov.

Now, if B has the Alexandrov topology relative to ≤, or more generally, ≤ is the specialization order on B, then Top↗B = Top/≤B and Top↙B = Top/≥B.

3.1. Proposition. The category Top/≤B has binary products preserved by the forgetful functor to Top if and only if B is a topological ∧-semilattice, i.e., B is a ∧-semilattice with order ≤ and the function ∧: B × B [figure] B is continuous.

Proof. Suppose Top/≤B has binary products preserved by the forgetful functor, and let ∧ = id_{B} × id_{B}, where id_{B}: B [figure]B is the identity map. Then ∧: B × B [figure]B is continuous, and the projections π_{i}: B × B [figure]B are morphisms in Top/≤B, i.e., ∧ ≤ id_{B} ∘ π_{i}. Thus, for all b_{1}, b_{2} ∈ B, b_{1} ∧ b_{2} ≤ b_{i}, for i = 1, 2. To see that b_{1} ∧ b_{2} is the meet in B, suppose that a ≤ b_{i}, for i = 1, 2. Since

[figure: 1; B; a; 1; B; b_{i}; B; B; ^{id^{B}}; ≤]

are morphisms in Top↗B, the unique map ⟨b_{1}, b_{2}⟩: 1 [figure] B × B satisfies

[figure: 1; B; a; 1; B × B; ⟨b_{1},b_{2}⟩; B × B; B; ∧; ≤]

and so a ≤ b_{1} ∧ b_{2}, as required.

For the converse, given p: X  [figure] B and q: Y  [figure] B, consider
p ∧ q: X × Y [figure: p×q] B × B [figure: ∧] B

Then, given a pair of morphisms,

[figure: X; Z; f; X; B; p; Z; Y; g; Z; B; r; Y; B; q; ≥; ≤]

one easily shows that the induced map ⟨f, g⟩: Z [figure]X × Y is a morphism in Top/≤B, i.e.,

[figure: Z; B; r; Z; X × Y; ⟨f,g⟩; X × Y; B; p∧q; ≤]

and so p ∧ q is the desired product.

One can show that the forgetful functor Top/≤B [figure]Top has a left adjoint if and only if B has a bottom element. Since right adjoints preserve products, we get:

3.2. Corollary. The following are equivalent for a partial order ≤ with a bottom element on a space B.

(a) Top/≤B has binary products.

(b) Top/≤B has binary products preserved by the forgetful functor.

(c) B is a topological ∧-semilattice (relative to ≤).

Taking ≤ to be the specialization order and its opposite we get, based on the remark at the beginning of the section, we get the following two corollaries:

3.3. Corollary. The following are equivalent for a T_{0} space B with a bottom element relative to the specialization order ≤.

(a) Top↗B has binary products.

(b) Top↗B has binary products preserved by the forgetful functor.

(c) B is a topological ∧-semilattice (relative to ≤).

3.4. Corollary. The following are equivalent for a T_{0} space B with a top element relative to the specialization order ≤.

(a) Top↙B has binary products.

(b) Top↙B has binary products preserved by the forgetful functor.

(c) B is a topological ∨-semilattice (relative to ≤).

Note that we could have stated a more general version of Corollary 3.3 (respectively, 3.4) omitting both (a) and the bottom (respectively, top) assumption, but every finite ∧-semilattice has a bottom (respectively, ∨-semilattice has a top), and the infinite ones we consider below are all complete.

4. Top↗B and Open Families

In this section, we introduce a category of B-indexed families of open sets that will be used to characterize exponentiable objects in Top↗B.

Given x ∈ X and p: X [figure] B, we get a morphism

[figure: 1; B; b; 1; X; x; X; B; p; ≤]

whenever b ≤ px. If B is a finite T_{0} space, or more generally, any Alexandrov space, then the sets Xb̂ = p^{−1}(↑b) are open in X. It will be useful to have conditions on this family of open sets that will make it possible to retrieve the original map p: X [figure] B.

Suppose B is any poset with a bottom element ⊥, and let OFam_{B} denote the category whose objects are families {Xb̂}_{b∈B} of open subsets of a space X_{ˆ⊥} satisfying

(O1) b ≤ c ⇒ Xb̂ ⊇ Xĉ (O2) ∀x ∈ X_{ˆ⊥}, the subposet {b ∈ B|x ∈ Xb̂} of B has a top element ⊤_{x}

Morphisms of are continuous maps f: X_{ˆ⊥} [figure] Y_{ˆ⊥} such that f(Xb̂) ⊆ Yb̂, for all b ∈ B.

For example, if 2 denotes the Sierpinski space, then OFam_{2} is the category of pairs (X, X_{U}) used by Funk [5] in his study of marked spheres. If B = 2 × 2, then objects of OFam_{B} are given by a space X together with a pair of open subsets X_{(0,1)} and X_{(1,0)}, and morphisms are continuous maps f: X [figure] Y such that f(X_{(0,1)}) ⊆ Y_{(0,1)} and f(X_{(1,0)}) ⊆ Y_{(1,0)}. If V is the subposet

[figure: (0,1); (0,0); (0,1); (1,0); (1,0); (0,0)]

then OFam_{V} is the full subcategory in which the open subsets are disjoint.

We will see, in Proposition 4.1 below, that if B has suprema of nonempty subsets which are bounded above, then (O2) is equivalent to

(O2^{⋆}) ∀\{b_{α}\} ⊆ B, ⋂ Xb̂_{α} = \{ \begin{matrix}Xb̂ & if ∀\{b_{α}\} ⊆ B, b = ∨b_{α} exists \\ ∅ & otherwise \end{matrix}

To establish an isomorphism Top↗B ≅ OFam_{B}, we will consider an alternate view of these families as certain relations, and obtained a bijection on objects from properties that hold in this more general setting. Although the bijection can be established directly in a straightforward manner, this approach sheds light on the role of condition (O2) above.

Given {Xb̂}, let R = {(x, b)|x ∈ Xb̂}. Using (O1) and the fact that each Xb̂ is upward closed, being an open subset of X_{ˆ⊥}, it follows that R is an order-ideal (i.e., a downward closed subset) of Xop ; _{ˆ⊥} × B.

Let Ord denote the category whose objects are posets and morphisms from X to B are order ideals of X^{op} × B. The identity on X is the ideal I_{X} = {(x, y)|x ≥ y} and composition is given by the usual composition of relations.

Following the custom for relations, morphisms will be denoted X− [figure]B. Every order-preserving map p: X [figure]B gives rise to a morphism R_{p}: X− [figure]B with a right adjoint R_{p}^{♯}: B− [figure] X defined by

R_{p} = {(x, b)|px ≥ b} and R_{p}^{♯} = {(b, x)|b ≥ px}

The following proposition shows that these are precisely the maps of the bicategory Ord, in the sense of [10]. Thus, to establish a relationship between families {Xb̂} and continuous maps p: X [figure] B, we would like to determine which families give rise to maps in Ord.

Given R: X− [figure] B and x ∈ X, let R[x] = {b ∈ B|(x, b) ∈ R}. Then R[x] is a downward closed subset of B and x ≤ y ⇒ R[x] ⊆ R[y]. Similarly, let R[b] = {x ∈ X|(x, b) ∈ R}. Then R[b] is an upward closed subset of X and b ≤ c ⇒ R[b] ⊇ R[c]. Note that if p: X [figure] B is an order-preserving map of posets, then R_{p}[b] = p^{−1}(↑ b), and if R is the order ideal related to a family {Xb̂}, then R[b] = Xb̂ and R[x] = {b ∈ B|x ∈ Xb̂}. Note that the latter says that condition (O2) is precisely (c) given below.

4.1. Proposition. The following are equivalent for R: X− [figure] B in Ord.

(a) R = R_{p}, for some order-preserving map p: X [figure] B.

(b) R has a right adjoint.

(c) R[x] has a top element ⊤_{x}, for all x ∈ X.

Moreover, if B has suprema of nonempty subsets which are bounded above, then (a) through (c) are equivalent to

(d) ∀\{b_{α}\} ⊆ B, ⋂ R[b_{α}] = \{ \begin{matrix}R[b] & if b = ∨b_{α} exists in B \\ ∅ & otherwise \end{matrix}

Proof. First, (a) ⇒ (b) since R_{p} ⊣ R_{p}^{♯}.

For (b) ⇒ (c), suppose R ⊣ S. Since (x, x) ∈ I_{X} and I_{X} ⊆ S ∘ R, there exists ⊤_{x} ∈ B such that (x, ⊤_{x}) ∈ R and (⊤_{x}, x) ∈ S. Given b ∈ R[x], since (⊤_{x}, x) ∈ S and (x, b) ∈ R implies (⊤_{x}, b) ∈ R ∘ S ⊆ I_{B}, it follows that ⊤_{x} ≥ b, as desired.

For (c) ⇒ (a), define p: X [figure]B by px = ⊤_{x}. Then p is order preserving since x ≤ y ⇒ R[x] ⊆ R[y] ⇒ ⊤_{x} ≤ ⊤_{y}, and R = R_{p} since

(x, b) ∈ R ⟺ ⊤_{x} ≥ b ⟺ px ≥ b ⟺ (x, b) ∈ R_{p}

where the first “⇐” holds since (x, ⊤_{x}) ∈ R and R is downward closed in X^{op} × B, and the third “⇒” from the definition of R_{p}.

It remains to show that (c) ⟺ (d), when B satisfies the suprema assumption. Suppose (c) holds and ⋂ R[b_{α}] ≠ ∅, say x ∈ ⋂ R[b_{α}]. Then for all α, we know b_{α} ≤ ⊤_{x} since b_{α} ∈ R[x], and so ∨b_{α} exists and ∨b_{α} ≤ ⊤_{x}. Since R[∨b_{α}] ⊇ R[⊤_{x}], it follows that x ∈ R[∨b_{α}]. Thus, ⋂ R[b_{α}] ⊆ R[∨b_{α}], and so ⋂ R[b_{α}] = R[∨b_{α}], as desired.

Conversely, suppose R satisfies (d), and let x ∈ X. Then t_{x} = ∨R[x] exists by (d), since ⋂ {R[b]|b ∈ R[x]} ≠ ∅, as x ∈ R[b] for all b ∈ R[x], and t_{x} ∈ R[x] since

x ∈ ⋂ {R[b]|b ∈ R[x]} = R[t_{x}]

Therefore, R[x] has a top element, to complete the proof.

4.2. Corollary. If B is an Alexandrov space on a poset with ⊥, then Top ↗ B ≅ OFam_{B}.

Proof. By Proposition 4.1 and its preceding remarks, families {Xb̂} correspond to orderpreserving maps p: X_{ˆ⊥} [figure] B such that R_{p}[b] is open in X_{ˆ⊥}. Since B has the Alexandrov topology and R_{p}[b] = p^{−1}(↑ b), it follows that these are precisely the continuous maps p: X_{ˆ⊥} [figure]B, thus defining a bijection between the objects of OFam_{B} and Top↗B. For morphisms, given p: X [figure] B and q: Y [figure] B, a continuous map f: X [figure] Y is in Top↗B if and only if f (p^{−1}(↑b)) ⊆ q^{−1}(↑b), for all b ∈ B if and only if f(Xb̂) ⊆ Yb̂, for all b ∈ B, and it follows that Top↗B ≅ OFam_{B}.

4.3. Corollary. If B is a poset with ⊥, then OFam_{B} has finite products if and only if B is a ∧-semilattice. Moreover, {Xb̂} × {Yb̂} = {Xb̂ × Yb̂}.

Proof. This follows directly from Corollaries 3.3 and 4.2.

5. Exponentiability in Top↗B

In this section, we characterize exponentiable objects of Top↗B when B is an Alexandrov space on a complete lattice. We begin with the first of four conditions (E1)–(E4) necessary for exponentiability using the presence of ⊥ to get information about exponentials when they exist. We prove the following in the more general setting of Top/≤B, so that we can use it when we consider exponentiability in Top↙B.

5.1. Lemma. If B is a topological ∧-semilattice with ⊥ and q: Y [figure] B is exponentiable in Top/≤B, then

(E1) Y is exponentiable in Top.

Moreover, if r: Z  [figure] B, then r^{q}: Z^{Y}  [figure] B and
(r^{q})^{−1}(↑b) = {σ ∈ Z^{Y} |b ∧ q ≤ rσ}

for all b ∈ B.

Proof. Given r: Z  [figure] B, let r^{q}: [q, r]  [figure] B denote the exponential in Top/≤B. Since

there are bijections

X × Y [figure: f] Z [figure][figure: X × Y; B; ⊥∧q=⊥; X × Y; Z; f; Z; B; r; ≤] [figure][figure: X; B; ⊥; X; [q, r]; f̂; [q, r]; B; r^{q}; ≤] [figure] X [figure: f̂] [q, r]

which are natural in X, it follows that the functors X [figure]Top(X × Y, Z) are representable, and so Y is exponentiable in Top, Z^{Y} = [q, r], and r^{q}: Z^{Y} [figure]B. Moreover, since

[figure: 1; B; b; 1; Z^{Y}; σ; Z^{Y}; B; r^{q}; ≤; 1 × Y; B; b∧q; 1 × Y; Z; σ; Z; B; r; ≤]

it follows that (r^{q})^{−1}(↑b) = {σ ∈ Z^{Y} |b ∧ q ≤ rσ}.

Following Corollary 4.2, to define exponentials the r^{q} when they exist in Top↗B, we will present Z^{Y} as a family {ZY ; b̂ } of open subsets of Z^{Y}. For simplicity, the reference to q and r has been omitted from the notation. Lemma 5.1 tells us how to define {ZY ; b̂ }, when B is any T_{0} space with ∧ and ⊥ relative to specialization order ≤.

5.2. Definition. Given a topological ∧-semilattice B and continuous maps q: Y [figure] B

and r: Z [figure] B, let ZY ; b̂ = {σ ∈ Z^{Y} |b ∧ q ≤ rσ}.

In view of the following proposition, it will be necessary to assume that B has all finite infima (i.e., the empty infimum ⊤, as well) if Top↗B is to have any exponentiable objects. This says that if one is interested in exponentiability in OFam_{V}, when V is the subspace of 2 × 2 considered in section four, then it is necessary to expand to the larger category OFam_{B}, where B = 2 × 2.

5.3. Proposition. If B is a topological ∧-semilattice such that id_{B}^{q} exists in Top/≤B, for some q: Y [figure] B, then B has a top element ⊤.

Proof. Since

[figure: 1 × Y; B; b∧q; 1 × Y; B; q; B; B; ^{id^{B}}; ≤; 1; B; b; 1; B^{Y}; q; B^{Y}; B; ^{id{q \atop B}}; ≤]

for all b, it follows that id_{B}^{q}(q) is the top element of B.

To show that ZY ; b̂ is open in Z^{Y}, we will assume that B is a complete lattice (an assumption that is already necessary in the finite case), and use Proposition 4.1(d), or equivalently, condition (O2^{⋆}), to prove that {ZY ; b̂ } is in OFam_{B}. However, to establish criteria for (O2^{⋆}), it is not necessary to assume that B is complete, only that B has finite infima.

5.4. Lemma. Suppose B is a ∧-semilattice with ⊤, c = ⋁ b_{α} exists in B, and B has the Alexandrov topology. Then q: Y [figure]B satisfies ⋂ ZY ; b̂_{α} = ZY ; ĉ, for all Z [figure]B if and only if

(E2) ⋁(b_{α} ∧ qy) exists and equals (⋁ b_{α}) ∧ qy, for all y ∈ Y.

Proof. Suppose ⋂ ZY ; b̂_{α} = ZY ; ĉ, for all r: Z [figure] B. To prove (E2), it suffices to fix y_{0} ∈ Y and show that

(⋁ b_{α}) ∧ qy_{0} ≤ d ⟺ b_{α} ∧ qy_{0} ≤ d, ∀α

for all d ∈ B.

Let d ∈ B, and define σ_{d}: Y [figure] B by

σ_{d}(y) = \{ \begin{matrix}⊤ & if qy ≰ qy_{0} \\ d & if qy ≤ qy_{0} \end{matrix}

Then σ_{d} is clearly continuous, and b ∧ qy_{0} ≤ d implies b ∧ qy ≤ σ_{d}y, for all y ∈ Y, since if qy ≰ qy_{0}, then σ_{d}y = ⊤, and if qy ≤ qy_{0}, then b ∧ qy ≤ b ∧ qy_{0} ≤ d = σ_{d}y. Thus,

b ∧ qy_{0} ≤ d ⟺ b ∧ q ≤ σ_{d}

for all b ∈ B. Taking r = id_{B}, since ⋂ BY ; b̂_{α} = BY ; ĉ, where

BY ; b̂ = {σ ∈ B^{Y} |b ∧ q ≤ σ}

we see that (⋁ b_{α}) ∧ qy_{0} ≤ d iff (⋁ b_{α}) ∧ q ≤ σ_{d} iff c ∧ q ≤ σ_{d} iff σ_{d} ∈ BY ; ĉ iff σ_{d} ∈ ⋂ BY ; b̂_{α}

iff σ_{d} ∈ BY ; b̂_{α}, for all α, iff b_{α} ∧ q ≤ σ_{d}, for all α, iff b_{α} ∧ qy_{0} ≤ d, for all α.

Conversely, suppose (E2) holds and r: Z [figure] B. Then σ ∈ ⋂ ZY ; b̂_{α} iff σ ∈ ZY ; b̂_{α}, for all α, iff b_{α} ∧ q ≤ rσ, for all α, iff b_{α} ∧ qy ≤ rσy, for all y ∈ Y and for all α, iff ⋁(b_{α} ∧ qy) ≤ rσy, for all y ∈ Y and for all α, iff (⋁ b_{α}) ∧ qy ≤ rσy, for all y ∈ Y iff (⋁ b_{α}) ∧ q ≤ rσ iff

c ∧ q ≤ rσ iff σ ∈ ZY ; ĉ, and so ⋂ ZY ; b̂_{α} = ZY ; ĉ.

Note that if B is a complete lattice, then (E2) holds if and only if − ∧ qy preserves suprema, for all y ∈ Y if and only if qy is exponentiable in B, i.e., − ∧ qy: B [figure] B has a right adjoint, for all y ∈ Y.

The following alternate description of ZY ; b̂ will be useful in determining when these sets are open in Z^{Y}.

5.5. Proposition. If q: Y  [figure] B and r: Z  [figure] B are continuous, then

ZY ; b̂ = {σ ∈ Z^{Y} |σ(Yĉ) ⊆ Zĉ, ∀c ≤ b} = {σ ∈ Z^{Y} |Yĉ ⊆ σ^{−1}(Zĉ), ∀c ≤ b}

where Yĉ = q^{−1}(↑c) and Zĉ = r^{−1}(↑c).

Proof. Suppose b ∧ q ≤ rσ, c ≤ b, and y ∈ Yĉ. Then c ≤ qy, and so c = c ∧ qy ≤ b ∧ qy ≤ rσy, and it follows that σy ∈ Zĉ. Thus, σ(Yĉ) ⊆ Zĉ. Conversely, suppose σ(Yĉ) ⊆ Zĉ, for all c ≤ b. To show b ∧ qy ≤ rσy, for all y, let y ∈ Y and take c = b ∧ qy. Then y ∈ Yĉ. Since c ≤ b, we know σ(Yĉ) ⊆ Zĉ, and so σy ∈ Zĉ. Thus, b ∧ qy = c ≤ rσy, as desired.

Suppose B is a complete lattice and A ⊆ B, and let ⟨A⟩ denote the finite (including empty) ∧ and ∨ closure of A in B.

5.6. Lemma. If B is a complete lattice with the Alexandrov topology and q: Y [figure] B is continuous, then ZY ; b̂ is open in Z^{Y}, for all b ∈ B, r: Z [figure] B, if and only if

(E3) q(Y) is finite, and

(E4) Yĉ is compact, for all ∨-irreducible c in ⟨q(Y)⟩.

Proof. Suppose q satisfies (E3) and (E4). Let J denote the set of ∨-irreducible elements of ⟨q(Y)⟩, and for each c ∈ J, let H_{c} = {U ∈ O(Y)|Yĉ ⊆ U}. Then

⟨H_{c}, Zĉ⟩ = {σ ∈ Z^{Y} |Yĉ ⊆ σ^{−1}(Zĉ)}

is open in Z^{Y}, since H_{c} is Scott-open (by compactness of Yĉ) and Zĉ is open in Z. To show that ZY ; b̂ is open in Z^{Y}, since J is finite, it suffices to show that

ZY ; b̂ = ⋂_{c∈J∩↓b}⟨H_{c}, Zĉ⟩

But, ZY ; b̂ = ⋂_{c≤b}⟨H_{c}, Zĉ⟩ by Proposition 5.5, and so it suffices to show that

⋂_{c∈J∩↓b}⟨H_{c}, Zĉ⟩ ⊆ ⋂_{c≤b} ⟨H_{c}, Zĉ⟩

So, suppose σ ∈ ⟨H_{c}, Zĉ⟩, for all c ∈ J∩ ↓ b, and c ≤ b. If c ∉ ⟨q(Y)⟩, then Yĉ ⊆ σ^{−1}(Zĉ), since Yĉ = ∅. If c ∈ ⟨q(Y)⟩, then writing c = ∨{c_{j} ∈ J|c_{j} ≤ c}, we see that

Yĉ = ∩Yĉ_{j} ⊆ ∩σ^{−1}(Zĉ_{j}) = σ^{−1}(∩Zĉ_{j}) = σ^{−1}(Zĉ)

Thus, σ ∈ ⟨H_{c}, Zĉ⟩, for all c ≤ b, as desired.

Conversely, suppose ZY ; b̂ is open in Z^{Y}, for all b ∈ B, r: Z [figure] B. To show that q(Y) is finite, take r = id_{B} and b = ⊤. Then q ∈ BY ; _{ˆ⊤}, since ⊤ ∧ q ≤ q, and so there exists H_{1}, …, H_{n} Scott-open in O(Y) and finite sets F_{1}, …, F_{n} ⊆ B such that

q ∈ ⟨H_{1}, ↑F_{1}⟩ ∩ ⋯ ∩ ⟨H_{n}, ↑F_{n}⟩ ⊆ BY ; _{ˆ⊤}

To show that q(Y) is finite, we will show that q(Y) ⊆ S, where

S = ⟨F_{1} ∪ ⋯ ∪ F_{n}⟩

Let σ = fq, where f: B [figure]B is defined by f(b) = ∨(S∩ ↓b). Then f ≤ id_{B} and f is order-preserving, hence, continuous. Also, f(↑s) ⊆↑s, for all s ∈ S, since s ≤ b ⇒ fs ≤ fb ⇒ s ≤ fb, as fs = s, and so ↑s ⊆ f^{−1}(↑s) which gives

q^{−1}(↑s) ⊆ q^{−1}f^{−1}(↑s) = σ^{−1}(↑s)

for all s ∈ S. Thus, for all i, q^{−1}(↑F_{i}) ⊆ σ^{−1}(↑F_{i}), and it follows that σ^{−1}(↑F_{i}) ∈ H_{i}, since H_{i} is Scott-open and q ∈ ⟨H_{i}, ↑F_{i}⟩ ⇒ q^{−1}(↑F_{i}) ∈ H_{i}, and so,

σ ∈ ⋂⟨H_{i}, ↑F_{i}⟩ ⊆ BY ; _{ˆ⊤} ⇒ ⊤ ∧ q ≤ σ ⇒ q ≤ σ ⇒ q ≤ fq

Since f ≤ id_{B}, it follows that q = fq, and so qy = fqy ∈ S, for all y ∈ Y. Therefore, q(Y) ⊆ S, as desired.

Let c be ∨-irreducible in ⟨q(Y)⟩. To show that Yĉ is compact, we will show that ↑Yĉ is Scott-open in O(Y) by showing ↑Yĉ = π^{−1}(BY ; ĉ), for some π: O(Y) [figure] B^{Y}. Take b < c maximal in ⟨q(Y)⟩, and let f: 2 [figure]B be the continuous map given by f(0) = b and f(1) = c. Then define π to be the composite of f^{Y}: 2^{Y} [figure] B^{Y} and the homeomorphism O(Y) ≅ 2^{Y} given by U [figure] χ_{U}, the characteristic function of U. Then

U ∈ π^{−1}(BY ; ĉ) ⟺ c ∧ q ≤ fχ_{U} ⟺ c ∧ qy ≤ fχ_{U}(y)

for all y ∈ Y. Note that if y ∉ Yĉ, then c∧qy < c, and so c∧qy ≤ b, since b ≤ (c∧qy)∨b ≤ c, c is ∨-irreducible and b is maximal in ⟨q(Y)⟩∩ ↓c. Thus,

c ∧ qy ≤ \{ \begin{matrix}b & if y ∉ Yĉ \\ c & if y ∈ Yĉ \end{matrix}

Since

fχ_{U}(y) = \{ \begin{matrix}b & if y ∉ U \\ c & if y ∈ U \end{matrix}

it follows that

U ∈ π^{−1}(BY ; ĉ) ⟺ Yĉ ⊆ U ⟺ U ∈↑Yĉ

and so ↑Yĉ = π^{−1}(BY ; ĉ), as desired.

5.7. Theorem. Suppose B is a complete lattice with the Alexandrov topology. Then q: Y [figure] B is exponentiable in Top↗B if and only if

(E1) Y is exponentiable in Top,

(E2) qy is exponentiable in B, i.e., − ∧ qy preserves ⋁, for all y ∈ Y,

(E3) q(Y) is finite, and

(E4) Yĉ is compact, for all ∨-irreducible c in ⟨q(Y)⟩.

Proof. Suppose q: Y [figure]B is exponentiable. Applying Lemma 5.1, we see that (E1) holds, and for all r: Z [figure] B, we have r^{q}: Z^{Y} [figure] B and the sets ZY ; b̂ defined in 5.2 agree with those in the family of open sets corresponding to r^{q} via Corollary 4.2. Then (E2) follows from Lemma 5.4, since the family satisfies (O2^{⋆}), and properties (E3) and (E4) follow from Lemma 5.6, since ZY ; b̂ is open in Z^{Y}, for all b ∈ B.

Conversely, suppose q satisfies (E1)–(E4). Given r: Z [figure] B, define

ZY ; b̂ = {σ ∈ Z^{Y} |b ∧ q ≤ rσ}

Then ZY ; _{ˆ⊥} = Z^{Y} and {ZY ; b̂ } is in OFam_{B} by Lemmas 5.4 and 5.6, since (E2)–(E4) hold, and so we get a continuous map r^{q}: Z^{Y} [figure]B. Since Y is exponentiable, we have the

usual bijection between f: X × Y  [figure] Z and f̂: X  [figure] Z^{Y}. It remains to show that

[figure: X × Y; B; p∧q; X × Y; Z; f; Z; B; r; ≤; X; B; p; X; Z^{Y}; f̂; Z^{Y}; B; q^{r}; ≤]

But, p ∧ q ≤ rf iff px ∧ qy ≤ rf(x, y), for all (x, y), iff px ∧ q ≤ rf̂x, for all x, iff f̂x ∈ ZY ; p̂x,

for all x, iff px ≤ r^{q}(f̂x), for all x, iff p ≤ r^{q}f̂, where the third “iff” follows from the definition of ZY ; b̂ and the fourth from the definition of r^{q} via the family {ZY ; b̂ }. Therefore, q: Y [figure] B is exponentiable in Top↗B.

Since every finite T_{0} space is an Alexandrov space, applying Propositions 3.3 and 5.3, we get:

5.8. Corollary. Suppose B is a finite T_{0} space with a bottom element ⊥ relative to the specialization order. Then Top↗B has finite products if and only if B is a topological ∧-semilattice. Moreover, q: Y [figure] B is exponentiable in Top↗B if and only if

(a) B has a top element ⊤ ,

(b) Y is exponentiable in Top,

(c) qy is exponentiable in B, i.e., − ∧ qy preserves ⋁, for all y ∈ Y, and

(d) Yĉ is compact, for all ∨-irreducible c ∈ B.

Taking B to be the Sierpinski space 2, the desired intrinsic characterization of exponentials alluded to in the introduction follows.

5.9. Corollary. (Y, Y_{U}) is exponentiable in Top↗2 if and only if Y_{U} is compact and Y is exponentiable in Top.

6. Exponentiability in Top↙B

To characterize exponentiable objects in Top ↙B, we define a category of B-indexed families of closed sets, and proceed as before. Note that we consider closed rather than open sets here since the opposite of the specialization order gives us information about the sets ↓b.

Given x ∈ X and p: X [figure] B, we get a morphism

[figure: 1; B; b; 1; X; x; X; B; p; ≥]

whenever b ≥ px. If B is a finite T_{0} space, or more generally, any Alexandrov space, then the sets Xb̌ = p^{−1}(↓b) are closed in X. As before, we would like to have conditions on the family of sets Xb̌ that will make it possible to retrieve the original map p: X [figure] B, but we need an additional assumption here since the sets B\ ↓ b do not necessarily generate the topology on B (in the sense that one cannot represent every open set as a union of a finite intersection of these open sets).

Suppose B is any poset with a top element ⊤, and let CFam_{B} denote the category whose objects are families {Xb̌}_{b∈B} of closed subsets of a space X_{ˇ⊤} satisfying (C1) b ≤ c ⇒ Xb̌ ⊆ Xč

(C2) ∀x ∈ X_{ˇ⊤}, the subposet {b ∈ B|x ∈ Xb̌} of B has a bottom element ⊥_{x}

Morphisms of CFam_{B} are continuous maps f: X_{ˇ⊤} [figure] Y_{ˇ⊤} such that f(Xb̌) ⊆ Yb̌, for all b ∈ B.

For example, if B = 2 × 2, then objects of CFam_{B} are given by a space X together with a pair of closed subsets X_{(0,1)} and X_{(1,0)}, and morphisms f: X [figure]Y such that f(X_{(0,1)}) ⊆ Y_{(0,1)} and f(X_{(1,0)}) ⊆ Y_{(1,0)}.

Given {Xb̌}, let R^{♯} = {(b, x)|x ∈ Xb̌}. Using (C1) and the fact that each Xb̌ is downward closed, being a closed subset of X_{ˇ⊤}, it follows that R^{♯}: B− [figure] X is a relation in Ord. Thus, by the dual of Proposition 4.1, if B has infima of nonempty subsets which are bounded below, then (C2) is equivalent to

(C2^{⋆}) ⋂ Xb̌_{α} = \{ \begin{matrix}Xb̌ & if b = ∧b_{α} exists \\ ∅ & otherwise \end{matrix} ∀\{b_{α}\} ⊆ B

If p: X [figure] B is an order-preserving map of posets, then R_{p}^{♯}[b] = p^{−1}(↓ b), and if R^{♯} is the order ideal related to a family {Xb̌}, then R^{♯}[b] = Xb̌ and R^{♯}[x] = {b ∈ B|x ∈ Xb̌}. The latter says that condition (C2) is precisely the dual of Proposition 4.1(c), and we get the following corollary to Proposition 4.1.

6.1. Corollary. If B a poset with ⊤ and the sets B\↓b generate the Alexandrov topology on B, then Top↙B ≅ CFam_{B}.

Proof. By the dual of Proposition 4.1, families {Xb̌} correspond to order-preserving maps p: X_{ˇ⊤} [figure]B such that R_{p}^{♯}[b] is closed in X_{ˇ⊤}. Since the sets B\ ↓ b generate the topology on B and R_{p}^{♯}[b] = p^{−1}(↓b), it follows that these are precisely the continuous maps p: X_{ˇ⊤} [figure] B, thus defining a bijection between the objects of CFam_{B} and Top↙B. For morphisms, given p: X [figure] B and q: Y [figure] B, a map f: X [figure] Y is in Top↙B if and only if

f (p^{−1}(↓b)) ⊆ q^{−1}(↓b), for all b ∈ B

if and only if f(Xb̌) ⊆ Yb̌, for all b ∈ B, and so Top↙B ≅ CFam_{B}.

6.2. Corollary. If B is as in Corollary 6.1, then CFam_{B} has finite products if and only if B is a ∨-semilattice. Moreover, {Xb̌} × {Yb̌} = {Xb̌ × Yb̌}.

Proof. This follows directly from Corollaries 3.4 and 6.1.

Following Corollary 6.1, to define exponentials r^{q} when they exist in Top↙B, we will present Z^{Y} as a family {ZY ; b̌ } of closed subsets of Z^{Y}. The dual of Lemma 5.1 tells us how to define {ZY ; b̌ }, when B is any T_{0} space with ∨ and ⊤ relative to specialization order ≤.

6.3. Definition. Given a topological ∨-semilattice B and continuous maps q: Y [figure] B

and r: Z [figure] B, let ZY ; b̌ = {σ ∈ Z^{Y} |b ∨ q ≥ rσ}.

6.4. Theorem. Suppose B is a complete lattice and the sets B\↓b generate the Alexandrov topology on B. Then q: Y [figure] B is exponentiable in Top↙B if and only if

(E1) Y is exponentiable in Top and

(E2) qy is exponentiable in B^{op}, i.e., − ∨ qy preserves ⋀, for all y ∈ Y.

Proof. Suppose q: Y [figure]B is exponentiable in Top ↙ B. Applying the dual of Lemma 5.1, we see that (E1) holds, and for all r: Z [figure]B, we have r^{q}: Z^{Y} [figure]B and the sets ZY ; b̌ defined in 6.3 agree with those in the family of closed sets corresponding to r^{q} via Corollary 6.1. In addition, (E2) follows from the dual of Lemma 5.4, since the family satisfies (C2^{⋆}).

Conversely, suppose q satisfies (E1) and (E2). Given r: Z [figure] B, define

ZY ; b̌ = {σ ∈ Z^{Y} |b ∨ q ≥ rσ}

Then ZY ; _{ˇ⊤} = Z^{Y} and {ZY ; b̌ } satisfies (C1) and (C2) by definition and the dual of Lemmas 5.4. To see that ZY ; b̌ is closed in Z^{Y}, we will show that its complement is open. Since

ZY ; b̌ = {σ ∈ Z^{Y} |Yč ⊆ σ^{−1} (Zč) , ∀c ≥ b}

by the dual of Proposition 5.5, it follows that

Z^{Y} \ ZY ; b̌ = ⋃_{b≤c}⟨H_{c}, Z \ Zč⟩

where H_{c} = {U ∈ O(Y)|U ⊈ Y \ Yč}. Since H_{c} is Scott-open, each ⟨H_{c}, Z \ Zč⟩ is open in Z^{Y}, and so Z^{Y} \ ZY ; b̌ is open, as desired. Finally, the proof that

[figure: X × Y; B; p∨q; X × Y; Z; f; Z; B; r; ≥; X; B; p; X; Z^{Y}; f̂; Z^{Y}; B; q^{r}; ≥]

is dual to that of Theorem 5.7. Therefore, q: Y [figure] B is exponentiable in Top↗B.

Since every finite T_{0} space is an Alexandrov space, applying Proposition 3.4, and the dual of Proposition 5.3, we get:

6.5. Corollary. Suppose B is a finite T_{0} space with a top element ⊤ relative to the specialization order. Then Top↙B has finite products if and only if B is a topological ∨-semilattice. Moreover, q: Y [figure] B is exponentiable in Top↙B if and only if

(a) B has a bottom element ⊥ ,

(b) Y is exponentiable in Top, and (c) qy is exponentiable in B^{op}, i.e., − ∨ qy preserves ⋀, for all y ∈ Y.

Thus, if B is a finite T_{0} space which is a co-Heyting algebra relative to the specialization order (so that − ∨ b preserves ⋀, for all b ∈ B), then q: Y [figure] B is exponentiable in Top↙B if and only if Y is exponentiable in Top. There are also infinite spaces satisfying the latter, for example, take B to be the negative integers with the natural order and a bottom adjoined. In fact:

6.6. Corollary. Suppose B^{op} is a spatial locale. Then the following are equivalent for the Alexandrov space on the poset B.

(a) B is Noetherian

(b) q: Y [figure] B is exponentiable in Top↙B ⟺ Y is exponentiable in Top

Proof. Suppose B^{op} is a spatial locale and B is Noetherian. Then every element is exponentiable in B^{op}. Since

↑c = ⋂_{b∈B\↑c} B\↓b

to apply Theorem 6.4, it suffices to show that the set of maximal elements of B\↑c is finite. Since B^{op} is a spatial, we know B^{op} = O(X), for some sober space X, and so it suffices to show that set M of minimal (nonempty) elements of {U ∈ O(X)|U ⊈ V } is finite, for all V ∈ O(X).

Suppose U_{1}, U_{2}, … are distinct elements of M. Since U_{1} ⊈ V, there exists x_{1} ∈ U_{1} such that x_{1} ∉ V. Then x_{1} ∉ U_{2}, since U_{1} ∩ U_{2} ⊆ V (by minimality of U_{2} ⊈ V). Since U_{2} ⊈ V, there exists x_{2} ∈ U_{2} such that x_{2} ∉ V. Then x_{1}, x_{2} ∉ U_{3}, since U_{1} ∩ U_{3}, U_{2} ∩ U_{3} ⊆ V (by minimality of U_{3} ⊈ V). Continuing we get x_{1}, x_{2}, … ∉ V such that x_{1}, … x_{n−1} ∉ U_{n} and x_{n} ∈ U_{n}, and so

x_{n} ∉ \overline{\{x_{1}, x_{2}, … x_{n−1}\}}

giving a descending sequence

X \ \overline{\{x_{1}\}} ⊇ X \ \overline{\{x_{1}, x_{2}\}} ⊇ … ⊇ X \ \overline{\{x_{1}, x_{2}, … x_{n}\}} ⊇ …

of proper subsets, contradicting that B = O(X)^{op} is Noetherian.

For the converse, we will show that if B is not Noetherian, say

y_{1} < y_{2} < y_{3} …

in B, then Y = {y_{1}, y_{2}, …} is exponentiable in Top but the inclusion j: Y [figure] B is not exponentiable in Top↙B. Exponentiability of Y is clear since the Alexandrov topology on Y is locally compact. So, assume for contradiction that j is exponentiable in Top↙B, and consider id_{B}^{j}: B^{Y} [figure] B.

Define σ: Y [figure] B by σ(y_{n}) = y_{n+1}, and let b = id_{B}^{j}(σ). Then σ ∈ (id_{B}^{j})^{−1}(↑ b), and so there exist Scott-open sets H_{1}, …, H_{k} of O(Y) and open sets W_{1}, …, W_{k} of B such that

σ ∈ ⟨H_{1}, W_{1}⟩ ∩ … ∩ ⟨H_{k}, W_{k}⟩ ⊆ (id_{B}^{j})^{−1}(↑b)

and ⟨H_{i}, W_{i}⟩ ≠ B^{Y}, for any i. Note that σ^{−1}(W_{i}) ≠ ∅ for otherwise ∅ ∈ H_{i} making ⟨H_{i}, W_{i}⟩ = B^{Y}.

First, we show that we can assume

H_{1} = H_{n_{1}} = {U ∈ O(Y)|y_{n_{1}} ∈ U} and W_{1} =↑y_{n_{1}+1}

where n_{1} is the least integer such that y_{n_{1}} ∈ σ^{−1}(W_{1}). Since σ^{−1}(W_{1}) is open in Y = {y_{1}, y_{2}, …}, the minimality of n_{1} implies that σ^{−1}(W_{1}) =↑ y_{n_{1}}. Since σ(y_{n_{1}}) = y_{n_{1}+1}, we know ↑ y_{n_{1}+1} ⊆ W_{1} and σ ∈ ⟨H_{n_{1}}, ↑ y_{n_{1}+1}⟩. Thus, ⟨H_{n_{1}}, ↑ y_{n_{1}+1}⟩ ⊆ ⟨H_{1}, W_{1}⟩, since τ^{−1}(↑ y_{n_{1}+1}) ∈ H_{n_{1}} ⇒ y_{n_{1}} ∈ τ^{−1}(↑ y_{n_{1}+1}) ⇒ y_{n_{1}} ∈ τ^{−1}(W_{1}) ⇒ ↑ y_{n_{1}} ⊆ τ^{−1}(W_{1}) ⇒ σ^{−1}(W_{1}) ⊆ τ^{−1}(W_{1}) ⇒ τ^{−1}(W_{1}) ∈ H_{1}, as H_{1} is Scott-open and σ^{−1}(W_{1}) ∈ H_{1}. Similarly, for i = 2, …, k, we can assume

H_{i} = H_{n_{i}} = {U ∈ O(Y)|y_{n_{i}} ∈ U} and W_{1} =↑y_{n_{i}+1}

for some y_{i} ∈ Y.

Thus, without loss of generality, we have n_{1} < … < n_{k} such that

σ ∈ ⟨H_{n_{1}}, ↑y_{n_{1}+1}⟩ ∩ … ∩ ⟨H_{n_{k}}, ↑y_{n_{k}+1}⟩ ⊆ (id_{B}^{j})^{−1}(↑b)

Let N = n_{k} and define τ: Y [figure] B by

τ(y_{n}) = \{ \begin{matrix}y_{n+1} & if n ≤ N \\ y_{N+1} & otherwise \end{matrix}

Then τ is continuous and

τ ∈ ⟨H_{n_{1}}, ↑y_{n_{1}+1}⟩ ∩ … ∩ ⟨H_{n_{k}}, ↑y_{n_{k}+1}⟩ ⊆ (id_{B}^{j})^{−1}(↑b)

and so id_{B}^{j}(τ) ≥ b. Now, y_{N+1} ∨j ≥ τ, since y_{N+1} ∨j(y_{n}) ≥ y_{N+1} ≥ τ(y_{n}), for all n. Thus, y_{N+1} ≥ id_{B}^{j}(τ) ≥ b. Since b = id_{B}^{j}(σ), we know that b∨j ≥ σ, Thus, b∨j(y_{N+1}) ≥ σ(y_{N+1}), or equivalently, b ∨ y_{N+1} ≥ y_{N+2}. But, the latter contradicts y_{N+1} < y_{N+2}, since y_{N+1} ≥ b would imply y_{N+1} = b∨y_{N+1} ≥ y_{N+2}. Therefore, the inclusion j in B is not exponentiable in Top↙B.

7. Concluding Remarks

As noted in the introduction, this work began as a search for an intrinsic characterization of exponentiability in Top ↗ B and Top ↙ B when the category has finite products preserved by the forgetful functor and B is a finite T_{0} space. It soon became apparent that the results could be extended to infinite Alexandrov spaces by adding an additional condition to obtain exponentiability in Top↗B and assuming that B has the topology generated by the sets B\↓b for Top↙B. In the process of writing this paper, it became clear that some of the results for Top↗B were dual to those in Top↙B, and did not require a separate proof, if they were cast in Top/≤B. Although the paper covers more

EXPONENTIABILITY IN LAX SLICES OF TOP  235

than was initially intended, many new questions arise with the relaxation of the finiteness assumption and the introduction of Top/≤B. Here are a few.

Does the characterization of exponentiability in Top↙B extend to all Alexandrov spaces without the assumption that the sets B\↓b generate the topology on B? What can be said about exponentiability in Top/≤B when B does not have the Alexandrov topology on ≤, or when ≤ is not the specialization order or its opposite? What happens if we relax the assumption that the forgetful functor preserves products?

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[3] R. H. Fox, On topologies for function spaces, Bull. Amer. Math. Soc. 51 (1945) 429–432.

[4] P. Freyd, Abelian Categories, Harper & Row, New York, 1964. Republished in: Reprints in Theory and Applications of Categories, No. 3 (2003) pp -23 - 164.

[5] J. Funk, The Hurwitz action and braid group orderings, Theory Appl. Categ. 9 (2001), 121–150.

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[7] S. B. Niefield, Cartesianness: topological spaces, uniform spaces, and affine schemes, J. Pure Appl. Algebra 23 (1982), 147–167.

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[10] R. Street, Cauchy characterization of enriched categories, Rend. Sem. Mat. e Fis. Milano 51 (1981), 217–233. Republished in: Reprints in Theory and Applications of Categories, No. 14 (2005), pp 1-18.

Union College Department of Mathematics Schenectady, NY 12308

Email: niefiels@union.edu

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

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Richard Blute, Université d’ Ottawa: rblute@mathstat.uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: r.brown@bangor.ac.uk Aurelio Carboni, Università dell Insubria: aurelio.carboni@uninsubria.it Valeria de Paiva, Xerox Palo Alto Research Center: paiva@parc.xerox.com

Ezra Getzler, Northwestern University: getzler(at)math(dot)northwestern(dot)edu

Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk 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 Stephen Lack, University of Western Sydney: s.lack@uws.edu.au 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 Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Brooke Shipley, University of Illinois at Chicago: bshipley@math.uic.edu 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: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca