Theory and Applications of Categories, Vol. 27, No. 2, 2012, pp. 10–26.

EXPONENTIABILITY VIA DOUBLE CATEGORIES

SUSAN NIEFIELD

Abstract. For a small category B and a double category 𝔻, let Lax_{N}(B, 𝔻) denote the category whose objects are vertical normal lax functors B [figure] 𝔻 and morphisms are horizontal lax transformations. It is well known that Lax_{N}(B, ℂat) ≃ Cat/B, where ℂat is the double category of small categories, functors, and profunctors. In [19], we generalized this equivalence to certain double categories, in the case where B is a finite poset. In [23], Street showed that Y [figure] B is exponentiable in Cat/B if and only if the corresponding normal lax functor B [figure] ℂat is a pseudo-functor. Using our generalized equivalence, we show that a morphism Y [figure] B is exponentiable in 𝔻_{0}/B if and only if the corresponding normal lax functor B [figure] 𝔻 is a pseudo-functor plus an additional condition that holds for all X [figure] !B in Cat. Thus, we obtain a single theorem which yields characterizations of certain exponentiable morphisms of small categories, topological spaces, locales, and posets.

1. Introduction

Suppose A is a category with finite limits. An object Y of A is called exponentiable if the functor − × Y: A [figure] A has a right adjoint, denoted by ( )^{Y}. A morphism is called exponentiable if it is exponentiable in A/Y.

Exponentiable morphisms in the category Cat of small categories were characterized independently by Giraud [6] and Conduché [2] as those functors satisfying a factorization lifting property now known as the Giraud-Conduché condition. Exponentiable morphisms in the category Top of topological spaces were characterized by the author in [14] (see also [15, 16, 17]) as those satisfying a somewhat technical condition (see Lemma 4.1 below) which was used to show that the inclusion of a subspace of B is exponentiable if and only if it is locally closed, and also to establish the exponentiability of perfect maps as well as locally compact spaces over a locally Hausdorff base.

The obstruction to exponentiability in each of these two categories is quite different. In Cat, the Giraud-Conduché condition is used to define composition of morphisms in the category that serves as the exponential, and the unit and counit follow. Whereas in Top, one can always define a candidate for the exponential for which the unit is continuous, but the extra condition is needed for the continuity of the counit.

There is a more recent characterization of exponentiability in Cat. In a 1986 handwritten manuscript, referenced in his 2001 unpublished note [23], Street used the equivalence (attributed to Bénabou) between Cat/B and a category Lax_{N}(B, P rof) to show that a functor Y [figure] B is exponentiable if and only if the corresponding normal lax functor B [figure] P rof is a pseudo-functor. Objects of Lax_{N}(B, P rof) are normal lax functors from B to the bicategory P rof of small categories and profunctors, and morphisms of Lax_{N}(B, P rof) are functor-valued lax transformations. Thus, Bénabou’s equivalence can by viewed as taking place in the realm of double categories (in the sense of [4] or [7]). In particular, for a double category 𝔻, we can consider the category Lax_{N}(B, 𝔻) whose objects are vertical normal lax functors and morphisms are horizontal lax transformations.

Received by the editors 2011-10-07 and, in revised form, 2012-04-29.

Published on 2012-05-01 in the volume of articles from CT2011.

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, 2012. Permission to copy for private use granted.

In [19], we established the equivalence between Lax_{N}(B, 𝔻) and 𝔻_{0}/Γ_{B}1, for certain double categories 𝔻, in the case where B is a finite poset and the constant functor 𝔻_{0} [figure] Lax_{N}(B, 𝔻) admits a left adjoint Γ_{B}. When B = 𝟚, we know Γ_{B}1 = 𝟚 in Cat and Pos. It is the Sierpinski space 𝟚 in Top and the Sierpinski locale S in Loc. The poset B = 𝟚 was also used in [19] to define open and closed inclusions in 𝔻_{0} and obtain a general construction of exponentials for locally closed inclusions over an arbitrary base, which we then applied to Cat, Top, Loc, and Pos.

In this paper, we characterize the exponentiable objects Y: B [figure] 𝔻 of Lax_{N}(B, 𝔻) when B is the 3-element linearly-ordered poset, Lax_{N}(B, 𝔻) has finite limits, and 𝔻 has, what we call, a zero object. Double categories 𝔻 with these properties include ℂat, ℙos, 𝕋op, 𝕃oc, and ℝel. In particular, we show that Y is exponentiable if and only if −×Y preserves pseudo-functors, Y_{b} is exponentiable, for all b ∈ B, and Y_{b} [figure: •] Y_{c} is exponentiable as an object of 𝔻_{1}, for all b < c in B. Using the equivalence established in [19], we thus obtain a characterization of exponentiability in 𝔻_{0}/Γ_{B}1, for a general poset B, which applies to Cat, Pos, Top, and Loc, with the additional assumption that B is finite in the latter two cases. Note that every vertical morphism is exponentiable in 𝔻_{1}, when 𝔻 is ℂat or ℙos.

We proceed as follows. In Section 2, we recall the definition of double category, and introduce zero objects as well as the double categories that will be considered throughout. The definition of Lax_{N}(B, 𝔻) and the characterization of its exponentiable objects are presented in Section 3. We conclude, in Section 4, by characterizing exponentiable objects of 𝔻_{1}, for the two remaining cases, namely, 𝕋op and 𝕃oc.

The author would like to thank the referee for carefully reading the preliminary versions of this paper and providing useful suggestions and corrections.

2. Double Categories

A double category 𝔻 is a weak internal category

[figure: 𝔻_{1} ×_{𝔻_{0}} 𝔻_{1}; 𝔻_{1}; c; 𝔻_{1}; 𝔻_{0}; d_{0}; 𝔻_{0}; 𝔻_{1}; Δ; 𝔻_{1}; 𝔻_{0}; d_{1}]

in CAT. It consists of objects (those of 𝔻_{0}), two types of morphisms: horizontal (morphisms of 𝔻_{0}) and vertical (objects of 𝔻_{1} with domain and codomain given by d_{0} and d_{1}), and cells (morphisms of 𝔻_{1}) of the form

[figure: X_{1}; Y_{1}; f_{1}; X_{0}; X_{1}; m; X_{0}; Y_{0}; f_{0}; Y_{0}; Y_{1}; n; •; •]  (1)

Composition of morphisms and identities are defined horizontally in 𝔻_{0} and vertically using c and Δ, respectively. Cell composition is given horizontally in 𝔻_{1} and vertically via c. Cells in which the horizontal morphisms are identities are called special cells.

There are five double categories of interest in this paper.

(E1) ℂat has small categories as objects, functors and profunctors as horizontal and vertical morphisms, respectively, and natural transformations m [figure] n(f_{0}−, f_{1}−) as cells of the form (1).

(E2) 𝕋op has topological spaces as objects and continuous maps as horizontal morphisms. Vertical morphisms m: X_{0} [figure: •] X_{1} are finite intersection-preserving maps m: O(X_{0}) [figure] O(X_{1}) on the open set lattices, and there is a cell of the form (1) if and only if f_{1}^{−1}n ⊆ mf_{0}^{−1}.

(E3) 𝕃oc has locales as objects, locale morphisms (in the sense of [11]) as horizontal morphisms, and finite meet-preserving maps as vertical morphisms. There is a cell of the form (1) if and only if f_{1}^{∗}n ≤ mf_{0}^{∗}.

(E4) ℙos has partially-ordered sets as objects and order-preserving maps as horizontal morphisms. Vertical morphisms m: X_{0} [figure: •] X_{1} are order ideals (i.e., up-sets) m ⊆ X_{0}^{op} × X_{1}, and there is a cell of the form (1) if and only if (x_{0}, x_{1}) ∈ m ⇒ (f_{0}(x_{0}), f_{1}(x_{1})) ∈ n.

(E5) ℝel has sets as objects, functions and relations as horizontal and vertical morphisms, respectively, and there is a cell of the form (1) if and only if (x_{0}, x_{1}) ∈ m ⇒ (f_{0}(x_{0}), f_{1}(x_{1})) ∈ n.

Our most general result, Theorem 3.3, will follow from properties shared by the five double categories. Although (E1)–(E5) are all framed bicategories (in the sense of [21]) and the first four have 𝟚-glueing (in the sense of [19]), these conditions will not be used until we apply the main theorem to obtain the exponentiability results in (E1)–(E5).

An object 0 of 𝔻 is called a zero object if it is horizontally initial, vertically both initial and terminal, and there exists a unique cell

[figure: X_{1}; Y_{1}; f_{1}; X_{0}; X_{1}; X_{0}; Y_{0}; f_{0}; Y_{0}; Y_{1}; n; •; •; X_{0}; 0; 0; X_{1}; •]

for all f_{0}, f_{1}, n. Note that the latter condition implies that X_{0} [figure: •] 0 [figure: •] X_{1} is an initial object in the category of vertical morphisms X_{0} [figure: •] X_{1} and special cells.

The double categories in the five examples each have an initial object which is a zero object. In fact, if 𝔻 is any framed bicategory, then any horizontal initial object which is vertically initial and terminal is easily seen to be a zero object.

3. Exponentiability of Normal Lax Functors

Suppose B is a poset and 𝔻 is a double category. A vertical normal lax functor X: B [figure] 𝔻 consists of an object X_{b} of 𝔻, for every b ∈ B, a vertical morphism X_{bc}: X_{b} [figure: •] X_{c}, for every b < c, and a special cell X_{cd}X_{bc} [figure] X_{bd}, called a comparison cell, for every b < c < d, satisfying the usual coherence conditions. In particular, we are assuming our normal lax functors are strict normal in the sense that X_{bb} is the vertical identity morphism on X_{b}, for all b. A normal lax functor for which the comparison cells are all isomorphisms is called a pseudo-functor. A horizontal lax transformation f: X [figure] Y: B [figure] 𝔻 consists of a horizontal morphism f_{b}: X_{b} [figure] Y_{b}, for all b ∈ B, and a cell

[figure: X_{c}; Y_{c}; f_{c}; X_{b}; X_{c}; X_{bc}; X_{b}; Y_{b}; f_{b}; Y_{b}; Y_{c}; Y_{bc}; •; •]

for every b < c, compatible with the comparison cells for X and Y. Vertical normal lax functors and horizontal transformations form a category which we denote by Lax_{N}(B, 𝔻). Note that Lax_{N}(𝟙, 𝔻) ≅ 𝔻_{0} and Lax_{N}(𝟚, 𝔻) ≅ 𝔻_{1}.

If 𝔻 has a zero object, then pseudo-functors X: 𝟛 [figure] 𝔻 can be described as follows. Given any normal lax functor X: 𝟛 [figure] 𝔻, there is a commutative diagram in Lax_{N}(𝟛, 𝔻) of the form

[figure: 0; X_{1}; ·; X_{1}; 0; ·; X_{0}; X_{1}; ·; X_{1}; 0; ·; 0; X_{1}; ·; X_{1}; X_{2}; ·; X_{1}; X_{1}; X_{1}; X_{1}; X_{1}; X_{1}; X_{1}; X; X_{1}; X_{1}; X_{1}; X]  (2)

where the morphisms and cells are identities or uniquely determined by the definition of zero object. If X is a functor, i.e., X_{12}X_{01} = X_{02} and the comparison cell is the identity, then one can show that (2) is a pushout in Lax_{N}(𝟛, 𝔻). Thus, we get:

3.1. Lemma. The diagram (2) is a pushout in Lax_{N}(𝟛, 𝔻) if and only if X is a pseudofunctor.

Proof. Since (2) is a pushout when X is a functor, it follows that such a diagram is a pushout if and only if there is an isomorphism from X to the corresponding functor X_{0} [figure: •] X_{1} [figure: •] X_{2} such that the cells X_{01} [figure] X_{01} and X_{12} [figure] X_{12} are identities if and only if X is a pseudo-functor.

3.2. Lemma. If 𝔻 has a zero object and A is a subposet of B, then the restriction functor ( )_{A}: Lax_{N}(B, 𝔻) [figure] Lax_{N}(A, 𝔻) has a left adjoint L_{A} such that ( )_{A}L_{A} = id.

Proof. Given X: A [figure] 𝔻, define

(L_{A}X)_{b} = \{ \begin{matrix}X_{b} & if b ∈ A \\ 0 & otherwise \end{matrix}

and given b < c, let (L_{A}X)_{bc} = X_{bc}, if b, c ∈ A, and let (L_{A}X)_{bc} be the unique vertical morphism to or from 0, otherwise. That L_{A}X is a normal lax functor follows directly from the definition of zero object, and the result easily follows.

We will use the following notation for the functors L_{A} and ( )_{A} in some special cases. We write L_{b} and ( )_{b} when A = {b} and L_{bc} when A is the subposet with two elements b < c. Similarly, we use the notation L_{bcd} in the case where b < c < d. We also write L_{b}: 𝔻_{0} [figure] 𝔻_{1} and ( )_{b}: 𝔻_{1} [figure] 𝔻_{0} for the functors induced by the isomorphisms Lax_{N}(𝟙, 𝔻) ≅ 𝔻_{0} and Lax_{N}(𝟚, 𝔻) ≅ 𝔻_{1}, where b = 0, 1.

Suppose Lax_{N}(B, 𝔻) has finite products. Then so does Lax_{N}(A, 𝔻), for all subposets A ⊆ B, since ( )_{A} has a left adjoint by Lemma 3.2. In particular, 𝔻_{0} has finite products and (X × Y)_{b} ≅ X_{b} × Y_{b} in 𝔻_{0}, for all b ∈ B and X, Y ∈ Lax_{N}(B, 𝔻).

Note that if 𝔻_{0} has chosen products, we do not necessarily know that we can take (X × Y)_{b} to be the chosen product. However, we will assume 𝔻 is horizontally invariant, in the sense of [7], and then this problem disappears. The five double categories (E1)–(E5) of interest all have companions and conjoints, and hence are horizontally invariant.

Given b < c < d, we have cells

[figure: X_{d} × Y_{d}; X_{d} × Y_{d}; X_{b} × Y_{b}; X_{d} × Y_{d}; X_{b} × Y_{b}; X_{b} × Y_{b}; X_{b} × Y_{b}; X_{d} × Y_{d}; X_{bd}×Y_{bd}; •; •; X_{b} × Y_{b}; X_{c} × Y_{c}; X_{bc}×Y_{bc}; X_{c} × Y_{c}; X_{d} × Y_{d}; X_{cd}×Y_{cd}; •; θ_{bcd}]

where π_{1}θ_{bcd} is π_{1} · π_{1}: (X_{cd} × Y_{cd}) · (X_{bc} × Y_{bc}) [figure] X_{bd} · X_{bc} followed by the comparison cell X_{bd} · X_{bc} [figure] X_{bd}, and π_{2}θ_{bcd} is defined similarly.

We will say − × Y_{bcd} preserves pseudo-functors if θ_{bcd} is invertible, whenever X is a pseudo-functor in Lax_{N}({b, c, d}, 𝔻). In the case where B is the 3-element totally-ordered set 𝟛 = {0, 1, 2}, we will say − × Y preserves pseudo-functors, when − × Y_{012} does.

Note that if Y_{bcd} preserves pseudo-functors, for all b < c < d, then Y itself is necessarily a pseudo-functor, provided that Lax_{N}(B, 𝔻) has a pseudo-functorial terminal object, e.g., 𝔻 has a double terminal (in the sense of [7]). This is the case for the five double categories under consideration.

3.3. Theorem. Suppose 𝔻 is a horizontally invariant double category with a zero object such that 0 × X ≅ 0, for all X in 𝔻, and Lax_{N}(𝟛, 𝔻) has finite limits. Then Y: 𝟛 [figure] 𝔻 is exponentiable in Lax_{N}(𝟛, 𝔻) if and only if

(i) Y_{b} is exponentiable in 𝔻_{0}, for all b;

(ii) Y_{b} [figure: •] Y_{c} is exponentiable in 𝔻_{1}, for all b < c; and

(iii) − × Y preserves pseudo-functors.

Proof. Suppose (i), (ii), and (iii) hold. By the remark following Lemma 3.2, we know that (X × Y)_{bc}: X_{b} × Y_{b} [figure: •] X_{c} × Y_{c} is the product of X_{bc} and Y_{bc} in 𝔻_{1}, for all b < c. Given Z: 𝟛 [figure] 𝔻, consider the exponential Z_{bc}^{Y_{bc}} in 𝔻_{1}. Then (Z_{bc}^{Y_{bc}})_{b} ≅ Z_{b}^{Y_{b}}, since there are natural bijections

𝔻_{0}(X × Y_{b}, Z_{b}) ≅ 𝔻_{0}(X × Y_{b}, (Z_{bc})_{0})
≅ 𝔻_{1}(L_{0}(X × Y_{b}), Z_{bc})
≅ 𝔻_{1}(L_{0}X × Y_{bc}, Z_{bc})
≅ 𝔻_{1}(L_{0}X, Z_{bc}^{Y_{bc}})
≅ 𝔻_{0}(X, (Z_{bc}^{Y_{bc}})_{0})

where the third bijection follows from the isomorphism 0 × Y_{c} ≅ 0. Similarly, (Z_{bc}^{Y_{bc}})_{c} ≅ Z_{c}^{Y_{c}}, and by horizontal invariance, we can assume these isomorphisms are equalities.

Consider the exponential Z_{bc}^{Y_{bc}}: Z_{b}^{Y_{b}} [figure: •] Z_{c}^{Y_{c}} in 𝔻_{1}. Thus, b ↦ Z_{b}^{Y_{b}} becomes a lax functor 𝟛 [figure] 𝔻 via the cell on the left which corresponds to the diagram on the right by exponentiability of Y_{02}: Y_{0} [figure: •] Y_{2}

[figure: Z_{0}^{Y_{0}}; Z_{1}^{Y_{1}}; Z_{01}^{Y_{01}}; Z_{1}^{Y_{1}}; Z_{2}^{Y_{2}}; Z_{12}^{Y_{12}}; Z_{2}^{Y_{2}}; Z_{2}^{Y_{2}}; id; Z_{0}^{Y_{0}}; Z_{2}^{Y_{2}}; Z_{0}^{Y_{0}}; Z_{0}^{Y_{0}}; id; Z_{0}^{Y_{0}}; Z_{2}^{Y_{2}}; Z_{02}^{Y_{02}}; •; •; •; Z_{0}^{Y_{0}} × Y_{0}; Z_{1}^{Y_{1}} × Y_{1}; Z_{1}^{Y_{1}} × Y_{1}; Z_{2}^{Y_{2}} × Y_{2}; Z_{2}^{Y_{2}} × Y_{2}; Z_{2}^{Y_{2}} × Y_{2}; id; Z_{0}^{Y_{0}} × Y_{0}; Z_{2}^{Y_{2}} × Y_{2}; (Z{Y_{12} \atop 12} ·Z{Y_{01} \atop 01} )×Y_{02}; Z_{0}^{Y_{0}} × Y_{0}; Z_{0}^{Y_{0}} × Y_{0}; id; Z_{0}^{Y_{0}} × Y_{0}; Z_{2}^{Y_{2}} × Y_{2}; •; •; •; Z_{1}^{Y_{1}} × Y_{1}; Z_{1}; Z_{0}^{Y_{0}} × Y_{0}; Z_{1}^{Y_{1}} × Y_{1}; Z{Y_{01} \atop 01} ×Y_{01}; Z_{0}^{Y_{0}} × Y_{0}; Z_{0}; ε_{0}; Z_{0}; Z_{1}; Z_{01}; Z_{2}^{Y_{2}} × Y_{2}; Z_{2}; ε_{2}; Z_{1}^{Y_{1}} × Y_{1}; Z_{2}^{Y_{2}} × Y_{2}; Z{Y_{12} \atop 12} ×Y_{12}; Z_{1}^{Y_{1}} × Y_{1}; Z_{1}; ε_{1}; Z_{1}; Z_{2}; Z_{12}; •; •; Z_{2}; Z_{2}; id; Z_{0}; Z_{2}; Z_{0}; Z_{0}; id; Z_{0}; Z_{2}; Z_{02}; •]

where there is a cell in the left rectangle of the right diagram, since − × Y preserves pseudo-functors. Note that we are applying the latter to the pseudo-functor b ↦ Z_{b}^{Y_{b}} with Z_{12}^{Y_{12}} · Z_{01}^{Y_{01}}: Z_{0}^{Y_{0}} [figure: •] Z_{2}^{Y_{2}} and the identity comparison cell. With this definition, it is not difficult to show that the unit and counit for ( )^{Y_{bc}} extend to ones for ( )^{Y}, and it follows that Y is exponentiable in Lax_{N}(𝟛, 𝔻).

Conversely, suppose Y is exponentiable in Lax_{N}(𝟛, 𝔻). Then arguments analogous to the one proving (Z_{bc}^{Y_{bc}})_{b} ≅ Z_{b}^{Y_{b}} in the first half of the proof, show (i) and (ii) hold. To see that − × Y preserves pseudo-functors, suppose X: 𝟛 [figure] 𝔻 is a pseudo-functor. Then

[figure: 0; X_{1}; ·; X_{1}; 0; ·; X_{0}; X_{1}; ·; X_{1}; 0; ·; 0; X_{1}; ·; X_{1}; X_{2}; ·; X_{1}; X_{1}; X_{1}; X_{1}; X_{1}; X_{1}; X_{1}; X; X_{1}; X_{1}; X_{1}; X]

is a pushout, by Lemma 3.1. Since − × Y preserves pushouts, it follows that

[figure: 0; X_{1}×Y_{1}; ·; X_{1}×Y_{1}; 0; ·; X_{0}×Y_{0}; X_{1}×Y_{1}; ·; X_{1}×Y_{1}; 0; ·; 0; X_{1}×Y_{1}; ·; X_{1}×Y_{1}; X_{2}×Y_{2}; ·; X_{1}×Y_{1}; X_{1}×Y_{1}; X_{1}×Y_{1}; X_{1}×Y_{1}; X_{1}×Y_{1}; X_{1}×Y_{1}; X_{1}×Y_{1}; X × Y; X_{1}×Y_{1}; X_{1}×Y_{1}; X_{1}×Y_{1}; X × Y]

is a pushout, as well, and the desired result follows.

Note that the obstruction to extending the above proof to a general poset B is that the construction b ↦ Z_{b}^{Y_{b}} may not be a lax functor since the coherence condition relative to associativity need not hold. We will see that this is not a problem in ℂat, or for certain double categories including the other four of interest here.

Using Bénabou’s equivalence Lax_{N}(B, ℂat) ≃ Cat/B, the fact that (i) and (ii) always hold, and − × Y_{bcd} preserves pseudo-functors, for all pseudo-functors Y: B [figure] ℂat, we get Street’s characterization [23], as a consequence of Theorem 3.3 as follows.

3.4. Corollary. A functor Y [figure] B is exponentiable in Cat if and only if the corresponding vertical normal lax functor B [figure] ℂat is a pseudo-functor.

Proof. Suppose Y [figure] B is exponentiable in Cat. Then so is 𝟛 ×_{B} Y [figure] 𝟛, for all functors f: 𝟛 [figure] B, since pulling back along f preserves exponentiability. Thus, the corresponding normal lax functor, denoted via abuse of notation by Y_{f}: 𝟛 [figure] ℂat, is exponentiable in Lax_{N}(𝟛, ℂat), and hence, satisfies (iii) of Theorem 3.3. Since ℂat has a double terminal object, it follows that that Y_{f} is a pseudo-functor, and so B [figure] ℂat is as well.

Conversely, suppose the corresponding vertical normal lax functor Y: B [figure] ℂat is a pseudo-functor. Then so is Y_{f}: 𝟛 [figure] ℂat, for all f: 𝟛 [figure] B. Since (i) and (ii) of Theorem 3.3 hold, in any case, and every pseudo-functor satisfies (iii), it follows that Y_{f} is exponentiable in Lax_{N}(𝟛, ℂat). To see that Y is exponentiable in Lax_{N}(B, ℂat), we need only show that the construction b ↦ Z_{b}^{Y_{b}} given in Theorem 3.3 is coherent relative to associativity.

For β: b [figure] c in B, the exponential Z_{β}^{Y_{β}}: Z_{b}^{Y_{b}} [figure: •] Z_{c}^{Y_{c}} can be described as follows. Identifying Z_{b}^{Y_{b}} with the usual functor category, elements of the set Z_{β}^{Y_{β}}(σ_{b}, σ_{c}) correspond to cells in 𝔻 of the form

[figure: Y_{c}; Z_{c}; σ_{c}; Y_{b}; Y_{c}; Y_{β}; Y_{b}; Z_{b}; σ_{b}; Z_{b}; Z_{c}; Z_{β}; •; •]

Unravelling the proof of Theorem 3.3, the comparison cells Z_{γ}^{Y_{γ}} · Z_{β}^{Y_{β}} [figure] Z_{γβ}^{Y_{γβ}} are induced by the diagram

[figure: Y_{d}; Y_{d}; id; Y_{b}; Y_{d}; Y_{γβ}; Y_{b}; Y_{b}; id; Y_{b}; Y_{d}; •; φ^{−1}; Y_{c}; Z_{c}; Y_{b}; Y_{c}; Y_{β}; Y_{b}; Z_{b}; σ_{b}; Z_{b}; Z_{c}; Z_{β}; •; •; Y_{d}; Z_{d}; σ_{d}; Y_{c}; Y_{d}; Y_{γ}; Y_{c}; Z_{c}; σ_{c}; Z_{c}; Z_{d}; Z_{γ}; •; •; Z_{d}; Z_{d}; id; Z_{b}; Z_{d}; Z_{b}; Z_{b}; id; Z_{b}; Z_{d}; Z_{γβ}; •; ψ]

where φ and ψ are the comparison cells for Y and Z, respectively, and so coherence easily follows.

Next, we prove a corollary that gives exponentiability results for ℙos, 𝕋op, 𝕃oc, and ℝel. First, we recall (from [7]) a property of double categories which is shared by these four examples and eliminates the coherence problem in the general version of Theorem 3.3.

A double category 𝔻 is called flat if its cells are determined by their domains and codomains. In this case, there is at most one cell

[figure: X_{1}; Y_{1}; f_{1}; X_{0}; X_{1}; m; X_{0}; Y_{0}; f_{0}; Y_{0}; Y_{1}; n; •; •]

for all f_{0}, m, n, f_{1}.

3.5. Corollary. Suppose B is a poset, 𝔻 is a flat horizontally invariant double category with a zero object such that 0 × X ≅ 0, for all X in 𝔻, and Lax_{N}(B, 𝔻) has finite limits. Then Y: B [figure] 𝔻 is exponentiable in Lax_{N}(B, 𝔻) if and only if

(i) Y_{b} is exponentiable in 𝔻_{0}, for all b;

(ii) Y_{b} [figure: •] Y_{c} is exponentiable in 𝔻_{1}, for all b < c; and

(iii) − × Y_{bcd} preserves pseudo-functors, for all b < c < d.

Proof. Given (i)–(iii), consider b ↦ Z_{b}^{Y_{b}}, as defined in Theorem 3.3. Since 𝔻 is flat, this construction is coherent relative to associativity. Thus, we get a normal lax functor B [figure] 𝔻, and it follows that Y: B [figure] 𝔻 is exponentiable in Lax_{N}(B, 𝔻).

Suppose Y: B [figure] 𝔻 is exponentiable in Lax_{N}(B, 𝔻). Then, by an argument analogous to the one proving (Z_{bc}^{Y_{bc}})_{b} ≅ Z_{b}^{Y_{b}} in the proof Theorem 3.3, we see that Y_{bcd} is exponentiable in Lax_{N}(𝟛, 𝔻), for all b < c < d, and the desired result follows.

When B is a poset, Bénabou’s equivalence is easily seen to restrict to ℙos, yielding Lax_{N}(B, ℙos) ≃ Pos/B. As in ℂat, we know (i) and (ii) always hold (see [16]), and −×Y_{bcd} preserves pseudo-functors, for all pseudo-functors Y: B [figure] ℙos. Applying Corollary 3.5, we get:

3.6. Corollary. A morphism Y [figure] B is exponentiable in Pos if and only if the corresponding vertical normal lax functor B [figure] ℙos is a pseudo-functor.

In [19], we showed that if 𝔻 is a double category satisfying certain conditions, then Bénabou’s equivalence generalizes to Lax_{N}(B, 𝔻) ≃ 𝔻_{0}/Γ_{B}1, for every finite poset B, where Γ_{B} is left adjoint to the constant functor 𝔻_{0} [figure] Lax_{N}(B, 𝔻). Examples include ℂat, ℙos, 𝕋op, and 𝕃oc. In any case, we know pulling back preserves exponentiability in 𝔻_{0}, and so using this equivalence, we can replace (i) and (ii) of Corollary 3.5 by the single condition that Y_{b} [figure: •] Y_{c} is exponentiable in 𝔻_{1}, for all b ≤ c.

As noted above, the finiteness condition is not necessary in ℂat and ℙos. Whether it is necessary in 𝕋op and 𝕃oc is an open question. However, we know that Γ_{B}1 is the Alexandroff space on B (in the sense of [1], i.e., open sets are downward closed) in 𝕋op, and Γ_{B}1 is the locale ↓Cl(B) of down-sets of B in 𝕃oc. Since every finite T_{0} space is the Alexandroff space of its poset of points with the specialization order (see [11]), we get the following two corollaries:

3.7. Corollary. The following are equivalent for a finite T_{0} space B and a continuous map q: Y [figure] B with corresponding vertical normal lax functor n: B [figure] 𝕋op.

(a) q: Y [figure] B is exponentiable in Top.

(b) n: B [figure] 𝕋op is exponentiable in Lax_{N}(B, 𝕋op).

(c) Y_{b} [figure: •] Y_{c} is exponentiable in 𝕋op_{1}, for all b ≤ c and − × n_{bcd} preserves pseudofunctors, for all b < c < d.

3.8. Corollary. The following are equivalent for a finite poset B and a locale morphism q: Y [figure]↓Cl(B) with corresponding vertical normal lax functor n: B [figure] 𝕃oc.

(a) q: Y [figure]↓Cl(B) is exponentiable in Loc.

(b) n: B [figure] 𝕃oc is exponentiable in Lax_{N}(B, 𝕃oc).

(c) Y_{b} [figure: •] Y_{c} is exponentiable in 𝕃oc_{1}, for all b ≤ c and − × n_{bcd} preserves pseudofunctors, for all b < c < d.

In the next section, we will characterize exponentiability in 𝕋op_{1} and 𝕃oc_{1}, to get more complete versions of Corollaries 3.7 and 3.8. In fact, in both cases, we will also show that − × n_{bcd} preserves pseudo-functors, for all b < c < d, whenever n is a pseudo-functor such that n_{bc}: Y_{b} [figure: •] Y_{c} is exponentiable, for all b ≤ c.

Now, we will use Lemma 3.1 to characterize pseudo-functors, when Γ/B is an equivalence of categories. As noted earlier, ℙos satisfies this condition for any poset B. Also, 𝕋op and 𝕃oc do when B is finite, and ℂat does for any small category B. Our proof below, assumes B is a poset, but can be adapted to apply to ℂat for a general B.

Suppose B is a poset and n: B [figure] 𝔻. We adopt the following abuse of notation. Given b < c, let Y_{bc} denote the image of n_{bc}: Y_{b} [figure: •] Y_{c} under Γ_{𝟚}: Lax_{N}(𝟚, 𝔻) [figure] 𝔻_{0}, and similarly, Y_{bcd} for b < c < d and Γ_{𝟛}. If Y [figure] Γ_{B}1 is the image of n under Γ_{B}, then Y_{bc} ≅ Γ_{B}L_{bc}1×_{Γ_{B}1}Y and Y_{bcd} ≅ Γ_{B}L_{bcd}1 ×_{Γ_{B}1} Y, when 𝔻_{0} is ℂat, ℙos, 𝕋op, and 𝕃oc. In any case, one can show that there is a commutative diagram

[figure: Y_{bc}; Y_{cd}; Y_{c}; Y_{bc}; Y_{c}; Y_{cd}; Y_{bc}; Y_{bcd}; Y_{bc}; Y_{cd}; Y_{cd}; Y_{bcd}]  (3)

in 𝔻_{0}, for all b < c < d.

3.9. Proposition. Suppose 𝔻 has 0 and 1, Γ/𝟛: Lax_{N}(𝟛, 𝔻) [figure] 𝔻_{0}/Γ_{𝟛}1 is equivalence of categories, and B is a poset. Then a normal lax functor n: B [figure] 𝔻 is a pseudo-functor if and only if the diagram (3) is a pushout in 𝔻_{0}, for all b < c < d.

Proof. First, n is a pseudo-functor if and only if n_{bcd}: 𝟛 [figure] 𝔻 is, for all b < c < d if and only if the diagram (2) from Lemma 3.1 is a pushout, for all b < c < d. Since Γ/𝟛: Lax_{N}(𝟛, 𝔻) [figure] 𝔻_{0}/Γ_{𝟛}1 is an equivalence, the latter holds if and only if (3) is a pushout, for all b < c < d.

We conclude this section by turning our attention to ℝel. In this case, the functor Γ/B: Lax_{N}(B, ℝel) [figure] ℝel_{0}/Γ_{B}1 is not an equivalence unless B = 𝟙, since it is not difficult to show that Γ_{B}1 is a one-point set so that ℝel_{0}/Γ_{B}1 ≅ Set. Thus, the theorem from [19] does not apply. However, Lax_{N}(B, ℝel) is equivalent to the category Pos_{d}/B of posets with discrete fibers over B (see [18]), and ℝel_{1} is easily seen to be cartesian closed. Applying Corollary 3.5, we get:

3.10. Corollary. Suppose B is a poset. Then Y [figure] B is exponentiable in Pos_{d}/B if and only if the corresponding vertical normal lax functor B [figure] ℝel is a pseudo-functor.

4. Exponentiability in 𝕋op_{1} and 𝕃oc_{1}

A space Y is exponentiable in Top if and only if O(Y) is a continuous lattice (in the sense of Scott [20]) if and only if 𝟚^{Y} exists in Top, where 𝟚 denotes the Sierpinski space {0, 1}, with {0} open but not {1}. In this case, 𝟚^{Y} ≅ O(Y) with the Scott topology, which is defined as follows.

Recall that a subset H of a complete lattice L is called Scott open if ↑H = H and ∨S ∈ H ⇒ ∨F ∈ H, for some finite F ⊆ S. The set ΣL of Scott open subsets is called the Scott topology on L. Given u, v ∈ L, we say u is way below v, and write u << v, if v ≤ ∨S ⇒ u ≤ ∨F, for some finite F ⊆ S. Then L is a continuous lattice if it satisfies v = ∨{u | u << v}. A locale which is a continuous lattice is also called locally compact.

The characterization of exponentiable spaces has appeared in many forms, but was first achieved in 1970 when Day and Kelly [3] proved that − × Y preserves quotient maps precisely when O(Y) is a continuous lattice. By Freyd’s Special Adjoint Functor Theorem, − × Y has a right adjoint if and only if it preserves quotient maps, for then it preserves all colimits (since coproducts are preserved in any case). The “technical condition” for exponentiability in Top/B, proved in [14] and referred to in the introduction, reduces to the Day/Kelly characterization when B = 1, and has the following form when B is a poset with the down-set topology.

Suppose q: Y [figure] B is a continuous map. Then H ⊆ ⨆_{b∈B} O(Y_{b}) is called fiberwise Scott open provided that H_{b} is Scott open, for all b ∈ B, and V_{c} ∈ H_{c} ⇒ V_{b} ∈ H_{b}, for all b < c and V ∈ O(Y). With this topology, ⨆_{b∈B} O(Y_{b}) becomes a space over B via the projection. Consider the (not necessarily continuous) function ε: (⨆_{b∈B} O(Y_{b})) ×_{B} Y [figure] 𝟚 defined by

ε(V_{b}, y) = \{ \begin{matrix}0 & if y ∈ V_{b} \\ 1 & if y ∉ V_{b} \end{matrix}

Then, from [14], we get:

4.1. Lemma. The following are equivalent for q: Y [figure] B in Top.

(a) q: Y [figure] B is exponentiable in Top.

(b) The map ε: (⨆_{b∈B} O(Y_{b})) ×_{B} Y [figure] 𝟚 is continuous.

(c) For all V_{b} ∈ O(Y_{b}) and y_{b} ∈ V_{b}, there exists H fiberwise Scott open such that V_{b} ∈ H_{b} and y_{b} is in the interior in Y of the set ⋃_{b∈B}(∩H_{b}).

We will use Lemma 4.1, in the case where B = 𝟚, to show that the continuous lattice characterization of exponentiable objects in Top and Loc generalizes to 𝕋op_{1} and 𝕃oc_{1}. But, first we use this lemma to prove that − × n_{bcd} preserves pseudo-functors, for all b < c < d, whenever n is a pseudo-functor such that n_{bc}: Y_{b} [figure: •] Y_{c} is exponentiable, for all b ≤ c, thus removing the extra condition in Corollary 3.7 (and hence, in Corollary 3.8, as well). We begin by recalling the equivalence between Lax_{N}(B, 𝕋op) and Top/B, for a finite poset B.

Given m: B [figure] 𝕋op, write m_{bc}: X_{b} [figure: •] X_{c}, for b < c. Let X = ⨆_{b∈B} X_{b}, with U open if U_{b} is open in X_{b}, for all b, and U_{c} ⊆ m_{bc}U_{b}, for all b < c. A horizontal transformation f: m [figure] n gives rise to a continuous map f: X [figure] Y in the obvious way. Note that B is the space associated with the terminal object of Lax_{N}(B, 𝕋op), and so the projection X [figure] B is continuous. Thus, we get Γ_{B}: Lax_{N}(B, 𝕋op) [figure] Top, which is left adjoint to the constant functor and induces an equivalence Γ/B: Lax_{N}(B, 𝕋op) [figure] Top/B, whose pseudo-inverse is defined as follows (see [19] for details).

For X [figure] B, let m_{b} = X_{b}, the fiber over X at b. Then the inclusion i_{b}: X_{b} [figure] X induces a locale morphism i_{b}: O(X_{b}) [figure] O(X) defined by i_{b}^{∗} = i_{b}^{−1} and i_{b}_{∗}(U_{b}) = [U_{b} ∪ (X \ X_{b})]^{∘}.

Given b < c, the composition O(X_{b}) [figure: ^{i_{b}^{∗}}]O(X) [figure: i_{c}^{∗}] O(X_{c}) is a vertical morphism in 𝕋op which we denote by m_{bc}: X_{b} [figure: •] X_{c}. Thus, we get a normal lax functor m: B [figure] 𝕋op, and hence a functor Φ_{B}: Top/B [figure] Lax_{N}(B, 𝕋op) which is pseudo-inverse to Γ/B.

Suppose q: Y [figure] B corresponds with n: B [figure] 𝕋op. We would like to describe the normal lax functor related to the space ⨆_{b∈B} O(Y_{b}) with the fiberwise Scott topology over B. Of course, the fiber over b is O(Y_{b}) with the Scott topology ΣO(Y_{b}). Using the description of open sets of Y arising from the equivalence given above, one can show that H is fiberwise Scott open if and only if H_{b} is Scott open, for all b, and n_{bc}U_{b} ∈ H_{c} ⇒ U_{b} ∈ H_{b}, for all b < c, if and only if H_{b} is Scott open, for all b, and n_{bc}^{−1}H_{c} ⊆ H_{b}, for all b < c.

Given b < c, consider n̂_{bc}: O(Y_{b}) [figure: •] O(Y_{c}), defined by

n̂_{bc}H_{b} = ⋃{H_{c} ∈ ΣO(Y_{c}) | n_{bc}^{−1}H_{c} ⊆ H_{b}}

It is not difficult to show that n̂_{bc} preserves finite intersections, but n̂ does not necessarily define a lax functor b ↦ O(Y_{b}), since n_{bc}^{−1}H_{c} need not be Scott open, when H_{c} is. The latter holds precisely when n_{bc}: O(Y_{b}) [figure] O(Y_{c}) preserves directed unions. Such a function is called Scott continuous.

4.2. Lemma. If n is a pseudo-functor and n_{bc}: Y_{b} [figure: •] Y_{c} preserves directed unions, for all b < c, then n̂: B [figure] 𝕋op is a pseudo-functor, and Γ_{B}n̂ = ⨆_{b∈B} O(Y_{b}), with the fiberwise Scott topology.

Proof. First, we show that n̂ is a pseudo-functor. Since n_{bc}: O(Y_{b}) [figure] O(Y_{c}) preserves directed unions, it is Scott continuous, and so n_{bc}^{−1}H_{c} is Scott open, for all H_{c} Scott open in O(Y_{c}). Then n_{bc}^{−1}: ΣO(Y_{b}) [figure] ΣO(Y_{c}) is left adjoint to n̂_{bc}, by definition of the latter. Since n is a pseudo-functor, we know that n_{bd} = n_{cd}n_{bc}, and so n_{bd}^{−1} = n_{bc}^{−1}n_{cd}^{−1}. Thus, it follows that n̂_{bd} = n̂_{cd}n̂_{bc}, as desired.

It remains to show that Γ_{B}n̂ = ⨆_{b∈B} O(Y_{b}) with the fiberwise Scott topology. By definition, H ⊆ Γ_{B}n̂ is open if and only if H_{b} is open in O(Y_{b}), for all b, and H_{c} ⊆ n̂_{bc}H_{b}, for all b < c, if and only if H_{b} is open in O(Y_{b}), for all b, and n_{bc}^{−1}H_{c} ⊆ H_{b}, for all b < c, and the desired result follows.

4.3. Lemma. If q: Y [figure] 𝟚 is exponentiable in Top, then the corresponding n: Y_{0} [figure: •] Y_{1} preserves directed unions.

Proof. Suppose {U_{α}} is directed, and consider ⋃ nU_{α} ⊆ n (⋃ U_{α}). Now, O(Y_{1}) is a continuous lattice, since pulling back preserves exponentiability, and so given y_{1} ∈ n (⋃ U_{α}), there exists V_{1} ∈ O(Y_{1}) such that y_{1} ∈ V_{1} << n (⋃ U_{α}). Since q is exponentiable, applying Lemma 4.1 with B = 𝟚, there exists H ⊆ O(Y_{0}) ⨆ O(Y_{1}) fiberwise Scott open such that V_{1} ∈ H_{1} and y_{1} ∈ W ⊆ (∩H_{0}) ∪ (∩H_{1}), for some W open in Y. Then H_{1} is Scott open, V_{1} ∈ H_{1}, and V_{1} ⊆ n (⋃ U_{α}), and so n (⋃ U_{α}) ∈ H_{1}. Since H is fiberwise Scott open, we know ⋃ U_{α} ∈ H_{0}, and so U_{α} ∈ H_{0}, for some α. Also, W_{0} ⊆ U_{α}, since W_{0} ⊆ ∩H_{0}. Thus, y_{1} ∈ W_{1} ⊆ nW_{0} ⊆ nU_{α} ⊆ ⋃ nU_{α}, and it follows that n (⋃ U_{α}) ⊆ ⋃ nU_{α}, as desired.

4.4. Theorem. The following are equivalent for a finite T_{0} space B and a continuous map q: Y [figure] B with corresponding vertical normal lax functor n: B [figure] 𝕋op.

(a) q: Y [figure] B is exponentiable in Top.

(b) n: B [figure] 𝕋op is exponentiable in Lax_{N}(B, 𝕋op).

(c) n: B [figure] 𝕋op is a pseudo-functor and Y_{b} [figure: •] Y_{c} is exponentiable in 𝕋op_{1}, for all b ≤ c.

Proof. As before, we know (a)⇒(b)⇒(c). To show that (c)⇒(a), suppose n is a pseudofunctor and Y_{b} [figure: •] Y_{c} is exponentiable in 𝕋op_{1}, for all b ≤ c. By Lemma 4.1, it suffices to show that ε: (⨆_{b∈B} O(Y_{b})) ×_{B} Y [figure] 𝟚 is continuous. By Lemmas 4.2 and 4.3, we know ⨆_{b∈B} O(Y_{b}) [figure] B corresponds to n̂: B [figure] 𝕋op defined above, and so ε is continuous. Since Y_{b} [figure: •] Y_{c} is exponentiable in 𝕋op_{1}, for all b ≤ c, we know Y_{b} is exponentiable in Top, and so ε_{b}: O(Y_{b}) × Y_{b} [figure] 𝟚 is continuous, for all b. Thus, we have a cell

[figure: O(Y_{c}) × Y_{c}; 𝟚; ε_{c}; O(Y_{b}) × Y_{b}; O(Y_{c}) × Y_{c}; n̂_{bc}×n_{bc}; O(Y_{b}) × Y_{b}; 𝟚; ε_{b}; 𝟚; 𝟚; ^{id^{•}}; •; •; ⊇]

for each b < c, since n_{bc} is exponentiable in 𝕋op_{1}. Applying Γ/B to the associated morphism n̂ × n [figure] 𝟚 in Lax_{N}(B, 𝕋op), the desired result follows.

To generalize continuity to vertical morphisms in 𝕋op and 𝕃oc, we first note that there is a connection between the way-below relation and the Scott topology ΣL on L, namely, u << v if and only if there exists H ∈ ΣL such that v ∈ H and u ≤ ∧H (since u << v, for all u ≤ ∧H and v ∈ H). It is this condition that we generalize.

Suppose n: L_{0} [figure: •] L_{1} is in 𝕃oc, and define n̂: ΣL_{0} [figure: •] ΣL_{1} by

n̂H_{0} = ⋃{H_{1} ∈ ΣL_{1} | n^{−1}H_{1} ⊆ H_{0}}

Although n is not necessarily continuous in the Scott topology, i.e., n^{−1}H_{1} need not be Scott open when H_{1} is, one can show that H_{1} ⊆ n̂H_{0} ⟺ n^{−1}H_{1} ⊆ H_{0}. Note that this definition of n̂ agrees with the one defined above for 𝕋op.

Given u_{1}, v_{1} ∈ L_{1} and H_{0} ∈ ΣL_{0}, we say u_{1} is way below v_{1} relative to H_{0}, written u_{1} <<_{H_{0}} v_{1}, if u_{1} << v_{1} in L_{1}, v_{1} ∈ n̂H_{0}, and u_{1} ≤ n(∧H_{0}). Then n: L_{0} [figure: •] L_{1} is called doubly continuous if L_{0} is continuous and L_{1} satisfies

v_{1} = ⋁{u_{1} | u_{1} <<_{H_{0}} v_{1}, for some H_{0} ∈ ΣL_{0}}

4.5. Lemma. n: Y_{0} [figure: •] Y_{1} is exponentiable in 𝕋op_{1} if and only if n: O(Y_{0}) [figure: •] O(Y_{1}) is doubly continuous in 𝕃oc.

Proof. Suppose n: Y_{0} [figure: •] Y_{1} corresponds to q: Y [figure] 𝟚 via 𝕋op_{1} ≃ Top/𝟚. It suffices to show that q: Y [figure] 𝟚 is exponentiable in Top if and only if n: Y_{0} [figure: •] Y_{1} is doubly continuous.

Suppose q: Y [figure] 𝟚 is exponentiable. Then Y_{0} is exponentiable in Top, since the pullback of an exponentiable map is exponentiable, and so O(Y_{0}) is a continuous lattice. To see that n: O(Y_{0}) [figure: •] O(Y_{1}) is doubly continuous, suppose V_{1} ∈ O(Y_{1}) and y_{1} ∈ V_{1}. Then, by Lemma 4.1, there exists H fiberwise Scott open such that V_{1} ∈ H_{1} and y_{1} ∈ U ⊆ (∩H_{0}) ∪ (∩H_{1}), for some U ∈ O(Y). We claim that U_{1} <<_{H_{0}} V_{1}. First, U_{1} << V_{1} in O(Y_{1}), since U_{1} ⊆ ∩H_{1} and V_{1} ∈ H_{1}. Also, U_{1} ⊆ n(U_{0}) ⊆ n(∧H_{0}), since U is open in Y and U_{0} ⊆ ∩H_{0}. Finally, since H is fiberwise Scott open, we know n^{−1}H_{1} ⊆ H_{0}, and so V_{1} ∈ H_{1} ⊆ n̂H_{0}. Thus, U_{1} <<_{H_{0}} V_{1}, as desired.

Conversely, suppose n: O(Y_{0}) [figure: •] O(Y_{1}) is doubly continuous. We will show that q: Y [figure] 𝟚 satisfies Lemma 4.1(c). Given y_{0} ∈ V_{0} ∈ O(Y_{0}), there exists U_{0} << V_{0} such that y_{0} ∈ U_{0}. Take H_{0} = {W_{0} | U_{0} << W_{0}} and H_{1} = ∅. Then H is fiberwise Scott open, V_{0} ∈ H_{0}, and y_{0} ∈ U_{0} ⊆ (∩H_{0}) ∪ (∩H_{1}). Given y_{1} ∈ V_{1} ∈ O(Y_{1}), since n is doubly continuous, there exist H_{0} Scott open and U_{1} ∈ O(Y_{1}) such that y_{1} ∈ U_{1} and U_{1} <<_{H_{0}} V_{1}. Take H_{1} = {W_{1} ∈ n̂H_{0} | U_{1} << W_{1}}. Then H = H_{0} ⊔ H_{1} is fiberwise Scott open, since H_{1} ⊆ n̂H_{0} ⇒ n^{−1}H_{1} ⊆ H_{0}; V_{1} ∈ H_{1}, since V_{1} ∈ n̂H_{0} and U_{1} << V_{1}; and y_{1} ∈ (∧H_{0}) ∪ U_{1} ⊆ (∩H_{0}) ∪ (∩H_{1}) and (∧H_{0}) ∪ U_{1} is open, since U_{1} ⊆ n(∧H_{0}) by definition of U_{1} <<_{H_{0}} V_{1}. Therefore, q: Y [figure] ! < 125 > 𝟚 is exponentiable in Top, as desired.

In [9], Hyland showed that a locale L is exponentiable in Loc if and only if L is locally compact (i.e., a continuous lattice) if and only if the exponential S^{Y} exists in Loc, where S denotes the Sierpinski locale. This result is constructive so it applies to internal locales in any topos, in particular, in the topos Sh(B) of set-valued sheaves on the locale B. Moreover, Joyal and Tierney [12] showed that q ↦ q_{∗}Ω_{L} sets up an equivalence between Loc/B and the category Loc(Sh(B)) of internal locales in Sh(B), where Ω_{L} is the subobject classifier of Sh(L) and q: Sh(L) [figure] Sh(B) is the geometric morphism induced by q: L [figure] B. Thus, q: L [figure] B is exponentiable in Loc if and only if q_{∗}Ω_{L} is locally compact in Loc(Sh(B))

4.6. Theorem. The following are equivalent for n: L_{0} [figure: •] L_{1} in 𝕃oc with corresponding morphism q: L [figure] S in Loc.

(a) n: L_{0} [figure: •] L_{1} is exponentiable in 𝕃oc_{1}.

(b) q: L [figure] S is exponentiable in Loc.

(c) n: L_{0} [figure: •] L_{1} is doubly continuous.

(d) q_{∗}Ω_{L} is locally compact in Loc(Sh(S)).

Proof. (a) ⇔ (b) ⇔ (d) follows from 𝕃oc_{1} ≃ Loc/S ≃ Loc(Sh(S)).

(b) ⇒ (c) Suppose q: L [figure] S is exponentiable in Loc. Then L_{0} and L_{1} are exponentiable in Loc, and so L_{0} ≅ O(Y_{0}) and L_{1} ≅ O(Y_{1}), for some locally compact sober spaces Y_{0} and Y_{1} such that L ≅ O(Y), where Y ≅ Γ_{𝟚}n and n also denotes the induced vertical morphism n: Y_{0} [figure: •] Y_{1} in 𝕋op. To show that n: L_{0} [figure: •] L_{1} is doubly continuous, by Lemma 4.5 it suffices to show that n: Y_{0} [figure: •] Y_{1} is exponentiable in 𝕋op_{1}, or equivalently, Y [figure] 𝟚 is exponentiable in Top.

First, we show that O(X) ×_{S} O(Y) has enough points so that O(X) ×_{S} O(Y) ≅ O(X ×_{𝟚} Y), for all X [figure] 𝟚. It is easy to see that each point of O(X) ×_{S} O(Y) factors through O(X_{0}) × O(Y_{0}) or O(X_{1}) × O(Y_{1}), and the latter locales are spatial since O(Y_{0}) and O(Y_{1}) are locally compact [10].

Then Y is exponentiable in Top/𝟚, by Lemma 4.1, since

Top/𝟚(X ×_{𝟚} Y, 𝟚 × 𝟚) ≅ Loc/S(O(X ×_{𝟚} Y), S × S)
≅ Loc/S(O(X) ×_{S} O(Y), S × S)
≅ Loc/S(O(X), (S × S)^{O(Y)})
≅ Top/𝟚(X, pt((S × S)^{O(Y)}))

where pt is right adjoint to O [11]. Thus, n: Y_{0} [figure: •] Y_{1} is exponentiable in 𝕋op_{1}, and it follows that n: L_{0} [figure: •] L_{1} is doubly continuous by Lemma 4.5.

(c) ⇒ (d) Suppose n: L_{0} [figure: •] L_{1} is doubly continuous. Then L_{0} and L_{1} are continuous lattices, and so n: O(Y_{0}) [figure: •] O(Y_{1}), for some sober spaces Y_{0} and Y_{1} such that L ≅ O(Y), where Y ≅ Γ_{𝟚}n. To see that q_{∗}Ω_{L} is locally compact in Loc(Sh(S)) it suffices to show that q_{∗}Ω_{Y} is locally compact in Loc(Sh(𝟚)), or equivalently, for all V open in Y, V = ∨I, where I is the ideal I = {U | U << V } in Sh(𝟚). Note that I({0}) = {U_{0} | U_{0} << V_{0}} and I(𝟚) = {U | U << V in O(Y) and U_{0} << V_{0} in O(Y_{0})}.

Suppose V is open in Y. If V_{1} = ∅, then V ∈ O(Y_{0}), and the result follows by continuity of O(Y_{0}). Otherwise, since n: O(Y_{0}) [figure: •] O(Y_{1}) is doubly continuous, for all y_{1} ∈ V_{1}, there exists H_{0} Scott open in O(Y_{0}) and U_{1} ∈ O(Y_{1}) such that y_{1} ∈ U_{1} and U_{1} <<_{H_{0}} V_{1}, i.e., V_{1} ∈ n̂H_{0} and U_{1} ⊆ n(∧H_{0}). Consider U = (∧H_{0}) ∪ U_{1}. It suffices to show that U << V in O(Y) and U_{0} << V_{0} in O(Y_{0}), for then V = ∨I, as desired.

To see that U_{0} << V_{0}, suppose V_{0} ⊆ ∪_{A}W_{α} in O(Y_{0}). Since V_{1} ⊆ n(V_{0}) ⊆ n(∪_{A}W_{α}) and n̂H_{0} is Scott open, we know n(∪_{A}W_{α}) ∈ n̂H_{0}, and so ∪_{A}W_{α} ∈ H_{0} by definition of n̂. Thus, ∪_{F} W_{α} ∈ H_{0}, for some finite F ⊆ A, and it follows that U_{0} ⊆ ∪_{F} W_{α}, as desired.

To see that U << V, suppose V ⊆ ∪_{A}W_{α} in O(Y). Then V_{0} ⊆ ∪_{A}(W_{α})_{0}, and so U_{0} ⊆ ∪_{F_{0}}(W_{α})_{0}, for some finite F_{0} ⊆ A, as above. Also, U_{1} ⊆ ∪_{F_{1}}(W_{α})_{1}, for some finite F_{1} ⊆ A, since U_{1} <<_{H_{0}} V_{1}. Taking F = F_{0} ∪ F_{1}, it follows that U ⊆ ∪_{F} W_{α}, as desired.

EXPONENTIABILITY VIA DOUBLE CATEGORIES  25

Note that a single theorem for exponentiability in Top/𝟚 can be obtained by combining the conditions of Lemma 4.1 for B = 𝟚 with Lemma 4.5, and adding “q_{∗}Ω_{Y} is locally compact in Loc(Sh(𝟚)).” from Theorem 4.6. We can also add “Y: B [figure] 𝕋op is a pseudofunctor and Y_{b} [figure: •] Y_{c} is doubly continuous, for all b ≤ c.” to the conditions in Theorem 4.4, and “Y: B [figure] 𝕃oc is a pseudo-functor and Y_{b} [figure: •] Y_{c} is doubly continuous, for all b ≤ c.” to those in Corollary 3.8.

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[18] S. B. Niefield, Lax presheaves and exponentiability, Theory Appl. Categ. 24, (2010), 288–301 [19] S. B. Niefield, The glueing construction and double categories, to appear in Proceedings of CT2010.

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[21] M. Shulman, Framed bicategories and monoidal fibrations, Theory Appl. Categ. 20 (2008), 650– 738.

[23] R. Street, Powerful functors, unpublished note, September 2001.

Department of Mathematics Union College Schenectady, NY 12308

Email: niefiels@union.edu

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