Theory and Applications of Categories, Vol. 19, No. 1, 2007, pp. 4–18.

EXPONENTIABILITY IN HOMOTOPY SLICES OF TOP AND PSEUDO-SLICES OF CAT

SUSAN NIEFIELD

Abstract. We prove a general theorem relating pseudo-exponentiable objects of a bicategory K to those of the Kleisli bicategory of a pseudo-monad on K. This theorem is applied to obtain pseudo-exponentiable objects of the homotopy slices Top//B of the category of topological spaces and the pseudo-slices Cat//B of the category of small categories.

1. Introduction

The 2-slice Top/B is the 2-category whose objects are continuous maps p: X [figure] B, morphisms are commutative triangles

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

and 2-cells are equivalence classes {F } of homotopies F: f [figure] f′ over B, i.e., commutative triangles

[figure: X × I; B; pπ_{1}; X × I; Y; F; Y; B; q]

such that F |_{X×0} = f and F |_{X×1} = f′, where F ∼ F′ if there is a homotopy Φ: F [figure] F′ over B such that Φ|_{X×0×I} = f and Φ|_{X×1×I} = f′. Note that the use of “fiberwise homotopies” here makes X ×_{B} Y into the 2-product of p: X [figure] B and q: Y [figure] B in Top/B.

Exponentiability results for the 1-category Top/B (see [14]) easily generalize to dimension 2 (and higher). In particular, given an exponentiable map q: Y [figure] B, the natural bijections

θ_{X,Z}: Top/B(X ×_{B} Y, Z)  [figure] Top/B(X, Z^{Y})  (1)

are 2-natural isomorphisms of categories, or equivalently, the adjunction − ×_{B} Y ⊣ ( )^{Y} is a 2-adjunction, in the sense of [10].

Received by the editors 2006-12-18 and, in revised form, 2007-03-29.

Transmitted by Walter Tholen. Published on 2007-04-18.

2000 Mathematics Subject Classification: 18A40, 18C20, 18A25, 18B30, 54C35.

Key words and phrases: pseudo-exponentiable, Kleisli bicategory, homotopy slice, pseudo-slice.

\begin{matrix}c \\ ◯ \end{matrix} Susan Niefield, 2007. Permission to copy for private use granted.

EXPONENTIABILITY IN HOMOTOPY SLICES OF TOP AND PSEUDO-SLICES OF CAT 5

Working with commutative triangles over B is at times too rigid, since constructions are made in the fibers of maps over B. One can relax this restriction by taking morphisms to be triangles

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

which commute up to specified homotopy φ. However, composition of triangles is then neither associative nor unital, since composition of homotopies is only associative and unital up to homotopy. This can be rectified by imposing an equivalence relation on the triangles, as is done in [9], but one loses the distinct homotopies. An alternative is to move to the realm of weak 2-categories, i.e., bicategories, in the sense of Benabou [1]. But, what is a suitable choice of 2-cell?

Given a triangle

[figure: X × I; B; pπ_{1}; X × I; Y; F; Y; B; q; Φ]

restricting to X × t, we get a triangle

[figure: X; B; p; X; Y; F_{t}; Y; B; q; Φ_{t}]

and hence, a continuous family of homotopies from (f, φ) = (F_{0}, Φ_{0}) to (f′, φ′) = (F_{1}, Φ_{1}). We will see that taking suitable equivalence classes of these families gives rise to a bicategory Top//B, which we call a homotopy slice of Top.

For exponentiability of q: Y [figure] B in Top//B, the role of the fiber product in Top/B will be played by the homotopy pullback X ×_{B} B^{I} ×_{B} Y, where the map B^{I} [figure] B is evaluation at 0 (denoted by ev_{0}) when B^{I} appears on the right of ×_{B}, and evaluation at 1 (denoted by ev_{1}) when B^{I} appears on the left. The existence of a right adjoint to the functor

(X [figure: p] B) ↦ (X ×_{B} B^{I} ×_{B} Y [figure: pπ_{1}] B)

is rare when considered as an endofunctor of Top/B (see [15]). However, in the context of bicategories, it is more appropriate to consider pseudo-adjoints (or equivalently, biadjoints in the sense of Street [17]), and thus to replace the isomorphisms θ_{X,Z} in (1) by pseudonatural equivalences of categories. We will see that there are many pseudo-exponentiable maps q: Y [figure] B in Top//B for which ev_{0}π_{1}: B^{I} ×_{B} Y [figure] B is pseudo-exponentiable in Top/B.

When considering Top//B, to avoid cumbersome verification of details, it is useful to work in a more general setting and call upon an analogy with pseudo-slices of the 2-category Cat of small categories. Since these pseudo-slices are themselves 2-categories, and not merely bicategories, it will be necessary to generalize some of the concepts involved.

Recall that Cat/B is the 2-category whose objects are functors p: X [figure] B, morphisms are commutative triangles

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

and 2-cells are natural transformations F: f [figure] f′ such that qF = id_{p}. The pseudo-slice Cat//B has the same objects but the morphisms are triangles

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

which commute up to a specified natural isomorphism, and 2-cells from (f, φ) to (f′, φ′) are natural transformations F: f [figure] f′ such that the following diagram commutes

[figure: qf; qf′; qF; p; qf; φ; p; qf′; φ′]

Using a variation of a construction by Street [16], one can show that Cat//B is the Kleisli 2-category of a 2-monad on Cat/B. This construction cannot be applied to the 2-category Top, since Top it is not representable and the 2-cells of Top/B differ from those arising in [16]. Moreover, changing the 2-cell would result in a loss of 2-products in Top/B. However, using an analogous construction, we will see that Top//B is the Kleisli bicategory of a pseudo-monad on Top/B.

Pseudo-exponentiability in Cat//B was considered by Johnstone in [6], where it is stated that q: Y [figure] B is pseudo-exponentiable if and only if q satisfies a certain factorization lifting property, called FPL in Section 5 below. Only a sketch of the sufficiency proof is given in [6], since it is analogous to that of Conduché [3] and Giraud [5] for Cat/B, and the necessity proof is completely omitted, as it is not relevant to the paper. Taking I to be the category with objects 0 and 1 and a single isomorphism between them, it turns out that q: Y [figure] B is pseudo-exponentiable in Cat//B if and only if B^{I} ×_{B} Y [figure: ev_{0}π_{1}]B is 2-exponentiable in Cat/B, and the latter is easily seen to be equivalent to the relevant factorization lifting property. Moreover, the sufficiency of FPL can be established via a general theorem about Kleisli bicategories of pseudo-monoids, which will also be applied to obtain examples of pseudo-exponentiable objects in Top//B.

We begin with a presentation of Top//B and Cat//B as the Kleisli bicategories of the related 2-slice categories. In section three, we show that if T is a pseudo-monad on a bicategory K (satisfying certain properties which hold in the examples under consideration), and T Y is pseudo-exponentiable in K, then Y is pseudo-exponentiable in the Kleisli bicategory K_{T}. We conclude in sections four and five with applications yielding pseudo-exponentiable of objects of Top//B and Cat//B. In particular, we show that every exponentiable (Hurewicz) fibration is pseudo-exponentiable in Top//B and every FPL functor is pseudo-exponentiable in Cat//B.

The author would like to thank André Joyal, Miles Tierney, and Mark Weber for comments that led to the use of Kleisli bicategories, and hence a cleaner approach to this work.

2. The Kleisli bicategory of a pseudo-monad

In this section, we exhibit Top//B and Cat//B as the Kleisli bicategories of pseudomonads on Top/B and Cat/B, respectively.

Recall that a 2-monad on a 2-category K consists of a 2-functor T: K [figure] K together with 2-natural transformations η: id_{K} [figure] T and μ: T^{2} [figure] T such that

μ(T η) = id_{T}  μ(ηT) = id_{T}  μ(T μ) = μ(μT)  (2)

The Kleisli 2-category K_{T} of T is the 2-category whose objects are the same as those of K, and K_{T} (X, Y) = K(X, T Y) with id_{X} = η_{X} and composition induced by μ. Moreover, the 2-functor U: K [figure] K_{T}, given by the identity on objects and composition with η_{Y} on morphisms and 2-cells, has a right 2-adjoint T: K_{T} [figure] K given by X ↦ T X and

K_{T} (X, Y) = K(X, T Y) [figure] K(T X, T^{2}Y) [figure: K(T X,μ_{Y} )] K(T X, T Y)

For example, take K = Cat/B, and let I be as in the introduction. Then there is an internal category (in the sense of [7])

[figure: B^{I} ×_{B} B^{I}; c; B^{I}; B; ev_{0}; B^{I}; B; i; B^{I}; B; ev_{1}]

in Cat, where ev_{0} and ev_{1} denote the evaluation functors at 0 and 1, respectively, i is the functor B ↦ id_{B}, and c is the composition functor. Note that, as in Top, we write B^{I} on the left of ×_{B} when ev_{1}: B^{I} [figure] B, and on the right when ev_{0}: B^{I} [figure] B. Define

T (X [figure: p] B) = B^{I} ×_{B} X [figure: ev_{0}π_{1}] B

with the induced maps on morphisms and 2-cells, and η and μ given by

X [figure: <i_{p},id_{X}>] B^{I} ×_{B} X and B^{I} ×_{B} B^{I} ×_{B} X [figure: c×id_{X}] B^{I} ×_{B} X

Then it is not difficult to show that Cat//B is (isomorphic to) the Kleisli 2-category of T.

To obtain the homotopy slices Top//B, a slight modification is necessary. We can start with K = Top/B and consider

[figure: B^{I} ×_{B} B^{I}; c; B^{I}; B; ev_{0}; B^{I}; B; i; B^{I}; B; ev_{1}]

Since composition is associative and unital only up to homotopy, this is not an internal category in Top. We still get a 2-functor T: Top//B [figure] Top//B given by

T (X [figure: p] B) = B^{I} ×_{B} X [figure: ev_{0}π_{1}] B

and 2-natural transformations

η_{p}: X [figure: <i_{p},id_{X}>] B^{I} ×_{B} X and μ_{p}: B^{I} ×_{B} B^{I} ×_{B} X [figure: c×id_{X}] B^{I} ×_{B} X

but this is not a 2-monad since the equations in (2) do not hold. Instead, we have invertible modifications

[figure: T; T^{2}; T η; T; T; id_{T}; T^{2}; T; ηT; T^{2}; T; μ; T; T; id_{T}; l; r; T^{2}; T; μ; T^{3}; T^{2}; μT; T^{3}; T^{2}; T μ; T^{2}; T; μ; t]  (3)

which satisfy the coherence conditions in the definition of a pseudo-monad on a bicategory (c.f., [12, 2]). Note that the definition of a pseudo-monad does not require that T, η, μ be strict, as they are here, just pseudo-functors and pseudo-natural transformations.

The Kleisli construction K_{T} is defined for any pseudo-monad on a bicategory K, but it is merely a bicategory (even when K is a 2-category and T, η, μ are strict) since the equations in (2) have been replaced by the modifications in (3). Moreover, we get a pseudo-adjoint pair

K [figure: U; T] K_{T}

defined as above.

3. Exponentiability in Kleisli bicategories

In this section, we discuss pseudo-exponentiability of the Kleisli bicategory of a pseudomonad T on a bicategory K. Through a series of lemmas (whose hypotheses are satisfied by the relevant pseudo-monads on Top//B and Cat//B), we show that Y is pseudoexponentiable in K_{T}, provided that T Y is pseudo-exponentiable in K.

Recall that a diagram

[figure: X; Y; X × Y; X; π_{1}; X × Y; Y; π_{2}]

is a called a pseudo-product in a bicategory K if the induced functor

π_{Z}: K(Z, X × Y) [figure] K(Z, X) × K(Z, Y)

is an equivalence of categories, for all objects Z. Note that since π_{Z} is pseudo-natural in Z, so is its pseudo-inverse.

3.1. Lemma. If K is a bicategory with binary pseudo-products and T, η, μ is a pseudomonad on K such that the canonical map

ρ : T (X × T Y) [figure: <T π_{1},μ_{Y} T π_{2}>] T X × T Y

is an equivalence in K, for all X, Y, then X × T Y is the pseudo-product of X and Y in K_{T}.

Proof. The functor K_{T} (Z, X×T Y) [figure] K_{T} (Z, X)×K_{T} (Z, Y) is an equivalence of categories since it factors as a composite

K(Z, T (X × T Y)) [figure: K(Z,ρ)] K(Z, T X × T Y) [figure: π_{Z}] K(Z, T X) × K(Z, T Y)

of equivalences.

Returning to our examples, the functor

ρ: B^{I} ×_{B} X ×_{B} B^{I} ×_{B} Y [figure] (B^{I} ×_{B} X) ×_{B} (B^{I} ×_{B} Y)

is an isomorphism in Cat/B, and the map

ρ: B^{I} ×_{B} X ×_{B} B^{I} ×_{B} Y [figure] (B^{I} ×_{B} X) ×_{B} (B^{I} ×_{B} Y)

is an equivalence in Top/B. In fact, both are defined by

(b [figure: α] px, x, px [figure: β] qy, y) ↦ ((b [figure: α] px, x), (b [figure: α] px [figure: β] qy, y))

Thus, X ×_{B} B^{I} ×_{B} Y is the product in Cat//B and X ×_{B} B^{I} ×_{B} Y is the pseudo-product in Top//B.

Recall that an object Y is pseudo-exponentiable in a bicategory K if the pseudo-functor − × Y: K [figure] K has a right pseudo-adjoint (i.e., a biadjoint in the sense of Street [17]), or equivalently, for every object Z, these is an object Z^{Y} together with equivalence

θ_{X,Z}: K(X × Y, Z) [figure] K(X, Z^{Y})

which are pseudo-natural in X and Z.

Note that any object equivalent to a pseudo-exponentiable one is necessarily pseudoexponentiable, where Y is equivalent to Y′, written Y ≃ Y′, in a bicategory if there exist f: Y [figure] Y′ and g: Y′ [figure] Y such that fg ≅ id_{Y′} and gf ≅ id_{Y}. Moreover, if K is a 2-category, Y ≃ Y′, and Y is 2-exponentiable in K, then composing with the natural isomorphisms K(X × Y, Z) [figure] K(X, Z^{Y}) with the equivalences K(X × Y′, Z) [figure] K(X × Y, Z) gives the pseudo-exponentiability of Y′ in K.

Returning to the general case, suppose T Y is pseudo-exponentiable in K, and consider the following pseudo-natural transformations

[figure: K(X × T Y, T Z); K(T (X × T Y), T^{2}Z); K(T (X × T Y), T^{2}Z); K(T (X×T Y),μ); K(T (X × T Y), T Z); K(T X × T Y, T Z); ≃; K(T X × T Y, T Z); K(T X, T Z^{T Y}); ≃; K(T X, T Z^{T Y}); K(T X,η); K(T X, T (T Z^{T Y})); K(T X, T^{2}(T Z^{T Y})); K(T X,μ); K(T X, T^{2}(T Z^{T Y})); K(X, T (T Z^{T Y}))]

where the first and last functors are given by T. If we can show that these are all equivalences of categories, then we will have an equivalence

K_{T} (X × T Y, Z)  [figure] K_{T} (X, T Z^{T Y})  (4)

which is pseudo-natural in X, giving the pseudo-exponentiability of Y in K_{T}.

3.2. Lemma. If T, η, μ is a pseudo-monad on a bicategory K and ηT ≅ T η, then

τ_{X,Y}: K(X, T Y) [figure] K(T X, T^{2}Y) [figure: K(T X,μ_{Y} )] K(T X, T Y)

is an equivalence of categories, for all X, Y.

Proof. Consider τ′_{X,Y}: K(T X, T Y) [figure: K(η_{X},T Y)] K(X, T Y). To see that τ′_{X,Y} is a pseudoinverse of τ_{X,Y}, given f: X [figure] T Y and g: T X [figure] T Y, let θ_{f}: f [figure] τ′τf and θ′_{g}: ττ′g [figure] g be defined by the invertible 2-cells given by the diagrams

[figure: T Y; T^{2}Y; η_{T Y}; X; T Y; f; X; T X; η_{X}; T X; T^{2}Y; T f; T Y; T^{2}Y; T Y; T Y; id; T^{2}Y; T Y; μ_{Y}; f; ≅; ≅; ≅; T^{2}X; T^{2}Y; T g; T X; T^{2}X; η_{T X}; T X; T^{2}Y; T Y; T^{2}Y; T X; T Y; g; T X; T^{2}Y; T^{2}Y; T Y; μ_{Y}; T Y; T^{2}Y; η_{T Y}; T Y; T Y; id; T (gη_{X}); T Y; g; T η_{X}; ≅; ≅; ≅; ≅; ≅; =]

Then naturality of θ and θ′ follows from coherence and pseudo-naturality of η and μ, and so τ_{X,Y} is an equivalence of categories.

Note that since τ_{X,Y}: K_{T} (X, Y) [figure] K(T X, T Y) is pseudo-natural in X, so is τ′_{X,Y}, and it follows that we have a pseudo-natural transformation

K_{T} (−, Y) [figure] K(T (−), T Y)

considered as pseudo-functors from K_{T} to Cat.

Returning to the examples, we see that ηT ≅ T η in both case. For Top/B, given p: X [figure] B, define F: B^{I} ×_{B} X × I [figure] B^{I} ×_{B} B^{I} ×_{B} X by

F (β, x, t) = (β|_{[0,t]}, β|_{[t,1]}, x)

where β|_{[0,t]}(u) = β(ut) and β|_{[t,1]}(u) = β(u + t − ut). Then F is clearly a continuous map over B, F (β, x, 0) = (i_{b}, β, x) = η_{T X}(β, x), and F (β, x, 1) = (β, i_{px}, x) = T η_{X}(β, x), and it follows that ηT ≅ T η. For Cat/B, given p: X [figure] B, note that

η_{T p}, T η_{p}: B^{I} ×_{B} X [figure] B^{I} ×_{B} B^{I} ×_{B} X

are given by

(β, X) ↦ (id_{B}, β, X) (β, X) ↦ (β, id_{pX}, X)

respectively. Then it is not difficult to show that

m_{p}(β, X) = ((id_{B}, β), (β, id_{pX}), id_{X})

defines an invertible modification m: ηT [figure] T η, as desired.

3.3. Lemma. If T is as in Lemma 3.2 and T Y is pseudo-exponentiable in K, then η: T Z^{T Y} [figure] T (T Z^{T Y}) is an equivalence in K, for all Z in K.

Proof. Let α_{X}: T X × T Y [figure] T (X × T Y) denote the pseudo-inverse of the canonical morphism, and let ε: T Z^{T Y} × T Y [figure] T Z the counit of the pseudo-adjunction. Then

T (T Z^{T Y}) × T Y [figure: α_{T Z^{T Y}}] T (T Z^{T Y} × T Y) [figure: T ε] T^{2}Z [figure: μ] T Z

induces a morphism θ: T (T Z^{T Y}) [figure] T Z^{T Y}.

To see that θη ≅ id_{T Z^{T Y}}, first note that α_{X}(η_{X} × id) ≅ η_{X×T Y}, for all X via the pseudo-adjunction since their composites with <T π_{1}, μ_{Y} T π_{2}> are isomorphic. Using the invertible 2-cells

[figure: T Z^{T Y} × T Y; T (T Z^{T Y}) × T Y; η×id; T Z^{T Y} × T Y; T (T Z^{T Y} × T Y); η; T (T Z^{T Y}) × T Y; T (T Z^{T Y} × T Y); α; T Z; T Z^{T Y} × T Y; θ×id; T Z^{T Y} × T Y; T Z; ε; T Z; ε; T^{2}Z; T ε; η; μ; ≅; ≅; ≅]

it follows that θη ≅ id_{T Z^{T Y}}.

Now, since η is pseudo-natural, we know

[figure: T Z^{T Y}; T (T Z^{T Y}); η; T (T Z^{T Y}); T Z^{T Y}; θ; T (T Z^{T Y}); T^{2}(T Z^{T Y}); ηT; T^{2}(T Z^{T Y}); T (T Z^{T Y}); T θ; ≅]

and ηT ≅ T η, by assumption. Thus, ηθ ≅ (T θ)(ηT) ≅ (T θ)(T η) ≅ T (θη) ≅ T (id_{T Z^{T Y}}) ≅ id_{T (T Z^{T Y} )}.

3.4. Theorem. Suppose that K is a bicategory with binary pseudo-products and T, η, μ is a pseudo-monad on K such that ηT ≅ T η and ρ: T (X ×T Y) [figure] T X ×T Y is an equivalence in K, for all X, Y. If T Y is pseudo-exponentiable in K, then Y is pseudo-exponentiable in the Kleisli bicategory K_{T}.

Proof. Applying the three lemmas, we get a pseudo-natural transformation

K_{T} (− × T Y, Z) [figure] K_{T} (−, T Z^{T Y})

defined in (4) above, and so Y is pseudo-exponentiable in K_{T}.

Since Top/B and Cat/B satisfy the hypotheses of this theorem, it will be applied in the following sections to obtain pseudo-exponentiable objects of the homotopy slice Top//B and the pseudo-slice Cat//B.

4. Exponentiability in Top//B

In this section, we apply Theorem 3.4 to obtain pseudo-exponentiable objects of Top//B, including all fibrations which are exponentiable in Top/B.

Exponentiable objects of Top/B were characterized in [14] as follows. Given a map q: Y [figure] B, a functor ( )^{q}: Top/B [figure] Top/B is defined together with natural transformations

θ_{X,Z}: Top/B(X ×_{B} Y, Z) [figure] Top/B(X, Z^{Y})

where, by abuse of notation, Z^{Y} denotes the domain of r^{q} for r: Z [figure] B. Then q is exponentiable precisely when this functor is right adjoint to − × q, i.e., when these functions θ_{X,Z} are bijections, if and only if q satisfies a certain technical condition. This condition yields examples of exponentiable maps including all local homeomorphisms, locally trivial maps with locally compact fibers, locally closed inclusions, and locally compact spaces over a locally Hausdorff space. It is not difficult to show that when q is exponentiable, each θ_{X,Z} is an isomorphism of categories, so that these are precisely the 2-exponentiable objects of the 2-slice Top/B. By the remarks following Lemma 3.1, these 2-exponentiable objects provide a source of potential pseudo-exponentiable objects of Top/B and hence, by Theorem 3.4, pseudo-exponentiable objects of the homotopy slice Top//B.

4.1. Proposition. If q: Y [figure] B is a (Hurewicz) fibration and q is exponentiable in Top/B, then ev_{0}π_{1}: B^{I} ×_{B} Y [figure] B is pseudo-exponentiable in Top/B.

Proof. It suffices to show that η_{Y}: Y [figure] B^{I} ×_{B} Y is an equivalence in Top/B, for then q ≃ ev_{0}π_{1} and q is 2-exponentiable is Top/B, and so ev_{0}π_{1} is pseudo-exponentiable in Top/B. Consider the commutative diagram

[figure: (B^{I} ×_{B} Y) × I; B; ev∘π_{13}; B^{I} ×_{B} Y; (B^{I} ×_{B} Y) × I; ⟨id,1⟩; B^{I} ×_{B} Y; Y; π_{2}; Y; B; q; H]

where H exists since q is a fibration. Define η′_{Y}: B^{I} ×_{B} Y [figure] Y by

η′_{Y} (β, y) = H(β, y, 0)

Then η′ is a pseudo-inverse of η and the desired result follows.

Thus, we get the following corollary of Theorem 3.4.

4.2. Corollary. If q: Y [figure] B is a fibration and q is exponentiable in Top/B, then q is pseudo-exponentiable in Top//B.

Since pseudo-exponentiability is preserved by pseudo-equivalence, the following proposition yields additional examples.

4.3. Proposition. If f: X [figure] Y is a homotopy equivalence in Top and q: Y [figure] B then qf ≃ q in Top//B.

Proof. Suppose f: X [figure] Y is a homotopy equivalence. Then there exists g: Y [figure] X such that gf ≅ id_{X} and fg ≅ id_{Y}. Moreover, F: gf [figure] id_{X} and G: id_{Y} [figure] fg can be chosen so that (fF)(Gf) ∼ id_{f} and (F g)(gG) ∼ id_{g}, i.e., f and g are adjoint equivalences. Then the triangles

[figure: X; B; qf; X; Y; f; Y; B; q; id_{qf}; Y; B; q; Y; X; g; X; B; qf; qG]

give rise to morphisms (f, id_{qf}) and (g, qG) of Top//B such that

(f, id_{qf})(g, qG) ≅ (fg, qG) and (g, qG)(f, id_{qf}) ≅ (gf, qGf) ≅ (gf, qfF^{−1})

where the latter isomorphism follows from (fF)(Gf) ∼ id_{f}. Thus, it suffices to show that (id_{Y} , id_{q}) ≅ (fg, qG) and (gf, qfF^{−1}) ≅ (id_{X}, id_{qf}).

To see that (id_{Y} , id_{q}) ≅ (fg, qG), consider

[figure: Y × I; B; qπ_{1}; Y × I; Y; G; Y; B; q; Ψ]

where Ψ(y, t, u) = qG(y, tu). Then (G_{0}, Ψ_{0}) = (id_{Y} , id_{q}), since

(G(y, 0), Ψ(y, 0, u)) = (y, qG(y, 0)) = (y, qy)

and (G_{1}, Ψ_{1}) = (fg, qG), since (G(y, 1), Ψ(y, 1, u)) = (fgy, qG(y, u)).

To see that (gf, qfF^{−1}) ≅ (id_{X}, id_{qf}), consider

[figure: X × I; B; qf; X × I; X; F; X; B; qf; Φ]

where Φ(x, t, u) = qfF (x, 1 − u + tu). Then (F_{0}, Φ_{0}) = (gf, qfF^{−1}), since

(F (x, 0), Φ(x, 0, u)) = (gfx, qfF (x, 1 − u))

and (F_{1}, Φ_{1}) = (id_{X}, id_{qf}), since

(F (x, 1), Φ(x, 1, u)) = (x, qfF (x, 1)) = (x, qfx)

as desired.

4.4. Corollary. If f: X [figure] Y is a homotopy equivalence in Top and q: Y [figure] B is an exponentiable fibration in Top/B, then qf is pseudo-exponentiable in Top//B.

Proof. Since qf ≃ q by Proposition 4.3 and q is pseudo-exponentiable by Corollary 4.2, it follows that qf is pseudo-exponentiable in Top//B.

4.5. Corollary. If f: X [figure] Y is a homotopy equivalence in Top and qf: X [figure] B is an exponentiable fibration in Top/B, then q is pseudo-exponentiable in Top//B.

Proof. Since qf ≃ q by Proposition 4.3 and qf is pseudo-exponentiable by Corollary 4.2, it follows that q is pseudo-exponentiable in Top//B.

5. Exponentiability in Cat//B

In this section, we prove Johnstone’s theorem [6] characterizing pseudo-exponentiable Y [figure] B of the pseudo-slice Cat//B, and show that this is also equivalent to the exponentiability of B^{I} ×_{B} Y [figure] B in the 2-slice Cat/B.

Conduché [3] and Giraud [5] independently showed that the exponentiable objects of Cat//B are those q: Y [figure] B satisfying the following factorization lifting property (FL). Given γ: Y [figure] Y′ in Y and a factorization qγ = β_{2}β_{1} in B, the following diagram can be completed

[figure: Y; B; q; Y; Y′′; γ_{1}; Y; Y′; γ; Y′; Y′′; γ_{2}; qY; B′′; β_{1}; qY; qY′; qγ; qY′; B′′; β_{2}]

i.e., there exists an object Y′′ over B′′ and a factorization γ = γ_{2}γ_{1} in Y such that qγ_{1} = β_{1} and qγ_{2} = β_{2}. Moreover, this factorization is unique in the sense that any two such are equivalent via the equivalence relation generated by the relation (Y′′, γ_{1}, γ_{2}) ∼ (Ȳ′′, γ̄_{1}, γ̄_{2}) if there exists a morphism θ: Y′′ [figure] Ȳ′′ over the identity making the following diagram commute

[figure: Y; Ȳ′′; γ̄_{1}; Y′′; Y; γ_{1}; Y′′; Ȳ′′; Y′; Ȳ′′; γ̄_{2}; Y′′; Y′; γ_{2}; Y′′; Ȳ′′; θ]

To Conduché and Giraud this was a 1-dimensional problem, but Johnstone [6] pointed out that FL also characterizes 2-exponentiable objects in the 2-slice Cat/B. As noted in the introduction above, he also defined the following factorization pseudo-lifting property (FPL) in [6] and sketched the proof of its sufficiency for pseudo-exponentiability in Cat//B. In this section, we prove Johnstone’s theorem using Theorem 3.4 for the sufficiency of FPL and a variation of the proof in [13] for its necessity.

A functor q: Y [figure] B satisfies the factorization pseudo-lifting property (FPL) if

[figure: Y; B; q; Y; Y′′; γ_{1}; Y; Y′; γ; Y′; Y′′; γ_{2}; qY; B′′; β_{1}; qY; qY′′; qγ_{1}; B′′; qY′; β_{2}; B′′; qY′′; δ; qY′; qY′′; qγ_{2}]

i.e., given γ: Y [figure] Y′ in Y and a factorization qγ = β_{2}β_{1} in B, there exists a factorization γ = γ_{2}γ_{1} in Y and an isomorphism δ: B′′ [figure] qY′′ such that the diagram in B commutes. Moreover, this factorization is unique in the sense that any two such are equivalent via the equivalence relation generated by

(Y′′, γ_{1}, γ_{2}, δ) ∼ (Ȳ′′, γ̄_{1}, γ̄_{2}, δ̄)

if there exists a morphism θ: Y′′ [figure] Ȳ′′ such the following diagram commutes

[figure: qY′′; qȲ′′; qθ; B′′; qY′′; δ; B′′; qȲ′′; δ̄]

5.1. Theorem. The following are equivalent.

(a) q: Y [figure] B satisfies FPL.

(b) ev_{0}π_{1}: B^{I} ×_{B} Y [figure] B satisfies FL.

(c) ev_{0}π_{1}: B^{I} ×_{B} Y [figure] B is 2-exponentiable in Cat/B.

(d) q: Y [figure] B is pseudo-exponentiable in Cat//B. Proof. It suffices to prove (a)⇒(b) and (d)⇒(a), for (b)⇒(c) is essentially the Conduché/Giraud theorem and (c)⇒(d) follows from Theorem 3.4 since 2-exponentiable objects of Cat/B are necessarily pseudo-exponentiable.

For (a)⇒(b), suppose q satisfies FPL. To show ev_{0}π_{1} satisfies FL, let

(β, qγ): (B [figure: α] qY, Y) [figure] (B′ [figure: α′] qY′, Y′)

be a morphism of B^{I} ×_{B} Y and suppose β = β_{2}β_{1}. Thus, we have a commutative diagram

[figure: qY; qY′; qγ; B; qY; α; B; B′; β; B′; qY′; α′]

Since

qY [figure: α^{−1}] B [figure: β_{1}] B′′ [figure: β_{2}] B′ [figure: α′] qY′

is a factorization of qγ, applying FPL for q we get a factorization

Y [figure: γ_{1}] Y′′ [figure: γ_{2}] Y′

of γ in Y and an isomorphism δ: B′′ [figure] qY′′ such that

[figure: qY; B′′; β_{1}α^{−1}; qY; qY′′; qγ_{1}; B′′; qY′; α′β_{2}; B′′; qY′′; δ; qY′; qY′′; qγ_{2}]

commutes. Then (B′′ [figure: δ] qY′′, Y′′) is an object of B^{I} ×_{B} Y and the commutative diagram

[figure: qY; qY′′; qγ_{1}; B; qY; α; B; B′′; β_{1}; B′′; qY′′; δ; qY′′; qY′; qγ_{2}; B′′; qY′′; δ; B′′; B′; β_{2}; B′; qY′; α′]

gives the desired factorization of (β, qγ) in B^{I} ×_{B} Y. Moreover, the uniqueness condition of FL follows from that of FPL.

For (d)⇒(a), suppose q: Y [figure] B is pseudo-exponentiable in Cat//B. Then −×_{B} B^{I} ×_{B} Y preserves pseudo-pushouts (i.e., cocomma objects) in Cat//B. To show that q satisfies FPL, suppose qγ = β_{2}β_{1} where γ: Y [figure] Y′. Then β_{2}β_{1} induces a functor p: X [figure] B, where X is defined by the pseudo-pushout

[figure: 2; X; 1; 2; 1; 1; 2; 0; 2; X]  (5)

and 2 = {0, 1} is the category with one morphism 0 [figure] 1. Thus, (5) becomes a pseudopushout in Cat//B via p: X [figure] B. Note that X can be constructed as the colimit of the diagram

[figure: I; I; 0; I; I; 1; I; 2; 1; I; 2; 0]

it follows that X is the category

[figure: ·; ·; α_{1}; ·; ·; α_{2}; ≅]

with p(α_{1}) = β_{1} and p(α_{2}) = β_{2}. Since − ×_{B} B^{I} ×_{B} Y preserves pseudo-pushouts, it follows that the diagram obtained by applying − ×_{B} B^{I} ×_{B} Y to (5) is a pseudo-pushout in Cat. Thus, X ×_{B} B^{I} ×_{B} Y is given by the colimit of

[figure: I ×_{B} B^{I} ×_{B} Y; I ×_{B} B^{I} ×_{B} Y; 0×id; I ×_{B} B^{I} ×_{B} Y; I ×_{B} B^{I} ×_{B} Y; 1×id; I ×_{B} B^{I} ×_{B} Y; 2 ×_{B} B^{I} ×_{B} Y; 1×id; I ×_{B} B^{I} ×_{B} Y; 2 ×_{B} B^{I} ×_{B} Y; 0×id]

and so FPL follows from the construction of colimits in Cat.

18  SUSAN NIEFIELD

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Union College Department of Mathematics Schenectady, NY 12308

Email: niefiels@union.edu

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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