ASPHERICITY STRUCTURES, SMOOTH FUNCTORS, AND FIBRATIONS

GEORGES MALTSINIOTIS

This paper was translated from French by Jonathan Chiche.

Abstract. The aim of this paper is to generalize Grothendieck’s theory of smooth functors in order to include within this framework the theory of fibered categories. We obtain in particular a new characterization of fibered categories.

Introduction

In “Pursuing Stacks” [6], Grothendieck defines the notions of proper functors and smooth functors by analogy with the cohomological properties of proper morphisms and smooth morphisms between schemes. If

D = \xymatrix{X′ \ar[r]^{g} \ar[d]_{f′} & X \ar[d]^{f} \\ Y′ \ar[r]_{h} & Y}

is a commutative square of schemes and M is an abelian étale sheaf on X, there is a canonical “base change” morphism

h^{∗}Rf_{∗}(M) [figure] Rf′_{∗} g^{∗}(M) .

Proper and smooth base change theorems state [1, lectures 12, 13 and 16] that if the square D is cartesian, then this morphism is an isomorphism in the following two cases:

(a) f is proper and M is of torsion;

(b) h is smooth, M is of torsion prime to the residual exponents of Y, and f is quasi-compact and quasi-separated.

We denote by Cat the category of small categories and by W_{∞} the class of arrows in Cat whose image under the nerve functor is a simplicial weak equivalence, i.e. a morphism of simplicial sets whose topological realization is a homotopy equivalence. The localized category W_{∞}^{−1}Cat, obtained by formally inverting the arrows which belong to W_{∞}, is then equivalent to the homotopy category Hot of CW-complexes. For every small category I, define 𝔻(I) by

𝔻(I) = (W_{∞}^{I^{∘}})^{−1} \underline{Hom}(I^{∘}, Cat) ,

where \underline{Hom}(I^{∘}, Cat) denotes the category of presheaves on I with values in Cat, i.e. the contravariant functors from I to Cat. Let W_{∞}^{I^{∘}} be the class of natural transformations between functors from I^{∘} to Cat which are componentwise in W_{∞}:

W_{∞}^{I^{∘}} = \{α ∈ Fl(\underline{Hom}(I^{∘}, Cat)) | α_{i} ∈ W_{∞}, i ∈ Ob(I)\} .

For every arrow u : I [figure] J of Cat, the inverse image functor

\underline{Hom}(J^{∘}, Cat) [figure]\underline{Hom}(I^{∘}, Cat) , F [figure] F ∘ u^{∘} ,

defines a functor

u^{∗} : 𝔻(J) [figure] 𝔻(I) .

By a result of Alex Heller [8], this functor has a left and a right adjoint

u_{!} : 𝔻(I) [figure] 𝔻(J)  and  u_{∗} : 𝔻(I) [figure] 𝔻(J)

respectively.

Grothendieck defines the notion of proper (resp. smooth) functor as follows.

Definition. A functor between small categories u : A [figure] B (resp. w : B′ [figure] B) is said to be proper (resp. smooth) if for every cartesian square

\xymatrix{A′ \ar[r]^{v} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{w} & B & ,}

the base change morphism

w^{∗}u_{∗} [figure] u′_{∗}v^{∗}

(or, equivalently,

v_{!}u′^{∗} [figure] u^{∗}w_{!} ,

transpose of the previous morphism) is an isomorphism, and this property remains true after any base change.

Grothendieck obtains simple characterizations of proper functors and smooth functors, and he notices that his theory relies only on a small number of formal properties of W_{∞}, the most important one being Quillen’s Theorem A [10]. This leads him to introduce the notion of basic localizer.

A basic localizer is a class W of arrows of Cat satisfying the following properties. Loc1 (Weak saturation.) a) Identities belong to the class W.

b) If two out of the three arrows in a commutative triangle belong to W, then so does the third.

c) If i is an arrow of Cat which has a retraction r and if ir belongs to W, then i belongs to W.

Loc2 If A is a small category which has a final object, then the functor A [figure] e

from A to the final category e belongs to W. Loc3 (Relative Quillen’s Theorem A.) For every commutative triangle

\xymatrix{A \ar[rr]^{u} \ar[dr]_{v} &  & B \ar[dl]^{w} \\  & C}

of Cat, if for every object c of C, the functor u/c : A/c [figure] B/c (where A/c = A×_{C} (C/c) and B/c = B ×_{C} (C/c) stand for the “comma categories” of objects over c) induced by u belongs to W, then so does u.

Grothendieck conjectures in [6] that W_{∞} is the smallest basic localizer. This conjecture is proved by D.-C. Cisinski in [5].

A small category A is said to be W-aspheric if the morphism A [figure] e belongs to W. A functor between small categories A [figure] B is said to be W-aspheric if for every object b of B, the category A/b is W-aspheric. It follows from Loc3 that a W-aspheric functor belongs to W. As in the case W = W_{∞}, we define, for every small category I, the category 𝔻(I) by

𝔻(I) = (W^{I^{∘}})^{−1} \underline{Hom}(I^{∘}, Cat) ,

and for every arrow u : I [figure] J of Cat, we can show by elementary methods [9] that the inverse image functor u^{∗} : 𝔻(J) [figure] 𝔻(I) has a left adjoint u_{!} : 𝔻(I) [figure] 𝔻(J).

We can therefore define the notion of W-proper and W-smooth functors as in the case of W = W_{∞}, using base change morphisms related to u_{!}.

In fact, the theory developed in D.-C. Cisinski’s thesis [4] shows also the existence of a right adjoint u_{∗} : 𝔻(I) [figure] 𝔻(J), assuming a mild hypothesis on W, namely that W is accessible, i.e. that it is generated by a set of arrows of Cat (it is the smallest basic localizer containing a set of arrows of Cat).

Grothendieck obtains the following characterization of W-smooth functors [7], [9].

Theorem. Let u : A [figure] B be a morphism of Cat. The following conditions are equivalent:

(a) u is W-smooth, i.e. for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d]_{u′′} & A′ \ar[r] \ar[d]^{u′} & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

the base change morphism u′′_{!}v^{∗} [figure] w^{∗}u′_{!} is an isomorphism;

(b) for every object b of B, the inclusion

A_{b}  [figure] b\A ,  a [figure] (a, 1_{b} : b [figure] u(a) = b)

(where A_{b} denotes the fiber of A over b and b\A = A ×_{B} (b\B) the “comma category” of objects under b) is a W-aspheric functor.

(c) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

if the functor w is W-aspheric, then so is v.

Grothendieck is filled with wonder by the following fact that he deduces from this theorem:

Corollary. A morphism u : A [figure] B of Cat is W-smooth if and only if the opposite functor u^{∘} : A^{∘} [figure] B^{∘} is W-proper.

This result is a straightforward consequence of the characterization above, of the “dual” characterization of W-proper morphisms, and of the fact that a functor between small categories belongs to W if and only if the opposite functor does [6], [7], [9].

Fibrations (morphisms u : A [figure] B of Cat such that A is a fibered category over B) are important examples of W-smooth functors. This follows from the fact that if u : A [figure] B is a fibration, then for every object b of B the inclusion

A_{b}  [figure] b\A ,  a [figure] (a, 1_{b} : b [figure] u(a) = b)

has a right adjoint, and from the fact that a functor between small categories which has a right adjoint is a W-aspheric functor.

The impetus for this work was the observation that the class of fibrations shares many formal properties with the class of W-smooth functors and that the class of functors which have a right adjoint shares many properties with the class of W-aspheric functors. Nevertheless, there is no basic localizer W such that the W-smooth morphisms are exactly the fibrations, or such that the W-aspheric morphisms are exactly the functors which have a right adjoint. Moreover, we notice that the notions of W-aspheric, of W-smooth and of W-proper morphisms depend on the sole class of W-aspheric categories. If two basic localizers have the same class of aspheric objects, then the corresponding notions of aspheric functors, smooth functors and proper functors are the same. So I tried to look for the minimal properties that a class of objects of Cat should fulfil in order to give rise to a theory of smooth functors. This has led me, at the cost of breaking the symmetry of passing to the opposite category, to introduce the notion of right asphericity structure. Since this notion is not self-dual, one needs also to consider the dual notion of left asphericity structure, giving rise to a theory of proper functors.

In the first section, we define the notion of right asphericity structure, implying a notion of aspheric functor. We notice that there exists a minimal right asphericity structure, defined by the class of small categories which have a final object, the corresponding aspheric functors being exactly the functors between small categories which have a right adjoint. For every basic localizer, we define a right asphericity structure with the same aspheric functors as the given localizer.

The second and the third sections are logically independent. In the second one, we consider localizations of categories of functors with domain a small category and codomain Cat, with respect to natural transformations that are componentwise aspheric, and we prove the existence of left homotopical Kan extensions.

In the third one, we introduce the notion of smooth functor associated with a right asphericity structure. In this framework, it splits into two notions: smooth functors and weakly smooth functors. We give several equivalent characterizations for each of these two notions and we study their main properties. We show that the smooth functors with respect to the minimal right asphericity structure are exactly the fibrations, and that the weakly smooth functors with respect to this structure are exactly the prefibrations. This result provides a new characterization of fibered categories. Finally, we observe that, when the right asphericity structure is defined by a basic localizer, the notions of smooth functor and weakly smooth functor are equivalent and equivalent to the notion of smooth functor with respect to the basic localizer.

In the last section, we combine the results of the two previous ones. We prove a characterization of smooth functors (with respect to a right asphericity structure) in terms of base change morphisms. As in this paper we consider covariant functors with values in Cat, rather than presheaves, we obtain a characterization which is dual to Grothendieck’s formulation.

Recently, Jonathan Chiche has generalized the notion of right asphericity structure, and some of the results of this paper, to the framework of 2-categories [3].

1. Right asphericity structures

1.1. A right asphericity structure is a class A of small categories satisfying the following two conditions.

As1 Every small category which has a final object is in A.

As2 For every functor between small categories u : A [figure] B, if B is in A, and if for every object b of B, A/b is in A, then A is also in A.

Example 1.2. The class A of small categories which have a final object is a right asphericity structure. Indeed, condition As1 being fulfilled by definition, it is enough to check condition As2. Let B be a category with a final object b_{0}, and u : A [figure] B a morphism in Cat such that for every object b of B, A/b has a final object. Then the category A, which is isomorphic to A/b_{0}, has a final object, hence the assertion. This right asphericity structure is the minimal right asphericity structure.

Example 1.3. Let W be a basic localizer. The class of W-aspheric categories is a right asphericity structure.

In the sequel, we fix, once and for all, a right asphericity structure A.

1.4. A small category C is said to be A-aspheric, or more simply aspheric, if it belongs to A. Using this definition, condition As1 states that a small category which has a final object is aspheric.

Proposition 1.5. The product of two small aspheric categories is an aspheric category.

Proof. Let A and B be two small aspheric categories. Let us show that their product A×B is aspheric too. Considering the first projection A×B [figure] A, it suffices by As2 to show that for every object a of A, the category (A × B)/a ≃ A/a × B is aspheric. Considering the second projection A/a × B [figure] B, it suffices by As2 to show that for every object b of B, the category (A/a × B)/b ≃ A/a × B/b is aspheric. Now, the latter has a final object, and the assertion follows from As1. □

1.6. A morphism u : A [figure] B is said to be A-aspheric, or more simply aspheric, if for every object b of B, the category A/b is aspheric. In terms of this definition, condition As2 states that if the codomain of an aspheric morphism is an aspheric category, then its domain is aspheric, too. Notice that a small category A is aspheric if and only if the functor A [figure] e from A to the final category is an aspheric morphism. For every small category A, the identity functor 1_{A} : A [figure] A is aspheric. Indeed, for every object a of A, the category A/a has a final object, and the assertion follows from condition As1.

Example 1.7. If A is the minimal right asphericity structure (cf. 1.2), then the aspheric morphisms are exactly the functors between small categories which have a right adjoint. Indeed, a functor between small categories u : A [figure] B has a right adjoint if and only if, for every object b of B, the category A/b has a final object.

Example 1.8. If A is the right asphericity structure associated with a basic localizer W (cf. 1.3), then the A-aspheric morphisms are exactly the W-aspheric functors.

Corollary 1.9. Let u : A [figure] B and u′ : A′ [figure] B′ be two aspheric morphisms of Cat. Then the functor u × u′ : A × A′ [figure] B × B′ is aspheric.

Proof. We have to show that for every object (b, b′) of B × B′, the category (A × A′)/(b, b′) is aspheric. Now, (A × A′)/(b, b′) is canonically isomorphic to (A/b) × (A′/b′) and by hypothesis A/b and A′/b′ are aspheric. Therefore the assertion follows from proposition 1.5. □

Proposition 1.10. Let

\xymatrix{A \ar[rr]^{u} \ar[dr] &  & B \ar[dl]^{v} \\  & C}

be a commutative triangle in Cat. The morphism u is aspheric if and only if, for every object c of C, the morphism u/c : A/c [figure] B/c, induced by u, is aspheric.

Proof. We check immediately that for every object c of C and for every object (b, p : v(b) [figure] c) of B/c, the category (A/c)/(b, p) is canonically isomorphic to A/b. We deduce that if u is aspheric, then so is u/c. Conversely, assume that for every object c of C the morphism u/c is aspheric. Then, for every object b of B, the category (A/v(b))/(b, 1_{v(b)}) ≃ A/b is aspheric, which proves that u is aspheric. □

Proposition 1.11. Let \xymatrix{A \ar[r]^{u} & B \ar[r]^{v} & C} be a pair of composable morphisms of Cat. If u and v are aspheric, then so is vu.

Proof. For every object c of C, since v is aspheric, the category B/c is aspheric, and by the previous proposition, since u is aspheric, so is the functor u/c : A/c [figure] B/c induced by u. From this we deduce that the category A/c is aspheric (As2), which proves that vu is aspheric. □

Proposition 1.12. Let u : A [figure] B be a morphism of Cat. If u has a right adjoint, then u is aspheric.

Proof. Let v : B [figure] A be a right adjoint to u. The functorial bijection

Hom_{B}(u(a), b)) ≃ Hom_{A}(a, v(b)), a ∈ Ob(A), b ∈ Ob(B),

implies that, for every object b of B, the category A/b is isomorphic to the category A/v(b), which has a final object. It follows that A/b is aspheric, hence the proposition. □

Corollary 1.13. An equivalence between small categories is an aspheric functor.

1.14. Let u : A [figure] B be a functor. We recall that the fiber of u over an object b of B is the subcategory (not full in general) A_{b} of A whose objects are the objects a of A such that u(a) = b, and whose morphisms are the arrows f of A such that u(f) = 1_{b}. An arrow c : a [figure] a′ of A is called cocartesian (with respect to u, or over B) if for every morphism f : a [figure] a′′ of A such that u(f) = u(c), there is a unique morphism g : a′ [figure] a′′ of A such that u(g) = 1_{u(a′)} and f = gc.

\xymatrix{ & a′′ \\ a \ar[ur]^{f} \ar[r]_{c} & a′ \ar@{-->}[u]_{g} \\ u(a) \ar[r]_{u(c)} & u(a′)}

The arrow c is called hypercocartesian (with respect to u, or over B) if for every morphism f : a [figure] a′′ of A and every morphism h : u(a′) [figure] u(a′′) of B such that u(f) = hu(c), there is a unique morphism g : a′ [figure] a′′ of A such that u(g) = h and f = gc.

\xymatrix{ &  & a′′ \\ a \ar[urr]^{f} \ar[r]_{c} & a′ \ar@{-->}[ur]_{g} \\ u(a) \ar[r]_{u(c)} & u(a′) \ar[r]_{h} & u(a′′)}

The functor u is called a precofibration if for every morphism p : b [figure] b′ of B, and for every object a of A over b (i.e. u(a) = b), there is a cocartesian morphism c : a [figure] a′ over p (i.e. u(c) = p). The functor u is called a cofibration if u is a precofibration, and if the class of cocartesian morphisms of A is stable under composition. It is easily checked that u is a cofibration if and only if for every morphism p : b [figure] b′ of B and for every object a of A over b, there is a hypercocartesian morphism c : a [figure] a′ over p.

Dually, a morphism of A is called cartesian (resp. hypercartesian) with respect to u, or over B, if the corresponding morphism of A^{∘} (the opposite category of A) is cocartesian (resp. hypercocartesian) with respect to u^{∘} : A^{∘} [figure] B^{∘}. The functor u is called a prefibration (resp. a fibration) if the functor u^{∘} is a precofibration (resp. a cofibration).

Lemma 1.15. A functor u : A [figure] B is a precofibration if and only if for every object b of B, the canonical functor A_{b} [figure] A/b, which sends an object a of the fiber A_{b} of u over b to the object (a, 1_{b}) of A/b, has a left adjoint.

The proof is left to the reader.

Proposition 1.16. Let u : A [figure] B be a morphism of Cat, and assume that u is a precofibration and that for every object b of B, the fiber A_{b} of u over b is aspheric. Then u is aspheric.

Proof. According to the previous lemma, for every object b of B, the functor

\begin{matrix}i_{b} : A_{b} & [figure] A/b \\ a & [figure] (a, 1_{b}) \end{matrix}

has a left adjoint

j_{b} : A/b [figure] A_{b} .

It follows from proposition 1.12 that the functor j_{b} is aspheric. Since A_{b} is aspheric, so is A/b (As2), hence u is aspheric. □

1.17. A morphism u : A [figure] B of Cat is said to be locally A-aspheric, or more simply locally aspheric, if for every object a of A, the morphism

A/a [figure] B/b , b = u(a) ,

induced by u, is aspheric.

Examples 1.18. a) A fully faithful aspheric functor is locally aspheric.

b) For any small category C, the functor C [figure] e is locally aspheric.

Proposition 1.19. a) A morphism u : A [figure] B of Cat is locally aspheric if and only if, for every object b of B, the functor u/b : A/b [figure] B/b, induced by u, is aspheric.

b) Let \xymatrix{A \ar[r]^{u} & B \ar[r]^{v} & C} be a pair of composable morphisms of Cat. If u and v are locally aspheric, then so is vu.

c) Let u : A [figure] B, u′ : A′ [figure] B′ be two locally aspheric morphisms of Cat. Then the functor u × u′ : A × A′ [figure] B × B′ is locally aspheric.

Proof. The first assertion is straightforward, the second one is a consequence of proposition 1.11, and the third one is a consequence of corollary 1.9. □

2. Homotopical Kan extensions

In this paragraph, we fix, once and for all, a right asphericity structure A.

2.1. We denote by Asph_{A}, or simply Asph, the class of aspheric morphisms of Cat, and we set

Hot = Hot_{A} = Asph^{−1}Cat .

More generally, for every small category I, we denote by Asph_{A}(I), or simply Asph(I), the class of morphisms of \underline{Hom}(I, Cat) which are componentwise aspheric, and we set

Hot(I) = Hot_{A}(I) = Asph(I)^{−1} \underline{Hom}(I, Cat) .

For every arrow w : J [figure] I of Cat, if we denote by

w^{∗} : \underline{Hom}(I, Cat) [figure]\underline{Hom}(J, Cat)

the inverse image functor, we have

w^{∗}(Asph(I)) ⊂ Asph(J) .

Therefore w^{∗} induces a functor between the localized categories, denoted

w^{∗} : Hot(I) [figure]Hot(J) ,

too, such that the following square is commutative:

\xymatrix{\underline{Hom}(I, Cat) \ar[r]^{w^{∗}} \ar[d]_{γ_{I}} & \underline{Hom}(J, Cat) \ar[d]^{γ_{J}} \\ Hot(I) \ar[r]_{w^{∗}} & Hot(J) & ,}

where the vertical arrows are the canonical localization functors.

Example 2.2. If A is the minimal right asphericity structure (cf. 1.2), then Hot is the category \overline{Cat} of small categories up to homotopy, i.e. the category with same objects as Cat, and such that for every pair of small categories A, B,

Hom_{\overline{Cat}} (A, B) = π_{0} \underline{Hom}(A, B) .

Indeed, let us denote by

γ : Cat  [figure]Hot ,  Q : Cat  [figure]\overline{Cat}

the canonical functors. By the universal properties of these functors, it is sufficient to show that:

a) the image under Q of an aspheric functor is a homotopism;

b) the images under γ of two homotopic functors are equal. In order to show the first assertion, we notice that, since aspheric morphisms of Cat with respect to the minimal right asphericity structure are exactly the functors between small categories which have a right adjoint (1.7), they are indeed homotopisms. In order to show the second assertion, it suffices to show that for every morphism h : Δ_{1} × A [figure] B of Cat, where Δ_{1} stands for the category {0 [figure] 1}, we have γ(h_{0}) = γ(h_{1}), where h_{ε} = h(∂_{ε} × 1_{A}), ε = 0, 1, and ∂_{ε} : e [figure] Δ_{1} is the morphism from the final category e to Δ_{1} defined by the object ε of Δ_{1}. Now, if we denote by p : Δ_{1} ×A [figure] A the second projection, we have p(∂_{0} ×1_{A}) = 1_{A} = p(∂_{1} ×1_{A}). Since the category Δ_{1} has a final object, it follows from As1 and corollary 1.9 that the functor p is aspheric. Therefore γ(p) is an isomorphism of Hot, and the equalities

γ(p)γ(∂_{0} × 1_{A}) = γ(p(∂_{0} × 1_{A})) = γ(p(∂_{1} × 1_{A})) = γ(p)γ(∂_{1} × 1_{A})

imply the relation

γ(∂_{0} × 1_{A}) = γ(∂_{1} × 1_{A}) ,

and hence we have

γ(h_{0}) = γ(h(∂_{0}×1_{A})) = γ(h)γ(∂_{0}×1_{A}) = γ(h)γ(∂_{1}×1_{A}) = γ(h(∂_{1}×1_{A})) = γ(h_{1}) .

Example 2.3. If A is the right asphericity structure associated with a basic localizer W (cf. 1.3), then it follows from the theory developed by D.-C. Cisinski [4] that

Hot ≃ W_{asph}^{−1} Cat ,

where W_{asph} stands for the basic localizer generated by the arrows A [figure] e of Cat, for A a W-aspheric category. Indeed, let us denote by

\xymatrix{Cat \ar[r]^{γ} & Hot ,  Cat \ar[r]^{γ′} & W_{asph}^{−1} Cat}

the canonical functors. By their universal properties, it suffices to show that: a) Asph ⊂ W_{asph};

b) γ(W_{asph}) is contained in the class of isomorphisms of Hot. The assertion (a) is obvious. In order to prove (b), we first observe that every basic localizer being a filtered union of accessible ones, an easy argument shows that we can assume that the basic localizer is accessible. Thomason-Cisinski theory [4], [11] then states that there exists a proper closed model category structure on Cat whose weak equivalences are the elements of W_{asph}. This implies that Cat has the structure of a category of fibrant objects [2], with weak equivalences the elements of W_{asph} and fibrations the arrows u : A [figure] B of Cat such that for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

if w is in W_{asph}, then so is v. Hence, by Ken Brown’s lemma [2, I.1 Factorization lemma], in order to prove that the elements of γ(W_{asph}) are isomorphisms of Hot, it suffices to show that if such a fibration is in W_{asph}, then its image under γ is an isomorphism. Now, such an arrow is universally in W_{asph}. It is therefore a W_{asph}-aspheric morphism, and in particular W-aspheric, which proves the assertion.

We can easily generalize this argument to show that for every small category I, the category Hot(I) is isomorphic to the localization of the category \underline{Hom}(I, Cat) by the arrows which are componentwise in W_{asph}.

2.4. For every object I of Cat, we denote by Cat/I the category of small categories over I, whose objects are the pairs (A, A [figure] I), consisting of a small category A and of a functor A [figure] I, and whose morphisms are commutative triangles

\xymatrix{A \ar[dr] \ar[rr] &  & A′ \ar[dl] \\  & I & .}

We denote by Asph/I the class of arrows

\xymatrix{A \ar[dr] \ar[rr] &  & A′ \ar[dl] \\  & I}

of Cat/I such that A [figure] A′ is an aspheric functor. For every arrow w : J [figure] I of Cat, we denote by

Cat/w : Cat/J [figure] Cat/I

the functor defined by

[figure: A; v; A; wv; J; I; .]

We have

(Cat/w)(Asph/J) ⊂ Asph/I ,

and thus the functor Cat/w induces a functor

\overline{Cat/w} : (Asph/J)^{−1}(Cat/J) [figure] (Asph/I)^{−1}(Cat/I) .

2.5. For every small category I, we define functors

\xymatrix{Cat/I \ar[r]^{Θ_{I}} & \underline{Hom}(I, Cat)} , \xymatrix{\underline{Hom}(I, Cat) \ar[r]^{Θ′_{I}} & Cat/I}

(A, A [figure] I) [figure](i [figure] A/i) ,  F [figure](∫ F, ∫ F  [figure] I) ,

where ∫ F stands for Grothendieck’s construction of the cofibered category over I defined by the functor F, and ∫ F [figure] I is the canonical functor. We recall that the objects of the category ∫ F are the pairs (i, a), where i is an object of I and a is an object of F (i). A morphism of ∫ F from (i, a) to (i′, a′) is a pair (k, f), where k : i [figure] i′ is an arrow of I and f : F (k)(a) [figure] a′ is an arrow of F (i′). The composition in ∫ F is defined, for a pair of composable morphisms

\xymatrix{(i, a) \ar[r]^{(k,f)} & (i′, a′) \ar[r]^{(k′,f′)} & (i′′, a′′) ,}

by the formula

(k′, f′) ∘ (k, f) = (k′k, f′ · F (k′)(f)) .

The canonical functor ∫ F [figure] I, which is a cofibration, is defined by

(i, a) [figure] i , (k, f) [figure] k .

For every object i of I, the fiber over i of the functor ∫ F [figure] I is canonically isomorphic to the category F (i).

For every morphism w : J [figure] I of Cat, the functors

\xymatrix{Cat/J \ar[r]^{Θ_{I}∘Cat/w} & \underline{Hom}(I, Cat)} , \xymatrix{\underline{Hom}(I, Cat) \ar[r]^{Θ′_{J} ∘w^{∗}} & Cat/J}

form an adjoint pair with counit and unit

ε : Θ_{I} ∘ Cat/w ∘ Θ′_{J} ∘ w^{∗}  [figure] 1_{\underline{Hom}(I,Cat)} , η : 1_{Cat/J}  [figure] Θ′_{J} ∘ w^{∗} ∘ Θ_{I} ∘ Cat/w

defined as follows.

a) Definition of ε. For every functor F : I [figure] Cat, and every object i of I, we need to define a functor

ε_{F,i} : (∫ F w) / i [figure] F (i) .

The objects of the category (∫ F w)/i are the triples (j, a, p : w(j) [figure] i), where j is an object of J, a is an object of F w(j), and p is an arrow of I. A morphism of (∫ F w)/i from (j, a, p : w(j) [figure] i) to (j′, a′, p′ : w(j′) [figure] i) is a pair (l, f), where l : j [figure] j′ is an arrow of J and f : F w(l)(a) [figure] a′ is an arrow of F w(j′) such that

\xymatrix{w(j) \ar[rr]^{w(l)} \ar[dr]_{p} & & w(j′) \ar[dl]^{p′} \\ & i} p = p′w(l) .

We define the functor ε_{F,i} by

\begin{matrix}ε_{F,i}(j, a, p : w(j) [figure] i) = F (p)(a) , \\ ε_{F,i}(l, f) = F (p′)(f) : F (p′)F w(l)(a) = F (p)(a) [figure] F (p′)(a′) . \end{matrix}

We leave it to the reader to check compatibility with respect to composition and identities, as well as functoriality in i and in F.

b) Definition of η. For every object (A, v : A [figure] J) of Cat/J, we need to define a functor

η_{(A,v)} : A [figure] ∫ A/w(j) ,

over J, where ∫ A/w(j) denotes, by a slight abuse, the cofibered category over J defined by the functor

J  [figure] Cat ,  j [figure] A/w(j) .

The objects of ∫ A/w(j) are the triples (j, a, p : wv(a) [figure] w(j)), where j is an object of J, a is an object of A and p is an arrow of I. A morphism

(j, a, p : wv(a) [figure] w(j)) [figure] (j′, a′, p′ : wv(a′) [figure] w(j′))

is a pair (l, f), where l : j [figure] j′ is an arrow of J and f : a [figure] a′ is an arrow of A such that the square

\xymatrix{wv(a) \ar[r]^{wv(f)} \ar[d]_{p} & wv(a′) \ar[d]^{p′} \\ w(j) \ar[r]_{w(l)} & w(j′)}

is commutative. For every object a of A, we define

η_{(A,v)}(a) = (v(a), a, 1_{wv(a)} : wv(a) [figure] wv(a)) ,

and for every arrow f : a [figure] a′ of A,

η_{(A,v)}(f) = (v(f), f) : (v(a), a, 1_{wv(a)}) [figure] (v(a′), a′, 1_{wv(a′)}).

Compatibility with respect to composition and identities, as well as functoriality in (A, v), are easily checked.

We leave it to the reader to check the triangle identities

((Θ′_{J}∘w^{∗})⋆ε)(η⋆(Θ′_{J}∘w^{∗})) = 1_{Θ′_{J} ∘w^{∗}} , (ε⋆(Θ_{I}∘Cat/w))((Θ_{I}∘Cat/w)⋆η) = 1_{Θ_{I}∘Cat/w}

\xymatrix{Θ′_{J} ∘ w^{∗} \ar[r]^{η⋆(Θ′_{J} ∘w^{∗})} & Θ′_{J} ∘ w^{∗} ∘ Θ_{I} ∘ Cat/w ∘ Θ′_{J} ∘ w^{∗} \ar[r]^{(Θ′_{J} ∘w^{∗})⋆ε} & Θ′_{J} ∘ w^{∗}}
\xymatrix{Θ_{I} ∘ Cat/w \ar[r]^{(Θ_{I}∘Cat/w)⋆η} & Θ_{I} ∘ Cat/w ∘ Θ′_{J} ∘ w^{∗} ∘ Θ_{I} ∘ Cat/w \ar[r]^{ε⋆(Θ_{I}∘Cat/w)} & Θ_{I} ∘ Cat/w}

which prove that (Θ_{I} ∘ Cat/w, Θ′_{J} ∘ w^{∗}) is indeed an adjoint pair.

As a special case, for J = I and w = 1_{I}, we get that (Θ_{I}, Θ′_{I}) is an adjoint pair.

Lemma 2.6. Let I be a small category, F, G : I [figure] Cat be two functors and u : F [figure] G be a natural transformation. If, for every object i of I, the functor u_{i} : F (i) [figure] G(i) is aspheric, then the functor

∫ u : ∫ F [figure] ∫ G

is aspheric.

Proof. We define a functor

H : ∫ G [figure] Cat

as follows. For every object (i, b) of ∫ G, i ∈ Ob(I), b ∈ Ob(G(i)), we set

H(i, b) = F (i)/b ,

and for every morphism (k, g) : (i, b) [figure] (i′, b′) of ∫ G, where k : i [figure] i′ is an arrow of I and g : G(k)(b) [figure] b′ is an arrow of G(i′), we define

H(k, g) : F (i)/b [figure] F (i′)/b′

by

(a, p : u_{i}(a) [figure] b) [figure] (F (k)(a), g · G(k)(p) : u_{i′}F (k)(a) [figure] b′) .

[figure: u_{i′}F (k)(a); b′; G(k)u_{i}(a); G(k)(p); G(k)(b); g]

Let us consider the category ∫ H. The objects of this category are the quadruples (i, b, a, p : u_{i}(a) [figure] b), i ∈ Ob(I), b ∈ Ob(G(i)), a ∈ Ob(F (i)), p ∈ Fl(G(i)). A morphism from (i, b, a, p) to (i′, b′, a′, p′) is a triple (k, g, f)

k : i [figure] i′ ∈ Fl(I) , g : G(k)(b) [figure] b′ ∈ Fl(G(i′)) , f : F (k)(a) [figure] a′ ∈ Fl(F (i′))

such that the square

[figure: u_{i′}F (k)(a); u_{i′} (f); u_{i′}(a′); p′; G(k)u_{i}(a); G(k)(p); G(k)(b); g; b′]

is commutative. For every pair of composable morphisms

\xymatrix{(i, b, a, p) \ar[r]^{(k, g, f)} & (i′, b′, a′, p′) \ar[r]^{(k′, g′, f′)} & (i′′, b′′, a′′, p′′)} ,

the composite morphism is defined by

(k′, g′, f′) ∘ (k, g, f) = (k′k, g′ · G(k′)(g), f′ · F (k′)(f)) .

The canonical functor θ_{H} : ∫ H [figure]∫ G is defined by

(i, b, a, p) [figure] (i, b) , (i, b, a, p) ∈ Ob(∫ H) , (k, g, f) [figure] (k, g) , (k, g, f) ∈ Fl(∫ H) .

We shall define a functor S : ∫ F [figure]∫ H such that the triangle

\xymatrix{ & ∫ H \ar[dr]^{θ_{H}} \\ ∫ F \ar[ur]^{S} \ar[rr]_{∫ u} &  & ∫ G}

is commutative. For every object (i, a) of ∫ F, i ∈ Ob(I), a ∈ Ob(F (i)), we set

S(i, a) = (i, u_{i}(a), a, 1_{u_{i}(a)}) ,

and for every morphism (k, f) : (i, a) [figure] (i′, a′) of ∫ F, where k : i [figure] i′ is an arrow of I and f : F (k)(a) [figure] a′ is an arrow of F (i′),

S(k, f) = (k, u_{i′}(f), f) : (i, u_{i}(a), a, 1_{u_{i}(a)}) [figure] (i′, u_{i′}(a′), a′, 1_{u_{i′} (a′)}) .

It is straightforward to check the compatibility of S with identities and composition, as well as to check the commutativity of the triangle above. We shall define a functor R : ∫ H [figure]∫ F which will be a right adjoint to S as well as a retraction to it. For every object (i, b, a, p) of ∫ H, we set R(i, b, a, p) = (i, a), and for every morphism (k, g, f) : (i, b, a, p) [figure] (i′, b′, a′, p′) of ∫ H, we set R(k, g, f) = (k, f). Compatibility of R with identities and composition is obvious, and so is the equality RS = 1_{∫ F}. We define a natural transformation ε : SR [figure] 1_{∫ H} as follows. For every object (i, b, a, p : u_{i}(a) [figure] b) of ∫ H, we set

ε_{(i,b,a,p)} = (1_{i}, p, 1_{a}) : SR(i, b, a, p) = S(i, a) = (i, u_{i}(a), a, 1_{u_{i}(a)}) [figure] (i, b, a, p) , and we notice immediately that we have thus defined a morphism of ∫ H. For every morphism (k, g, f) : (i, b, a, p) [figure] (i′, b′, a′, p′) of ∫ H, the square

\xymatrix{SR(i, b, a, p) \ar[r]^{ε_{(i,b,a,p)}} \ar[d]_{SR(k,g,f)} & (i, b, a, p) \ar[d]^{(k,g,f)} \\ SR(i′, b′, a′, p′) \ar[r]^{ε_{(i′,b′,a′,p′)}} & (i′, b′, a′, p′)}

is commutative, as expected. Indeed,

(k, g, f)ε_{(i,b,a,p)} = (k, g, f)(1_{i}, p, 1_{a}) = (k, g G(k)(p), f) ,

ε_{(i′,b′,a′,p′)}SR(k, g, f) = (1_{i′}, p′, 1_{a′})S(k, f) = (1_{i′}, p′, 1_{a′})(k, u_{i′}(f), f) = (k, p′u_{i′}(f), f)

and p′u_{i′}(f) = g G(k)(p), since (k, g, f) is a morphism of ∫ H. From this we deduce that ε : SR [figure] 1_{∫ H} is indeed a natural transformation. Finally, for every object (i, b, a, p) of ∫ H, we have

R(ε_{(i,b,a,p)}) = R(1_{i}, p, 1_{a}) = (1_{i}, 1_{a}) ,

and for every object (i, a) of ∫ F, we have

ε_{S(i,a)} = ε_{(i,u_{i}(a),a,1_{u_{i}(a)})} = (1_{i}, 1_{u_{i}(a)}, 1_{a}) ,

which proves that

ε : SR [figure] 1_{∫ H}  and  1_{1_{∫ F}} : 1_{∫ F}  [figure] RS = 1_{∫ F}

satisfy the triangle identities and that R is a right adjoint to S. Therefore, it follows from proposition 1.12 that S is an aspheric functor. Since, for every object i of I, u_{i} is an aspheric functor by hypothesis, θ_{H} is a cofibration whose fibers are aspheric. From this we deduce that the functor θ_{H} is aspheric (proposition 1.16), and therefore that so is the composite morphism ∫ u = θ_{H}S (proposition 1.11), which finishes the proof. □

Theorem 2.7. For every small category I, we have

Asph/I = Θ_{I}^{−1}(Asph(I)) , Θ′_{I}(Asph(I)) ⊂ Asph/I

and the functors

\overline{Θ}_{I} : (Asph/I)^{−1}(Cat/I) [figure] Asph(I)^{−1} \underline{Hom}(I, Cat) = Hot(I)

and

\overline{Θ′_{I}} : Hot(I) = Asph(I)^{−1} \underline{Hom}(I, Cat) [figure] (Asph/I)^{−1}(Cat/I) ,

induced by Θ_{I} and Θ′_{I} respectively, are equivalences of categories, quasi-inverse to each other.

Proof. The equality Asph/I = Θ_{I}^{−1}(Asph(I)) follows from proposition 1.10, and the inclusion Θ′_{I}(Asph(I)) ⊂ Asph/I from the previous lemma. From this we deduce an adjoint pair of functors

\overline{Θ}_{I} : (Asph/I)^{−1}(Cat/I) [figure]Hot(I) , \overline{Θ′_{I}} : Hot(I) [figure] (Asph/I)^{−1}(Cat/I)

the adjunction morphisms being induced by the adjunction morphisms

ε : Θ_{I}Θ′_{I}  [figure] 1_{\underline{Hom}(I,Cat)} ,  η : 1_{Cat/I}  [figure] Θ′_{I}Θ_{I}

(cf. 2.5). Therefore, it is sufficient to show that for every functor F : I [figure] Cat, the natural transformation ε_{F} is in Asph(I) and that for every object (A, v : A [figure] I) of Cat/I, the morphism η_{(A,v)} of Cat/I is in Asph/I. a) By 2.5, for every object i of I, ε_{F,i} is the morphism

(∫ F ) / i [figure] F (i)

which sends an object (j, a, p : j [figure] i) of (∫ F ) / i, j ∈ Ob(I), a ∈ Ob(F (j)), p ∈ Fl(I), to the object F (p)(a) of F (i). It is easy to check that this functor is a left adjoint to the inclusion functor

F (i) [figure] (∫ F ) / i , a [figure] (i, a, 1_{i}) ,

(cf. lemma 1.15), which proves, by proposition 1.12, that it is aspheric. From this we deduce that ε_{F} is componentwise aspheric, which proves the assertion relative to ε_{F}.

b) By 2.5, the morphism η_{(A,v)} is the inclusion

A [figure] ∫ A/i , a [figure] (v(a), a, 1_{v(a)}) ,

where, by an abuse of notation, ∫ A/i stands for the cofibered category over I defined by the functor

I [figure] Cat , i [figure] A/i .

It is easy to check that this functor is a left adjoint to the functor

∫ A/i [figure] A , (i, a, p : v(a) [figure] i) [figure] a

(cf. proof of the lemma 2.6), which proves that it is aspheric (1.12) and finishes the proof of the theorem. □

2.8. For every arrow w : J [figure] I of Cat, we denote by

w_{!} : \underline{Hom}(J, Cat) [figure]\underline{Hom}(I, Cat)

the composite functor w_{!} = Θ_{I} ∘ Cat/w ∘ Θ′_{J}.

\xymatrix{\underline{Hom}(J, Cat) \ar[r]^{Θ′_{J}} & Cat/J \ar[r]^{Cat/w} & Cat/I \ar[r]^{Θ_{I}} & \underline{Hom}(I, Cat)}

Since, by theorem 2.7 and the considerations of paragraph 2.4, we have

Θ′_{J}(Asph(J)) ⊂ Asph/J , (Cat/w)(Asph/J) ⊂ Asph/I, Θ_{I}(Asph/I) ⊂ Asph(I) ,

this functor induces a functor between the localized categories, also denoted, by an abuse of notation, by

w_{!} : Hot(J) [figure]Hot(I) ,

which is the composite of the functors

\xymatrix{Hot(J) \ar[r]^{\overline{Θ′_{J}}} & (Asph/J)^{−1}(Cat/J) \ar[r]^{\overline{Cat/w}} & (Asph/I)^{−1}(Cat/I) \ar[r]^{\overline{Θ_{I}}} & Hot(I) ,}

induced by Θ′_{J}, Cat/w and Θ_{I}. We have a commutative square

\xymatrix{\underline{Hom}(J, Cat) \ar[r]^{w_{!}} \ar[d]_{γ_{J}} & \underline{Hom}(I, Cat) \ar[d]^{γ_{I}} \\ Hot(J) \ar[r]_{w_{!}} & Hot(I)}

whose vertical arrows are the localization functors.

Theorem 2.9. For every morphism w : J [figure] I of Cat, the functors

w_{!} : Hot(J) [figure]Hot(I) , w^{∗} : Hot(I) [figure]Hot(J)

form an adjoint pair.

Proof. It follows from 2.1, 2.4, 2.5 and from theorem 2.7 that the pair of functors (\overline{Θ}_{I} ∘ \overline{Cat/w} , \overline{Θ′_{J}} ∘ w^{∗})

\xymatrix{(Asph/J)^{−1}(Cat/J) \ar[r]^{\overline{Cat/w}} & (Asph/I)^{−1}(Cat/I) \ar[r]^{^{\overline{Θ}^{I}}} & Hot(I)}

\xymatrix{Hot(I) \ar[r]^{w^{∗}} & Hot(J) \ar[r]^{\overline{Θ′_{J}}} & (Asph/J)^{−1}(Cat/J)}

is an adjoint pair. By theorem 2.7, the functors \overline{Θ}_{J} and \overline{Θ′_{J}} are equivalences of categories quasi-inverse to each other, therefore w^{∗} ≃ \overline{Θ}_{J} ∘ (\overline{Θ′_{J}} ∘ w^{∗}) is a right adjoint to (\overline{Θ}_{I} ∘ \overline{Cat/w}) ∘ \overline{Θ′_{J}} = w_{!}, which proves the theorem. □

Remark 2.10. If A is the right asphericity structure associated with a basic localizer W (cf. 1.3), it follows from the theory developed by D.-C. Cisinski [4] that the functor

w_{!} : Hot(J) [figure]Hot(I)

is canonically isomorphic to the left derived functor of the left adjoint of the inverse image functor

w^{∗} : \underline{Hom}(I, Cat) [figure]\underline{Hom}(J, Cat) .

3. Smooth functors

In this paragraph, we fix, once and for all, a right asphericity structure A.

Definition 3.1. A morphism u : A [figure] B of Cat is said to be weakly A-smooth, or more simply weakly smooth, if for every object b of B, the canonical morphism

j_{b} : A_{b}  [figure] b\A ,  a [figure] (a, 1_{b} : b [figure] u(a)) , a ∈ Ob(A_{b}) ,

is aspheric.

Example 3.2. A prefibration is a weakly smooth morphism. Indeed, for every object b of B, the functor j_{b} has a right adjoint (dual of lemma 1.15), and therefore it is aspheric by proposition 1.12. If A is the minimal right asphericity structure (1.2), it follows from the characterization of aspheric functors for this structure (1.7) and from the dual of lemma 1.15 that the weakly smooth morphisms with respect to the minimal right asphericity structure are exactly the prefibrations.

Proposition 3.3. Let u : A [figure] B be a morphism of Cat. The following conditions are equivalent:

(a) u is weakly smooth;

(b) for every object a of A, the fibers of the morphism

A/a [figure] B/b , b = u(a) ,

induced by u, are aspheric;

(c) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r] \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ Δ_{0} \ar[r] & Δ_{1} \ar[r] & B & ,}

where Δ_{0} [figure] Δ_{1} stands for the inclusion {0} [figure] {0 [figure] 1}, the morphism A′′ [figure] A′ is aspheric;

(d) for every arrow g : b_{0} [figure] b_{1} of B, and every object a_{1} of A_{b_{1}}, the category A(a_{1}, g), the objects of which are the arrows f : a [figure] a_{1} of A whose target is a_{1} and which lift g (i.e. u(f) = g), and the morphisms of which are the commutative triangles in A

\xymatrix{a \ar[dr]^{f} \ar[dd]_{h} \\  & a_{1} \\ a′ \ar[ur]_{f′}}

such that h is a morphism of A_{b_{0}} (i.e. u(h) = 1_{b_{0}}), is aspheric.

Proof. We leave it to the reader to check that for every arrow g : b_{0} [figure] b_{1} of B, and every object a_{1} of A over b_{1}, the category A_{b_{0}}/(a_{1}, g) (defined by the functor j_{b_{0}} : A_{b_{0}} [figure] b_{0}\A and the object (a_{1}, g : b_{0} [figure] u(a_{1}) = b_{1}) of b_{0}\A), as well as the fiber of A/a_{1} [figure] B/b_{1} over the object (b_{0}, g) of B/b_{1}, are isomorphic to the category A(a_{1}, g), which proves the equivalence of conditions (a), (b) and (d). Let us show the equivalence between (c) and (d). Observe that there is a one-to-one correspondence between arrows g of B, as in (d), and functors Δ_{1} [figure] B, as in (c). Using the notations of (c), the inclusion A′′ [figure] A′ is, by definition, aspheric if and only if for every object a′ of A′, the category A′′/a′ is aspheric. If a′ is an object of the fiber A′_{0} ≃ A′′ of A′ over 0, this is true without any hypothesis on u, for in that case A′′/a′ has a final object. Therefore, it is sufficient to check this for a′ in the fiber A′_{1} of A′ over 1. In that case, a′ corresponds to an object a_{1} of A_{b_{1}}, and we check that A′′/a′ is isomorphic to the category A(a_{1}, g) of (d). This completes the proof. □

Corollary 3.4. Weakly smooth morphisms are stable under base change, i.e. for every cartesian square in Cat

\xymatrix{A′ \ar[r] \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r] & B & ,}

if u is weakly smooth, then so is u′.

Proof. The corollary follows from the previous proposition, since the condition (c) is stable under base change. □

Proposition 3.5. Let u : A [figure] B be a morphism of Cat. The following conditions are equivalent:

(a) u is weakly smooth;

(b) for every object a of A, the morphism A/a [figure] B/b, b = u(a), induced by u, is weakly smooth.

Proof. Assume that the functor u is weakly smooth, and let a be an object of A, and b = u(a). By condition (b) of proposition 3.3, in order to prove that the functor A/a [figure] B/b, induced by u, is weakly smooth, it is sufficient to show that for every object (a′, f : a′ [figure] a) of A/a, the fibers of the functor (A/a)/(a′, f) [figure] (B/b)/(b′, g), where (b′, g) = (u(a′), u(f)), are aspheric. Now, this functor is canonically isomorphic to the functor A/a′ [figure] B/b′, induced by u, the fibers of which are aspheric, by condition (b) of proposition 3.3, hence the assertion. Conversely, assume that for every object a of A, the functor A/a [figure] B/b, b = u(a), induced by u, is weakly smooth. Let us show that, in that case, so is u. By condition (b) of proposition 3.3, the hypothesis that A/a [figure] B/b is weakly smooth implies that the fibers of the functor (A/a)/(a, 1_{a}) [figure] (B/b)/(b, 1_{b}) are aspheric. Now, the latter is canonically isomorphic to the functor A/a [figure] B/b, induced by u, which proves that u is weakly smooth, by condition (b) of proposition 3.3. □

Definition 3.6. A morphism u : A [figure] B of Cat is said to be A-smooth, or more simply smooth, if for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

if the morphism w is aspheric, then so is v.

Proposition 3.7. The class of smooth morphisms is stable under composition and base change.

Proof. It is a formal consequence of the definition. □

Proposition 3.8. A smooth morphism is weakly smooth.

Proof. It follows from condition (c) of proposition 3.3. □

Proposition 3.9. A local isomorphism is a smooth morphism.

Proof. Let u : A [figure] B be a local isomorphism, i.e. an arrow u : A [figure] B of Cat such that, for every object a of A, the functor A/a [figure] B/b, b = u(a), induced by u, is an isomorphism, and consider the diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v′} \ar[d]_{u′′} & A′ \ar[r]^{v} \ar[d]^{u′} & A \ar[d]^{u} \\ B′′ \ar[r]_{w′} & B′ \ar[r]_{w} & B & ,}

where we assume that the morphism w′ is aspheric. For every object a′ of A′, we deduce a diagram of cartesian squares

\xymatrix{A′′/a′ \ar[r] \ar[d] & A′/a′ \ar[r] \ar[d] & A/a \ar[d] \\ B′′/b′ \ar[r] & B′/b′ \ar[r] & B/b & ,}

where a = v(a′), b′ = u′(a′), b = u(a) = w(b′), the vertical arrows of which are isomorphisms. The category B′′/b′ being aspheric by hypothesis, so is A′′/a′, which proves the proposition. □

Proposition 3.10. Let u : A [figure] B be a morphism of Cat. The following conditions are equivalent:

(a) u is smooth;

(b) for every object a of A, the morphism A/a [figure] B/b, b = u(a), induced by u, is smooth.

Proof. Assume that u is smooth, and let a be an object of A, b = u(a). Stability of smooth morphisms under base change (3.7) implies that the functor A/b [figure] B/b, induced by u, is smooth. The canonical functor A/a [figure] A/b being a local isomorphism, it follows from the previous proposition that it is smooth. Stability of smooth morphisms under composition (3.7) thus implies that the composite A/a [figure] B/b is smooth.

Conversely, assume that for every object a of A, the functor A/a [figure] B/b, b = u(a), induced by u, is smooth, and consider a diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

where w is an aspheric functor. For every object a of A, we deduce a diagram of cartesian squares

\xymatrix{A′′/a \ar[r] \ar[d] & A′/a \ar[r] \ar[d] & A/a \ar[d] \\ B′′/b \ar[r] & B′/b \ar[r] & B/b & ,}

where b = u(a). The functor w being aspheric, so is the morphism B′′/b [figure] B′/b (1.10). The functor A/a [figure] B/b being smooth, it follows that A′′/a [figure] A′/a is aspheric. Since this is true for every object a of A, it follows from proposition 1.10 that v is aspheric, which finishes the proof. □

Corollary 3.11. Locally aspheric functors are stable under smooth base change, i.e. for every cartesian square in Cat

\xymatrix{A′ \ar[r]^{v} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{w} & B & ,}

if u is smooth and w is locally aspheric, then v is locally aspheric.

Proof. For every object a′ of A′, we have a cartesian square in Cat

\xymatrix{A′/a′ \ar[r] \ar[d] & A/a \ar[d] \\ B′/b′ \ar[r] & B/b & ,}

where a = v(a′), b′ = u′(a′), and b = u(a) = w(b′). By the previous proposition, if u is smooth, then so is A/a [figure] B/b, and if w is locally aspheric, then the functor B′/b′ [figure] B/b is aspheric, which proves that A′/a′ [figure] A/a is aspheric and finishes the proof. □

3.12. Let

\xymatrix{ & A \ar[d]^{u} \\ B′ \ar[r]_{w} & B}

be a diagram in Cat. We can form the cartesian square

\xymatrix{A′ \ar[r]^{v} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{w} & B & ,}

where A′ = B′ ×_{B} A. We can also form the “2-square”

[figure: A′_{0}; v_{0}; u′_{0}; A; u; B′; w; B; α; ,]

where A′_{0} is the comma category whose objects are triples

(b′, a, g : w(b′) [figure] u(a)) , b′ ∈ Ob(B′) , a ∈ Ob(A) , g ∈ Fl(B) ,

a morphism from (b′_{0}, a_{0}, g_{0}) to (b′_{1}, a_{1}, g_{1}) being a pair (g′, f), where g′ : b′_{0} [figure] b′_{1} is an arrow of B′, and f : a_{0} [figure] a_{1} is an arrow of A, such that the diagram

\xymatrix{w(b′_{0}) \ar[r]^{w(g′)} \ar[d]_{g_{0}} & w(b′_{1}) \ar[d]^{g_{1}} \\ u(a_{0}) \ar[r]_{u(f)} & u(a_{1})}

is commutative. The functors u′_{0}, v_{0} are defined by

\begin{matrix}u′_{0}(b′, a, g) = b′ , v_{0}(b′, a, g) = a ,  (b′, a, g) ∈ Ob(A′_{0}) , \\ u′_{0}(g′, f) = g′ , v_{0}(g′, f) = f ,  (g′, f) ∈ Fl(A′_{0}) , \end{matrix}

and the natural transformation α : wu′_{0} [figure] uv_{0} is defined by

α_{(b′,a,g)} = g : wu′_{0}(b′, a, g) = w(b′) [figure] u(a) = uv_{0}(b′, a, g) , (b′, a, g) ∈ Ob(A′_{0}) .

We fix an object b′_{0} of B′, an object a_{1} of A and a morphism g : w(b′_{0}) [figure] u(a_{1}). We set b_{0} = w(b′_{0}), b_{1} = u(a_{1}). We shall associate to these data three categories C_{0}, C_{1}, C_{2}, and leave it to the reader to check that they are isomorphic.

a) Definition of C_{0}. The triple (b′_{0}, a_{1}, g) is an object of A′_{0}, and there is a canonical functor A′ [figure] A′_{0} which sends an object (b′, a) of A′, b′ ∈ Ob(B′), a ∈ Ob(A), w(b′) = u(a), to the object (b′, a, 1_{u(a)}) of A′_{0}. The category C_{0} is the category A′/(b′_{0}, a_{1}, g).

b) Definition of C_{1}. The pair (b′_{0}, g) is an object of B′/b_{1}, and there is a cartesian square

\xymatrix{A′/a_{1} \ar[r] \ar[d] & A/a_{1} \ar[d] \\ B′/b_{1} \ar[r] & B/b_{1} & .}

The category C_{1} is the category (A′/a_{1})/(b′_{0}, g), defined by the left vertical arrow.

c) Definition of C_{2}. Let us consider the category (B′/b′_{0})^{∗} obtained by adding a new final object to B′/b′_{0}, and the inclusion B′/b′_{0} [figure] (B′/b′_{0})^{∗}. By the universal property of this construction, there is a unique morphism (B′/b′_{0})^{∗} [figure] B/b_{1} of Cat such that the image of the final object of (B′/b′_{0})^{∗} under this functor is the final object (b_{1}, 1_{b_{1}}) of B/b_{1}, and such that the square

\xymatrix{B′/b′_{0} \ar[r] \ar@{^{(}->}[d] & B/b_{0} \ar[d] \\ (B′/b′_{0})^{∗} \ar[r] & B/b_{1}}

is commutative, where the horizontal upper arrow is induced by w, and the vertical arrow on the right is defined by g. Let us consider the composite

(B′/b′_{0})^{∗}  [figure] B/b_{1}  [figure] B

of this functor with the forgetful functor from B/b_{1} to B, and let us form the diagram of cartesian squares

\xymatrix{A′′ \ar[d] \ar[r] & \overline{A′} \ar[d] \ar[r] & A \ar[d] \\ B′/b′_{0} \ar[r] & (B′/b′_{0})^{∗} \ar[r] & B & .}

The category C_{2} is the category A′′/a′_{1}, where a′_{1} is the object of \overline{A′} = (B′/b′_{0})^{∗}×_{B} A which is over the final object of (B′/b′_{0})^{∗} and whose projection in A is a_{1}.

Theorem 3.13. Let u : A [figure] B be an arrow of Cat. The following conditions are equivalent:

(a) u is smooth;

(b) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

if the functor w is aspheric with respect to the minimal right asphericity structure, i.e. if it has a right adjoint, then the morphism v is aspheric;

(c) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

where B′′ is a category which has a final object, B′ is obtained by adding a new final object to B′′, and w is the canonical inclusion, the morphism v is aspheric;

(d) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r] \ar[d]_{u′′} & A′ \ar[r] \ar[d]^{u′} & A \ar[d]^{u} \\ B′′ \ar[r] & B′ \ar[r] & B & ,}

and for every object a′ of A′, the morphism

A′′/a′  [figure] B′′/b′ ,  b′ = u′(a′) ,

induced by u′′, is aspheric.

Proof. The implications (a) ⇒ (b) ⇒ (c) are clear. Let us show the implication (c) ⇒ (d). As the condition (c) is stable under base change, it is sufficient to show that if

\xymatrix{A′ \ar[r]^{v} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{w} & B}

is a cartesian square, a_{1} an object of A, b_{1} = u(a_{1}), and (b′_{0}, g : w(b′_{0}) [figure] b_{1}) an object of B′/b_{1}, then the category (A′/a_{1})/(b′_{0}, g) is aspheric. By 3.12, this category is isomorphic to the category A′′/a′_{1}, where

\xymatrix{A′′ \ar[d] \ar[r] & \overline{A′} \ar[d] \ar[r] & A \ar[d] \\ B′/b′_{0} \ar[r] & (B′/b′_{0})^{∗} \ar[r] & B}

is the diagram of cartesian squares considered in 3.12, (c), and a′_{1} is the object of \overline{A′} over the final object of (B′/b′_{0})^{∗} whose image in A is the object a_{1}. Now, by condition (c), the functor A′′ [figure] \overline{A′} is aspheric. From this we deduce that the category A′′/a′_{1} is aspheric, therefore so is the category (A′/a_{1})/(b′_{0}, g).

It remains to show the implication (d) ⇒ (a). As the condition (d) is stable under base change, it is sufficient to show that if

\xymatrix{A′ \ar[r]^{v} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{w} & B}

is a cartesian square, where w is an aspheric morphism, then v is aspheric, too. Let a be an object of A. We need to show that the category A′/a is aspheric. Let b = u(a). By condition (d), the morphism A′/a [figure] B′/b, induced by u′, is aspheric. By hypothesis, the functor w is aspheric, hence the category B′/b is aspheric, and therefore so is A′/a, which concludes the proof. □

Corollary 3.14. A smooth functor is locally aspheric.

Proof. It is an immediate consequence of condition (d) of the previous theorem. □

Proposition 3.15. Let u : A [figure] B be a morphism of Cat. The following conditions are equivalent:

(a) u is a fibration;

(b) u is smooth with respect to the minimal right asphericity structure;

(c) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r]^{v} \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ B′′ \ar[r]_{w} & B′ \ar[r] & B & ,}

if the functor w has a right adjoint, then so has v;

(d) for every diagram of cartesian squares

\xymatrix{A′′ \ar[r] \ar[d] & A′ \ar[r] \ar[d] & A \ar[d]^{u} \\ Δ_{1} \ar[r] & Δ_{2} \ar[r] & B & ,}

where Δ_{1} [figure] Δ_{2} stands for the inclusion {0 [figure] 1} [figure] {0 [figure] 1 [figure] 2}, the functor A′′ [figure] A′ has a right adjoint.

Proof. Since aspheric functors with respect to the minimal right asphericity structure are exactly those which have a right adjoint (1.7), equivalence between conditions (b) and (c) is tautological. As the implication (c) ⇒ (d) is clear, it is sufficient to prove the implications (a) ⇒ (b) and (d) ⇒ (a).

Since fibrations are stable under base change, in order to show the implication (a) ⇒ (b) it is sufficient to show that for every cartesian square

[figure: A′ = B′ ×_{B} A; A; u; B′; w; B; ,]

if u is a fibration, and if for every object b of B, the category B′/b has a final object, then for every object a of A, the category A′/a has a final object.

Accordingly, let a_{1} be an object of A, b_{1} its image in B, and (b′_{0}, g : w(b′_{0}) [figure] b_{1}) a final object of B′/b_{1}. Since u is a fibration, there exists a hypercartesian morphism k : a_{0} [figure] a_{1} of A such that u(k) = g. We shall show that ((b′_{0}, a_{0}), k : a_{0} [figure] a_{1}) is a final object of A′/a_{1}. Let ((b′, a), f : a [figure] a_{1}) be an object of A′/a_{1}, w(b′) = u(a). We have to show that there exists an arrow of A′/a_{1} whose domain is ((b′, a), f) and whose codomain is ((b′_{0}, a_{0}), k), and that it is unique. In other words, we have to show that there exists a unique pair (g′, h), where g′ : b′ [figure] b′_{0} is an arrow of B′ and h : a [figure] a_{0} is an arrow of A, such that w(g′) = u(h) and f = kh. For such a pair, we have u(f) = u(k)u(h) = gw(g′), which means that

g′ : (b′, u(f) : w′(b′) [figure] b_{1}) [figure] (b′_{0}, g : w(b′_{0}) [figure] b_{1})

is an arrow of B′/b_{1}. Now, the hypothesis that (b′_{0}, g) is a final object of B′/b_{1} implies that such a g′ exists and is unique. Since the morphism k is hypercartesian, there exists a unique arrow h : a [figure] a_{0} of A such that u(h) = w(g′) and f = kh, which proves the assertion.

It remains to show the implication (d) ⇒ (a). Let Δ_{2} [figure] B be a functor defined by a pair of composable arrows

\xymatrix{b_{0} \ar[r]^{g_{0}} & b_{1} \ar[r]^{g_{1}} & b_{2}}

of B. Let us form the diagram of cartesian squares which appears in the statement of condition (d). This condition means that for every object a′_{2} of A′ whose image a_{2} in A is over b_{2}, the category A′′/a′_{2} has a final object. Let us describe this category. The set of objects of A′′/a′_{2} can be canonically identified with the disjoint sum A′′_{0}∐A′′_{1}, where

A′′_{0} = {(a_{0}, f_{0}) | a_{0} ∈ Ob(A), u(a_{0}) = b_{0}, f_{0} : a_{0} [figure] a_{2} ∈ Fl(A), u(f_{0}) = g_{1}g_{0}} ,

A′′_{1} = {(a_{1}, f_{1}) | a_{1} ∈ Ob(A), u(a_{1}) = b_{1}, f_{1} : a_{1} [figure] a_{2} ∈ Fl(A), u(f_{1}) = g_{1}} ,

and for every (a_{0}, f_{0}), (a′_{0}, f′_{0}) ∈ A′′_{0}, (a_{1}, f_{1}), (a′_{1}, f′_{1}) ∈ A′′_{1}, we have

Hom_{A′′/a′_{2}}((a_{0}, f_{0}), (a′_{0}, f′_{0})) = {g | g : a_{0} [figure] a′_{0} ∈ Fl(A), u(g) = 1_{b_{0}}, f_{0} = f′_{0}g} ,

Hom_{A′′/a′_{2}}((a_{0}, f_{0}), (a_{1}, f_{1})) = {g | g : a_{0} [figure] a_{1} ∈ Fl(A), u(g) = g_{0}, f_{0} = f_{1}g} ,

Hom_{A′′/a′_{2}}((a_{1}, f_{1}), (a_{0}, f_{0})) = ∅ ,

Hom_{A′′/a′_{2}}((a_{1}, f_{1}), (a′_{1}, f′_{1})) = {g | g : a_{1} [figure] a′_{1} ∈ Fl(A), u(g) = 1_{b_{1}}, f_{1} = f′_{1}g} .

Since the category A′′/a′_{2} has a final object, it is non-empty. The special case b_{0} = b_{1} and g_{0} = 1_{b_{1}} then shows that for every arrow b_{1} [figure] b_{2} of B and for every object a_{2} of A over b_{2}, there exists a morphism a_{1} [figure] a_{2} of A over b_{1} [figure] b_{2}. Coming back to the general case ( \xymatrix{b_{0} \ar[r]^{g_{0}} & b_{1} \ar[r]^{g_{1}} & b_{2}} being arbitrary), this implies that A′′_{0} and A′′_{1} are non-empty sets. Since there is no morphism in A′′/a′_{2} from an object of A′′_{1} to an object of A′′_{0}, we deduce that the final object (a_{1}, f_{1}) of A′′/a′_{2} belongs to A′′_{1}. This implies that (a_{1}, f_{1}) is also a final object of the full subcategory of A′′/a′_{2} whose objects are the objects of A′′_{1}. This means exactly that f_{1} is a cartesian morphism over g_{1} and shows that f_{1} is determined by the sole g_{1} and does not depend on g_{0}. As g_{1} is an arbitrary arrow of B and a_{2} an arbitrary object of the fiber of A over the target of g_{1}, this implies already that u is a prefibration. Now, (a_{1}, f_{1}) is a final object of A′′/a′_{2}. Therefore, for every object (a_{0}, f_{0}) of A′′/a′_{2} which belongs to the set A′′_{0}, f_{0} : a_{0} [figure] a_{2} ∈ Fl(A), u(f_{0}) = g_{1}g_{0}, there exists a unique morphism g : (a_{0}, f_{0}) [figure] (a_{1}, f_{1}) of A′′/a′_{2}. In other words, there is a unique arrow g : a_{0} [figure] a_{1} of A such that f_{1}g = f_{0} and u(g) = g_{0}. Since g_{0} is an arbitrary arrow of B whose codomain is the domain of g_{1}, and since f_{1} does not depend on g_{0}, it follows that the morphism f_{1} is hypercartesian, which finishes the proof. □

Corollary 3.16. Fibrations are smooth functors (with respect to any right asphericity structure).

Proof. The corollary follows from condition (c) of the previous proposition, condition (b) of theorem 3.13, and from proposition 1.12. □

Example 3.17. If A is the right asphericity structure associated with a basic localizer W (example 1.3), Grothendieck’s characterization of W-smooth functors and theorem 3.13 imply that, if u is a morphism of Cat, the following conditions are equivalent:

(a) u is W-smooth;

(b) u is weakly A-smooth;

(c) u is A-smooth. On the other hand, since there exist prefibrations which are not fibrations, proposition 3.15 and example 3.2 show that if A is the minimal right asphericity structure, then the class of A-smooth functors is a proper subclass of the class of weakly A-smooth functors.

4. Smooth functors and base change morphisms

Lemma 4.1. Let w : J [figure] I be a morphism of Cat and F : I [figure] Cat be a functor. We then have a cartesian square

\xymatrix{∫ F w \ar[r]|{w̃} \ar[d]_{θ_{Fw}} & ∫ F \ar[d]^{θ_{F}} \\ J \ar[r]_{w} & I & ,}

where θ_{F w} and θ_{F} stand for the cofibrations associated with the functors F w and F respectively, and w̃ stands for the functor (j, a) [figure] (w(j), a), (j, a) ∈ Ob(∫ F w), induced by w.

Proof. The lemma follows from an easy verification which is left to the reader. □

4.2. Let

D = \xymatrix{A′ \ar[r]^{w} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{v} & B}

be a cartesian square of Cat. For every functor F : A [figure] Cat, we deduce a composite cartesian square

\xymatrix{∫ F w \ar[r]|{w̃} \ar[d]_{θ_{F w}} & ∫ F \ar[d]^{θ_{F}} \\ A′ \ar[r]^{w} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{v} & B}

(cf. 4.1). For every object b′ of B′, the functor w̃ induces a functor

(∫ F w)/b′ [figure] (∫ F)/v(b′) ,

and we notice that

(∫ F w)/b′ = (u′_{!}w^{∗}(F))(b′)  and  (∫ F)/v(b′) = (v^{∗}u_{!}(F))(b′)

(cf. 2.8). We deduce a morphism

κ_{D} : u′_{!}w^{∗} [figure] v^{∗}u_{!}

of \underline{Hom}(\underline{Hom}(A, Cat), \underline{Hom}(B′, Cat)), called base change morphism associated with the square D.

Proposition 4.3. Let

D = \xymatrix{A′ \ar[r]^{w} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{v} & B}

be a cartesian square of Cat, where v is a smooth functor. Then the base change morphism κ_{D} : u′_{!}w^{∗} [figure] v^{∗}u_{!} is componentwise aspheric. In other words, for every functor F : A [figure] Cat and for every object b′ of B′, the morphism

κ_{D,F} (b′) : (u′_{!}w^{∗}(F))(b′) [figure] (v^{∗}u_{!}(F))(b′)

is aspheric.

Proof. Let F : A [figure] Cat be a functor, and consider the cartesian square

\xymatrix{∫ F w \ar[r]|{w̃} \ar[d]_{u′θ_{F w}} & ∫ F \ar[d]^{uθ_{F}} \\ B′ \ar[r]_{v} & B & .}

Since v is a smooth functor, it follows from condition (d) of theorem 3.13 that for every object b′ of B′, the functor κ_{D,F} (b′) : (∫ F w)/b′ [figure] (∫ F)/v(b′), induced by w̃, is aspheric, which proves the proposition. □

Theorem 4.4. Let u : A [figure] B be a morphism of Cat. The following conditions are equivalent :

(a) u is smooth;

(b) for every diagram of cartesian squares in Cat

\xymatrix{A′′ \ar[r] \ar[d] & B′′ \ar[d] \\ A′ \ar[r] \ar[d] & B′ \ar[d] \\ A \ar[r]_{u} & B & ,}

the base change morphism associated with the upper square is componentwise an aspheric functor.

Proof. Implication (a) ⇒ (b) follows from proposition 4.3 and from stability of smooth morphisms under base change (proposition 3.7). Conversely, we notice that condition (b), applied to the constant functor B′′ [figure] Cat with value the final category e, implies that for every object a′ of A′, if we denote its image in B′ by b′, the morphism A′′/a′ [figure] B′′/b′ is aspheric. We deduce that u fulfils condition (d) of theorem 3.13, which proves the theorem. □

Remark 4.5. Let

D = \xymatrix{A′ \ar[r]^{w} \ar[d]_{u′} & A \ar[d]^{u} \\ B′ \ar[r]_{v} & B}

be a cartesian square of Cat, and

[figure: \underline{Hom}(A, Cat); κ_{D}; w^{∗}; u_{!}; \underline{Hom}(A′, Cat); u′_{!}; u′_{!}w^{∗}; κ_{D}; v^{∗}u_{!}; \underline{Hom}(B, Cat); v^{∗}; \underline{Hom}(B′, Cat)]

the base change morphism (4.2). One checks easily that the natural transformation

[figure: Hot(A); _{\overline{κ}_{D}}; w^{∗}; u_{!}; Hot(A′); u′_{!}; u′_{!}w^{∗}; _{\overline{κ}_{D}}; v^{∗}u_{!} ,; Hot(B); v^{∗}; Hot(B′)]

induced by κ_{D} by localization, is the “base change morphism” formally defined by the adjunctions (cf. theorem 2.9): the natural transformation \overline{κ}_{D} is the composite

\xymatrix{u′_{!}w^{∗} \ar[r]^{u′_{!}w^{∗}⋆η} & u′_{!}w^{∗}u^{∗}u_{!} = u′_{!}u′^{∗}v^{∗}u_{!} \ar[r]^{ε′⋆v^{∗}u_{!}} & v^{∗}u_{!}} ,

where

η : 1_{Hot(A)} [figure] u^{∗}u_{!} , ε′ : u′_{!}u′^{∗} [figure] 1_{Hot(B′)}

denote the adjunction morphisms. Proposition 4.3 implies that if the functor v is smooth, then the natural transformation \overline{κ}_{D} is an isomorphism.

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