Exposé VI. Fibered Categories and Descent

0. Introduction

Contrary to what had been announced in the introduction to the preceding exposé, it has turned out to be impossible to do descent in the category of preschemes, even in particular cases, without first having developed with sufficient care the language of descent in general categories.

The notion of “descent” supplies the general framework for all procedures of “gluing” objects, and consequently of “gluing” categories. The most classical case of gluing is relative to the data of a topological space $X$ and a covering of $X$ by open subsets Xᵢ. Suppose one is given, for every $i$, a fiber space, say, Eᵢ over Xᵢ, and for every pair $(i,j)$ an isomorphism $f_{ji}$ from $E_i|X_{ij}$ to $E_j|X_{ij}$, where $X_{ij}=X_i\cap X_j$, satisfying the well-known transitivity condition, written in abbreviated form $f_{kj}{f}_{ji}=f_{ki}$. One knows that there exists a fiber space $E$ on $X$, defined up to isomorphism by the condition that one have isomorphisms $f_i:E|X_i\simeq E_i$ satisfying the relations $f_{ji}=f_j{f}_i^{-1}$, with the usual abuse of notation.

Let $X^{\prime }$ be the sum space of the Xᵢ; it is therefore a fiber space over $X$, i.e. endowed with a continuous map $X^{\prime }\to X$. The data of the Eᵢ can be interpreted more concisely as a fiber space $E^{\prime }$ over $X^{\prime }$, and the data of the $f_{ji}$ as an isomorphism between the two inverse images, by the two canonical projections, $E_1^{\prime \prime }$ and $E_2^{\prime \prime }$ of $E^{\prime }$ on $X^{\prime \prime }=X^{\prime }{\times }_X{X}^{\prime }$. The gluing condition can then be written as an identity between isomorphisms of fiber spaces $E_1^{\prime \prime \prime }$ and $E_3^{\prime \prime \prime }$ over the triple fiber product $X^{\prime \prime \prime }=X^{\prime }{\times }_X{X}^{\prime }{\times }_X{X}^{\prime }$, where $E_i^{\prime \prime \prime }$ denotes the inverse image of $E^{\prime }$ on $X^{\prime \prime \prime }$ by the canonical projection of index $i$. The construction of $E$ from $E^{\prime }$ and $f$ is a typical case of a “descent” procedure.

Moreover, starting from a fiber space $E$ on $X$, one says that $X$ is “locally trivial”, with fiber $F$, if there exists an open covering $(X_i)$ of $X$ such that the $E|X_i$ are isomorphic to $F\times X_i$, or what amounts to the same thing, such that the inverse image $E^{\prime }$ of $E$ on $X^{\prime }={\coprod }_i{X}_i$ is isomorphic to $X^{\prime }\times F$.

Thus the notion of “gluing” objects, like that of “localization” of a property, is tied to the study of certain types of “base changes” $X^{\prime }\to X$. In algebraic geometry, many other types of base change, and notably faithfully flat morphisms $X^{\prime }\to X$, must be regarded as corresponding to a procedure of “localization” relative to preschemes, or other objects, “over” $X$. This type of localization is used just as much as ordinary topological localization, of which it is moreover a special case. The same is true, to a lesser extent, in analytic geometry.

Most of the proofs, reducing to verifications, are omitted or merely sketched. Where appropriate, we specify the less evident diagrams that enter into a proof.

1. Universes, Categories, Equivalence of Categories

To avoid certain logical difficulties, we shall admit here the notion of a Universe, which is a set “large enough” that one does not leave it under the usual operations of set theory; an “axiom of Universes” guarantees that every object lies in a Universe. For details, see a book in preparation by C. Chevalley and the speaker.¹ Thus the symbol Set denotes not the category of all sets, a notion that has no sense, but the category of sets that lie in a given Universe, which we shall not specify in the notation. Similarly, Cat will denote the category of categories lying in the Universe in question; the “morphisms” from one object $X$ of Cat to another $Y$ are, by definition, the functors from $X$ to $Y$.

If $\mathcal{C}$ is a category, we denote by $Ob(\mathcal{C})$ the set of objects of $\mathcal{C}$, and by $Fl(\mathcal{C})$ the set of arrows of $\mathcal{C}$, or morphisms of $\mathcal{C}$. We shall therefore write $X\in Ob(\mathcal{C})$, avoiding the common abuse of notation $X\in \mathcal{C}$. If $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$ are two categories, a functor from $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$ will always mean what is commonly called a covariant functor from $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$. Its data include both the target category and the source category, $\mathcal{C}$ and ${\mathcal{C}}^{\prime }$. The functors from $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$ form a set, denoted $\mathrm{Hom}(\mathcal{C},{\mathcal{C}}^{\prime })$, which is the set of objects of a category denoted Hom̲(𝒞,𝒞′). By definition, a contravariant functor from $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$ is a functor from the opposite category ${\mathcal{C}}^{\circ }$ of $\mathcal{C}$ to ${\mathcal{C}}^{\prime }$.

We shall admit the notions of projective limit and inductive limit of a functor $F:\mathcal{I}\to \mathcal{C}$, and in particular the most common special cases of these notions: cartesian products and fiber products, the dual notions of direct sums and amalgamated sums, and the usual formal properties of these operations.

For example, in the category Cat introduced above, projective limits, relative to categories $\mathcal{I}$ lying in the chosen Universe, exist. The set of objects, respectively the set of arrows, of the projective-limit category $\mathcal{C}$ of the ${\mathcal{C}}_i$ is obtained by taking the projective limit of the sets of objects, respectively the sets of arrows, of the categories ${\mathcal{C}}_i$. The best-known case is that of the product of a family of categories. We shall constantly use, in what follows, the fiber product of two categories over a third.

For everything concerning categories and functors, pending the book in preparation already mentioned, see [VI.1], which is necessarily quite incomplete, even as concerns the generalities sketched in the present number.

Let us take this occasion to spell out the notion of equivalence of categories, which is not presented satisfactorily in [VI.1]. A functor $F:\mathcal{C}\to {\mathcal{C}}^{\prime }$ is said to be faithful, respectively fully faithful, if for every pair of objects $S$, $T$ of $\mathcal{C}$, the map $u\mapsto F(u)$ from $\mathrm{Hom}(S,T)$ to $\mathrm{Hom}(F(S),F(T))$ is injective, respectively bijective. One says that $F$ is an equivalence of categories if

$F$ is fully faithful and, moreover, every object $S^{\prime }$ of ${\mathcal{C}}^{\prime }$ is isomorphic to an object of the form $F(S)$. One shows that this is the same as saying that there exists a functor $G$ from ${\mathcal{C}}^{\prime }$ to $\mathcal{C}$ quasi-inverse to $F$, i.e. such that GF is isomorphic to $i{d}_{\mathcal{C}}$ and FG is isomorphic to $i{d}_{\mathcal{C}}^{\prime }$.

When this is so, giving a functor $G:{\mathcal{C}}^{\prime }\to \mathcal{C}$ and an isomorphism $\varphi :FG\to i{d}_{\mathcal{C}}^{\prime }$ is equivalent to giving, for every $S^{\prime }\in Ob({\mathcal{C}}^{\prime })$, a pair $(S,u)$ formed by an object $S$ of $\mathcal{C}$ and an isomorphism $u:F(S)\to S^{\prime }$, namely $(G(S^{\prime }),\varphi (S^{\prime }))$. With this notation, there exists a unique functor ${\mathcal{C}}^{\prime }\to \mathcal{C}$ having the given map $S^{\prime }\mapsto G(S^{\prime })$ as its object map, and such that the map $S^{\prime }\mapsto \varphi (S^{\prime })$ is a homomorphism of functors $FG\to i{d}_{\mathcal{C}}^{\prime }$.

Finally, if $G$ is a functor quasi-inverse to $F$, and if one chooses isomorphisms $\varphi :FG\simeq i{d}_{\mathcal{C}}^{\prime }$ and $\psi :GF\simeq i{d}_{\mathcal{C}}$, then the two compatibility conditions on $\varphi$ and $\psi$ stated in [VI.1, I.1.2] are in fact equivalent to one another; and for every chosen isomorphism $\varphi$, there exists a unique isomorphism $\psi$ such that those conditions are satisfied.

2. Categories over Another Category

Let $\mathcal{E}$ be a category in the chosen Universe. It is therefore an object of Cat, and one may consider the category $Ca{t}_{/}\mathcal{E}$ of “objects of Cat over $\mathcal{E}$”. An object of this category is therefore a functor

$$p:\mathcal{F}\to \mathcal{E}.$$

One also says that the category $\mathcal{F}$, endowed with such a functor, is a category over $\mathcal{E}$, or an $\mathcal{E}$-category. Thus an $\mathcal{E}$-functor from a category $\mathcal{F}$ over $\mathcal{E}$ to a category $\mathcal{G}$ over $\mathcal{E}$ will mean a functor

$$f:\mathcal{F}\to \mathcal{G}$$

such that

qf = p,

where $p$ and $q$ are the projection functors for $\mathcal{F}$ and $\mathcal{G}$ respectively. The set of $\mathcal{E}$-functors $f$ from $\mathcal{F}$ to $\mathcal{G}$ is therefore in bijective correspondence with the set of arrows with source $\mathcal{F}$ and target $\mathcal{G}$ in $Ca{t}_{/}\mathcal{E}$,

without this being an identity, since the data of an $f$ as above does not determine $\mathcal{F}$ and $\mathcal{G}$ as categories over $\mathcal{E}$. Of course, as in any other category ${\mathcal{C}}_{/}S$, we shall routinely make the abuse of language that identifies $\mathcal{E}$-functors, in the sense just explained, with arrows in a category $Ca{t}_{/}\mathcal{E}$.

We shall denote by

$${\mathrm{Hom}}_{\mathcal{E}}(\mathcal{F},\mathcal{G})$$

the set of $\mathcal{E}$-functors from $\mathcal{F}$ to $\mathcal{G}$. Of course, a composite of $\mathcal{E}$-functors is an $\mathcal{E}$-functor, the composition in question corresponding by definition to the composition of arrows in $Ca{t}_{/}\mathcal{E}$.

Now consider two $\mathcal{E}$-functors

$$f,g:\mathcal{F}\to \mathcal{G}$$

and a homomorphism of functors

$$u:f\to g.$$

One says that $u$ is an $\mathcal{E}$-homomorphism, or a “homomorphism of $\mathcal{E}$-functors”, if for every $\xi \in Ob(\mathcal{F})$, one has

$$q(u(\xi ))=i{d}_{p(\xi )}.$$

In words: putting $S=p(\xi )=qf(\xi )=qg(\xi )\in Ob(\mathcal{E})$, the morphism

$$u(\xi ):f(\xi )\to g(\xi )$$

in $\mathcal{G}$ is an $i{d}_S$-morphism. In general, for every morphism $\alpha :T\to S$ in $\mathcal{E}$ and every category $\mathcal{G}$ over $\mathcal{E}$, a morphism $v$ in $\mathcal{G}$ is called an $\alpha$-morphism if $q(v)=\alpha$, where $q$ denotes the projection functor $\mathcal{G}\to \mathcal{E}$. If one has a third $\mathcal{E}$-functor $h:\mathcal{F}\to \mathcal{G}$ and an $\mathcal{E}$-homomorphism $v:g\to h$, then vu is again an $\mathcal{E}$-homomorphism.

Thus the $\mathcal{E}$-functors from $\mathcal{F}$ to $\mathcal{G}$, and the $\mathcal{E}$-homomorphisms between them, form a subcategory of the category Hom̲(ℱ,𝒢) of all functors from $\mathcal{F}$ to $\mathcal{G}$; it will be called the category of $\mathcal{E}$-functors from $\mathcal{F}$ to $\mathcal{G}$ and denoted

Hom̲_{ℰ/-}(ℱ,𝒢).

It is also the kernel subcategory of the pair of functors

R,S: Hom̲(ℱ,𝒢) ⇉ Hom̲(ℱ,ℰ),

where $R$ is the constant functor defined by the object $p$ of Hom̲(ℱ,ℰ), and $S$ is the functor $f\mapsto q\circ f$ defined by $q:\mathcal{G}\to \mathcal{E}$.

To finish these generalities, it remains to define the natural pairings between the categories Hom̲_{ℰ/-}(ℱ,𝒢) by composition of $\mathcal{E}$-functors. In other words, one wants to define a “composition functor”

(i)  Hom̲_{ℰ/-}(ℱ,𝒢) × Hom̲_{ℰ/-}(𝒢,ℋ) → Hom̲_{ℰ/-}(ℱ,ℋ)

when $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are three categories over $\mathcal{E}$, in such a way that this functor induces, on objects, the composition map $(f,g)\mapsto gf$ for $\mathcal{E}$-functors $f:\mathcal{F}\to \mathcal{G}$ and $g:\mathcal{G}\to \mathcal{H}$. For this, recall that one defines a canonical functor

(ii) Hom̲(ℱ,𝒢) × Hom̲(𝒢,ℋ) → Hom̲(ℱ,ℋ),

which on objects is just the composition map $(f,g)\mapsto gf$ of functors, and which transforms an arrow $(u,v)$, where

u: f → f′,    v: g → g′,

are arrows in Hom̲(ℱ,𝒢), respectively in Hom̲(𝒢,ℋ), into the arrow

v ∗ u: gf → g′f′

defined by the relation

(v ∗ u)(ξ) = v(f′(ξ)) · g(u(ξ)) = g′(u(ξ)) · v(f(ξ)).

It is well known that one indeed obtains in this way a homomorphism from gf to $g^{\prime }{f}^{\prime }$, and that, for variable $f$, $g$ and $u$, $v$, one obtains the functor (ii); that is, one has

(I)   id_g ∗ id_f = id_{gf},

and

(II)  (v′ ∗ u′) ∘ (v ∗ u) = (v′ ∘ v) ∗ (u′ ∘ u).

Recall also that one has an associativity formula for the canonical pairings (ii), expressed on the one hand by the associativity $(hg)f=h(gf)$ of the composition of functors, and on the other hand by the formula

(III) (w ∗ v) ∗ u = w ∗ (v ∗ u)

for the composition products of homomorphisms of functors, where $u:f\to f^{\prime }$ and $v:g\to g^{\prime }$ are as above, and where one supposes given in addition a homomorphism $w:h\to h^{\prime }$ of functors $h,h^{\prime }:\mathcal{H}\to \mathcal{K}$.

I now say that when $\mathcal{F}$ and $\mathcal{G}$ are $\mathcal{E}$-categories, the canonical composition functor (ii) induces a functor (i). Since we already know that the composite of two $\mathcal{E}$-functors is an $\mathcal{E}$-functor, this amounts to saying that when $u:f\to f^{\prime }$ and $v:g\to g^{\prime }$ are homomorphisms of $\mathcal{E}$-functors, then $v\ast u:gf\to g^{\prime }{f}^{\prime }$ is also a homomorphism of $\mathcal{E}$-functors. This follows trivially from the definitions. Since the pairings (i) are induced by the pairings (ii), they satisfy the same associativity property, also expressed in the formulas $(hg)f=h(gf)$ and $(w\ast v)\ast u=w\ast (v\ast u)$, for $\mathcal{E}$-functors and $\mathcal{E}$-homomorphisms of $\mathcal{E}$-functors.

To complete the formulary (I), (II), (III), recall also the formulas

(IV)  v ∗ id_ℱ = v    and    id_𝒢 ∗ u = u,

where, for simplicity, one writes $v\ast f$ or $u\ast g$ instead of $v\ast u$ when $u$, respectively $v$, is the identity automorphism of $f$, respectively $g$.

It follows from the definition of the pairings (i) that Hom̲_{ℰ/-}(ℱ,𝒢) is a functor in $\mathcal{F}$ and $\mathcal{G}$, from the product category $Ca{t}_{/}{\mathcal{E}}^{\circ }\times Ca{t}_{/}\mathcal{E}$ to the category Cat. Indeed, if $g:\mathcal{G}\to {\mathcal{G}}_1$ is an $\mathcal{E}$-functor, i.e. an object of Hom̲_{ℰ/-}(𝒢,𝒢₁), then by taking $\mathcal{H}={\mathcal{G}}_1$ in (i), there corresponds to it a functor

g_*: Hom̲_{ℰ/-}(ℱ,𝒢) → Hom̲_{ℰ/-}(ℱ,𝒢₁).

One defines analogously, for an $\mathcal{E}$-functor $f:{\mathcal{F}}_1\to \mathcal{F}$, a functor

f^*: Hom̲_{ℰ/-}(ℱ,𝒢) → Hom̲_{ℰ/-}(ℱ₁,𝒢).

For short, these functors are also denoted by the symbols $f\mapsto g\circ f$ and $g\mapsto g\circ f$ respectively; these in fact denote only the corresponding maps on the sets of objects. It follows from the associativity property indicated above that one does indeed obtain in this way, as announced, a functor Cat_/ℰ° × Cat_/ℰ → Cat.

3. Base Change in Categories over ℰ

Since projective limits exist in Cat, relative to categories $\mathcal{I}$ belonging to the Universe, the same is true in $Ca{t}_{/}\mathcal{E}$. In particular, cartesian products exist there; these are interpreted as fiber products in Cat. In accordance with the general notation, if $\mathcal{F}$ and $\mathcal{G}$ are categories over $\mathcal{E}$, we denote by

$$\mathcal{F}{\times }_{\mathcal{E}}\mathcal{G}$$

their product in $Ca{t}_{/}\mathcal{E}$, i.e. their fiber product over $\mathcal{E}$ in Cat, regarded as a category over $\mathcal{E}$. Thus $\mathcal{F}{\times }_{\mathcal{E}}\mathcal{G}$ is endowed with two $\mathcal{E}$-functors $p{r}_1$ and $p{r}_2$, which define, for every category $\mathcal{H}$ over $\mathcal{E}$, a bijection

Hom_ℰ(ℋ, ℱ ×_ℰ 𝒢) ≃ Hom_ℰ(ℋ,ℱ) × Hom_ℰ(ℋ,𝒢).

This bijection moreover comes from an isomorphism of categories

Hom̲_{ℰ/-}(ℋ, ℱ ×_ℰ 𝒢) ≃ Hom̲_{ℰ/-}(ℋ,ℱ) × Hom̲_{ℰ/-}(ℋ,𝒢),

by taking the sets of objects of the two sides. The displayed functor is the one whose components are the functors $h\mapsto p{r}_1\circ h$ and $h\mapsto p{r}_2\circ h$ from the first member to the two factors of the second. We leave to the reader the verification that one indeed obtains an isomorphism in this way; the analogous fact is true more generally whenever one has a projective limit of categories, and not only in the case of a fiber product.

Recall moreover, as was said in VI.1, that

Ob(ℱ ×_ℰ 𝒢) = Ob(ℱ) ×_{Ob(ℰ)} Ob(𝒢),
Fl(ℱ ×_ℰ 𝒢) = Fl(ℱ) ×_{Fl(ℰ)} Fl(𝒢),

the composition of arrows being carried out componentwise.

In what follows, we consider a functor

$$\lambda :{\mathcal{E}}^{\prime }\to \mathcal{E},$$

and, for every category $\mathcal{F}$ over $\mathcal{E}$, we regard $\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }$ as a category over ${\mathcal{E}}^{\prime }$ by means of $p{r}_2$. In other words, we interpret the “fiber product” operation as an operation of “base change”, the functor $\lambda :{\mathcal{E}}^{\prime }\to \mathcal{E}$ being called the “base-change functor.” In accordance with the well-known general facts, one obtains in this way a functor, called the base-change functor for $\lambda$:

$$\lambda \ast :Ca{t}_{/}\mathcal{E}\to Ca{t}_{/}{\mathcal{E}}^{\prime }.$$

It is adjoint to the “restriction of the base” functor, which sends every category ${\mathcal{F}}^{\prime }$ over ${\mathcal{E}}^{\prime }$, with projection functor $p^{\prime }$, to ${\mathcal{F}}^{\prime }$ regarded as a category over $\mathcal{E}$ by the functor $p=\lambda p^{\prime }$. As is well known for a base-change functor in a category, the base-change functor “commutes with projective limits”, and in particular “transforms” fiber products over $\mathcal{E}$ into fiber products over ${\mathcal{E}}^{\prime }$.

Let $\mathcal{F}$ and $\mathcal{G}$ be two categories over $\mathcal{E}$. We shall define a canonical isomorphism

(i)  Hom̲_{ℰ′/-}(ℱ′,𝒢′) ≃ Hom̲_{ℰ/-}(ℱ ×_ℰ ℰ′,𝒢),
     where ℱ′ = ℱ ×_ℰ ℰ′ and 𝒢′ = 𝒢 ×_ℰ ℰ′.

For this, consider the functor

pr₁: 𝒢′ = 𝒢 ×_ℰ ℰ′ → 𝒢,

and define (i) by

$$F\mapsto p{r}_1\circ F,$$

which a priori denotes a functor

(ii) Hom̲(ℱ′,𝒢′) → Hom̲(ℱ′,𝒢).

It remains only to verify that this latter functor induces a functor on the subcategories in (i), and that this induced functor is an isomorphism. That (ii) induces a bijection

Hom_{ℰ′/-}(ℱ′,𝒢′) ≃ Hom_{ℰ/-}(ℱ ×_ℰ ℰ′,𝒢)

is the characteristic property of the base-change functor. It remains therefore

to prove that if $F$, $G$ are ${\mathcal{E}}^{\prime }$-functors ${\mathcal{F}}^{\prime }\to {\mathcal{G}}^{\prime }$, then the map

$$u\mapsto p{r}_1\circ u$$

induces a bijection

Hom_{ℰ′}(F,G) ≃ Hom_ℰ(pr₁ ∘ F, pr₁ ∘ G).

The verification of this fact is immediate and is left to the reader.

It follows from this isomorphism (i), and from the end of the preceding number, that

Hom̲_{ℰ′/-}(ℱ ×_ℰ ℰ′, 𝒢 ×_ℰ ℰ′)

may be regarded as a functor in ${\mathcal{E}}^{\prime }$, $\mathcal{F}$, $\mathcal{G}$, from the category Cat_/ℰ° × Cat_/ℰ° × Cat_/ℰ to the category Cat, isomorphic to the functor defined by the expression

Hom̲_{ℰ/-}(ℱ ×_ℰ ℰ′,𝒢).

In particular, for fixed $\mathcal{F}$ and $\mathcal{G}$, one obtains a functor in ${\mathcal{E}}^{\prime }$. Thus the $\mathcal{E}$-functor of projection $\lambda :{\mathcal{E}}^{\prime }\to \mathcal{E}$ defines a morphism, i.e. a functor

λ*_{ℱ,𝒢}: Hom̲_{ℰ/-}(ℱ,𝒢) → Hom̲_{ℰ′/-}(ℱ′,𝒢′),

which we now spell out. On the sets of objects of the two sides, it is the map

f ↦ f′ = f ×_ℰ ℰ′,

expressing the functorial dependence of $\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }$ on the object $\mathcal{F}$ over $\mathcal{E}$. On the other hand, consider two $\mathcal{E}$-functors

$$f,g:\mathcal{F}\to \mathcal{G}$$

and a homomorphism of $\mathcal{E}$-functors

$$u:f\to g.$$

We shall spell out the corresponding homomorphism of ${\mathcal{E}}^{\prime }$-functors

$$u^{\prime }:f^{\prime }\to g^{\prime }.$$

For every

$${\xi }^{\prime }=(\xi ,S^{\prime })\in Ob({\mathcal{F}}^{\prime })$$

with

ξ ∈ Ob(ℱ),    S′ ∈ Ob(ℰ′),    p(ξ) = λ(S′) = S,

the morphism

u′(ξ′): f′(ξ′) = (f(ξ),S′) → g′(ξ′) = (g(ξ),S′)    in 𝒢′

is defined by the formula

$$u^{\prime }({\xi }^{\prime })=(u(\xi ),i{d}_{S^{\prime }}).$$

This is indeed an $S^{\prime }$-morphism in ${\mathcal{G}}^{\prime }$, since $q(u(\xi ))=\lambda (i{d}_{S^{\prime }})=i{d}_S$.

Now consider any $\mathcal{E}$-functor

$${\lambda }^{\prime }:{\mathcal{E}}^{\prime \prime }\to {\mathcal{E}}^{\prime }$$

and the corresponding functor

Hom̲_{ℰ′/-}(ℱ ×_ℰ ℰ′, 𝒢 ×_ℰ ℰ′)
  → Hom̲_{ℰ″/-}(ℱ ×_ℰ ℰ″, 𝒢 ×_ℰ ℰ″).

I say that this functor is none other than the one obtained by the preceding process, starting from ${\mathcal{F}}^{\prime }$ and ${\mathcal{G}}^{\prime }$ over ${\mathcal{E}}^{\prime }$ and regarding ${\mathcal{E}}^{\prime \prime }$ as a category over ${\mathcal{E}}^{\prime }$, taking into account the isomorphisms of “transitivity of base change”

ℱ′ ×_ℰ′ ℰ″ ≃ ℱ″ = ℱ ×_ℰ ℰ″,
𝒢′ ×_ℰ′ ℰ″ ≃ 𝒢″ = 𝒢 ×_ℰ ℰ″,

which imply a canonical isomorphism

Hom̲_{ℰ″/-}(ℱ′ ×_ℰ′ ℰ″, 𝒢′ ×_ℰ′ ℰ″)
  ≃ Hom̲_{ℰ″/-}(ℱ ×_ℰ ℰ″, 𝒢 ×_ℰ ℰ″).

The verification of this compatibility is immediate and is left to the reader.

The functors just defined are compatible with the pairings defined in the preceding number. More precisely, if $\mathcal{F}$, $\mathcal{G}$, $\mathcal{H}$ are categories over $\mathcal{E}$ and if one puts

ℱ′ = ℱ ×_ℰ ℰ′,    𝒢′ = 𝒢 ×_ℰ ℰ′,    ℋ′ = ℋ ×_ℰ ℰ′,

one has commutativity in the following diagram of functors:

Hom̲_{ℰ/-}(ℱ,𝒢) × Hom̲_{ℰ/-}(𝒢,ℋ)  →  Hom̲_{ℰ/-}(ℱ,ℋ)
        ↓ λ*_{ℱ,𝒢} × λ*_{𝒢,ℋ}              ↓ λ*_{ℱ,ℋ}
Hom̲_{ℰ′/-}(ℱ′,𝒢′) × Hom̲_{ℰ′/-}(𝒢′,ℋ′) → Hom̲_{ℰ′/-}(ℱ′,ℋ′),

where the horizontal arrows are the composition functors defined in the preceding number. This commutativity is expressed by the formulas

$$(gf{)}^{\prime }=g^{\prime }{f}^{\prime }$$

for $f\in {\mathrm{Hom}}_{\mathcal{E}}(\mathcal{F},\mathcal{G})$, $g\in {\mathrm{Hom}}_{\mathcal{E}}(\mathcal{G},\mathcal{H})$, a formula which simply expresses the functoriality of base change, and

(v ∗ u)′ = v′ ∗ u′

when $u:f\to f_1$ is an arrow of Hom̲_{ℰ/-}(ℱ,𝒢) and $v:g\to g_1$ is an arrow of Hom̲_{ℰ/-}(𝒢,ℋ). The verification of this formula follows easily from the definitions.

In what follows, we shall be chiefly interested in Hom̲_ℰ(ℱ,𝒢), and certain remarkable subcategories of it, when $\mathcal{F}=\mathcal{E}$. For this reason we introduce a special notation:

Γ̲(𝒢/ℰ) = Hom̲_ℰ(ℰ,𝒢),
Γ(𝒢/ℰ) = Ob(Γ̲(𝒢/ℰ)) = Hom_ℰ(ℰ,𝒢).

Remarks. When $\mathcal{E}$ is a point category, i.e. $Ob(\mathcal{E})$ and $Fl(\mathcal{E})$ are reduced to a single element, which also means that $\mathcal{E}$ is a final object of the category Cat, then the data of a category over $\mathcal{E}$ is equivalent to the data of a category in the ordinary sense, since there will be a unique functor from $\mathcal{F}$ to $\mathcal{E}$. More precisely, $Ca{t}_{/}\mathcal{E}$ is then isomorphic to Cat. Moreover, the categories Hom̲_{ℰ/-}(ℱ,𝒢) are then none other than the Hom̲(ℱ,𝒢).

Recall then that the fundamental formula

Hom(ℋ, Hom̲(ℱ,𝒢)) ≃ Hom(ℱ × ℋ, 𝒢),

functorial in the three arguments appearing in it, allows Hom̲(ℱ,𝒢) to be interpreted axiomatically, in terms internal to the category Cat. Thus the familiar formulary for Hom̲-categories appears as a special case of a formulary valid in categories such as Cat, where “Hom̲-objects”, defined by the preceding formula, exist. There is an analogous interpretation of Hom̲_{ℰ/-}(ℱ,𝒢), when $\mathcal{E}$ is again arbitrary, by the formula

Hom(ℋ, Hom̲_{ℰ/-}(ℱ,𝒢)) ≃ Hom_ℰ(ℱ × ℋ, 𝒢),

functorial in the three arguments. In this way, the formal properties set out in VI.2 and VI.3 are special cases of more general results, valid in categories where the objects Hom̲_{ℰ/-}(ℱ,𝒢), for $\mathcal{F}$ and $\mathcal{G}$ two objects of the category over a third object $\mathcal{E}$, exist.

4. Fiber Categories; Equivalence of ℰ-Categories

Let $\mathcal{F}$ be a category over $\mathcal{E}$, and let $S\in Ob(\mathcal{E})$. The fiber category of $\mathcal{F}$ at $S$ is the subcategory ${\mathcal{F}}_S$ of $\mathcal{F}$ that is the inverse image of the point subcategory of $\mathcal{E}$ defined by $S$.

Thus the objects of ${\mathcal{F}}_S$ are the objects $\xi$ of $\mathcal{F}$ such that $p(\xi )=S$, and its morphisms are the morphisms $u$ of $\mathcal{F}$ such that $p(u)=i{d}_S$, i.e. the $S$-morphisms in $\mathcal{F}$. Of course, ${\mathcal{F}}_S$ is canonically isomorphic to the fiber product $\mathcal{F}{\times }_{\mathcal{E}}S$, where $S$ denotes the point subcategory of $\mathcal{E}$ defined by $S$, endowed with its inclusion functor into $\mathcal{E}$. It follows, taking the transitivity of base change into account, that if one makes a base change $\lambda :{\mathcal{E}}^{\prime }\to \mathcal{E}$, then for every $S^{\prime }\in Ob({\mathcal{E}}^{\prime })$, the projection $p{r}_1:{\mathcal{F}}^{\prime }=\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }\to \mathcal{F}$ induces an isomorphism

ℱ′_{S′} → ℱ_S,    where S = λ(S′).

Proposition.

Let $f:\mathcal{F}\to \mathcal{G}$ be an $\mathcal{E}$-functor. If $f$ is fully faithful, then for every base change ${\mathcal{E}}^{\prime }\to \mathcal{E}$, the corresponding functor

f′: ℱ′ = ℱ ×_ℰ ℰ′ → 𝒢′ = 𝒢 ×_ℰ ℰ′

is fully faithful.

The verification is immediate. More generally, one can show that every projective limit of fully faithful functors, here $f$ and the identity functors in $\mathcal{E}$ and ${\mathcal{E}}^{\prime }$, is a fully faithful functor.

One should note that the assertion analogous to 4.1, with “fully faithful” replaced by “equivalence of categories”, is false, already for $\mathcal{G}=\mathcal{E}$. However:

Proposition.

Let $f:\mathcal{F}\to \mathcal{G}$ be an $\mathcal{E}$-functor. The following conditions are equivalent:

1. There exists an $\mathcal{E}$-functor $g:\mathcal{G}\to \mathcal{F}$ and $\mathcal{E}$-isomorphisms

gf ≃ id_ℱ,    fg ≃ id_𝒢.

1. For every category ${\mathcal{E}}^{\prime }$ over $\mathcal{E}$, the functor

f′ = f ×_ℰ ℰ′: ℱ′ = ℱ ×_ℰ ℰ′ → 𝒢′ = 𝒢 ×_ℰ ℰ′

is an equivalence of categories.

1. $f$ is an equivalence of categories, and for every $S\in Ob(\mathcal{E})$, the functor $f_S:{\mathcal{F}}_S\to {\mathcal{G}}_S$ induced by $f$ is an equivalence of categories.

2. $f$ is fully faithful, and for every $S\in Ob(\mathcal{E})$ and every $\eta \in Ob({\mathcal{G}}_S)$, there exist $\xi \in Ob({\mathcal{F}}_S)$ and an $S$-isomorphism $u:f(\xi )\to \eta$.

Proof. Evidently (1) implies that $f$ is an equivalence of categories, a notion defined by the same condition but without requiring the isomorphisms of functors to be $\mathcal{E}$-morphisms. On the other hand, it follows from the functorialities of the preceding number that condition (1) is preserved after base change ${\mathcal{E}}^{\prime }\to \mathcal{E}$. Hence (1) ⇒ (2). Evidently (2) ⇒ (3), since it is enough to take ${\mathcal{E}}^{\prime }=\mathcal{E}$ and ${\mathcal{E}}^{\prime }=S$. It is still more trivial that (3) ⇒ (4). It remains to prove (4) ⇒ (1).

For this, choose for every $\eta \in Ob(\mathcal{G})$ an object $g(\eta )\in Ob(\mathcal{F})$ and an isomorphism $u(\eta ):f(g(\eta ))\to \eta$ such that $q(u(\eta ))=i{d}_S$, where $S=q(\eta )$. This is possible by the second condition in (4). As is known and immediate, the fact that $f$ is fully faithful implies that $g$ can be regarded in a unique way as a functor from $\mathcal{G}$ to $\mathcal{F}$, so that the $u(\eta )$ define a functorial homomorphism, hence isomorphism,

$$u:fg\simeq i{d}_{\mathcal{G}}.$$

Moreover, by construction, $g$ is an $\mathcal{E}$-functor and $u$ an $\mathcal{E}$-homomorphism. To the preceding data there then corresponds a functorial isomorphism $v:gf\to i{d}_{\mathcal{F}}$, defined by the condition $f\ast v=u\ast f$, and one sees at once that it is also an $\mathcal{E}$-morphism. This proves the assertion.

Definition.

If the preceding conditions are satisfied, one says that $f$ is an equivalence of categories over $\mathcal{E}$, or an $\mathcal{E}$-equivalence.

Corollary.

Suppose that the projection functor $p:\mathcal{F}\to \mathcal{E}$ is a transportable functor, i.e. that for every isomorphism $\alpha :T\to S$ in $\mathcal{E}$ and every object $\xi$ in ${\mathcal{F}}_T$, there exists an isomorphism $u$ in $\mathcal{F}$ with source $\xi$ such that $p(u)=\alpha$. Then every $\mathcal{E}$-functor $f:\mathcal{F}\to \mathcal{G}$ that is an equivalence of categories is an $\mathcal{E}$-equivalence.

This follows

from criterion (4).

Corollary.

Let $f:\mathcal{F}\to \mathcal{G}$ be an $\mathcal{E}$-equivalence. Then for every category $\mathcal{H}$ over $\mathcal{E}$, the corresponding functors

Hom̲_{ℰ/-}(𝒢,ℋ) → Hom̲_{ℰ/-}(ℱ,ℋ),
Hom̲_{ℰ/-}(ℋ,ℱ) → Hom̲_{ℰ/-}(ℋ,𝒢)

are equivalences of categories.

This follows from criterion (1) by the usual argument.

5. Cartesian Morphisms, Inverse Images, Cartesian Functors

Let $\mathcal{F}$ be a category over $\mathcal{E}$, with projection functor $p$.

Definition.

Consider a morphism

$$\alpha :\eta \to \xi$$

in $\mathcal{F}$, and let

S = p(ξ),    T = p(η),    f = p(α).

One says that $\alpha$ is a cartesian morphism if, for every ${\eta }^{\prime }\in Ob({\mathcal{F}}_T)$ and every $f$-morphism $u:{\eta }^{\prime }\to \xi$, there exists a unique $T$-morphism $\bar{u}:{\eta }^{\prime }\to \eta$ such that $u=\alpha \circ \bar{u}$.

This therefore means that, for every ${\eta }^{\prime }\in Ob({\mathcal{F}}_T)$, the map $v\mapsto \alpha \circ v$

$$(i){\mathrm{Hom}}_T({\eta }^{\prime },\eta )\to {\mathrm{Hom}}_f({\eta }^{\prime },\xi )$$

is bijective. It also means that the pair $(\eta ,\alpha )$ represents, as a functor in ${\eta }^{\prime }$, the functor ${\mathcal{F}}_T^{\circ }\to Set$ given by the second member.

If, for a given morphism $f:T\to S$ in $\mathcal{E}$ and a given $\xi \in Ob({\mathcal{F}}_S)$, such a pair $(\eta ,\alpha )$ exists, i.e. a cartesian morphism $\alpha$ in $\mathcal{F}$ with target $\xi$ and with $p(\alpha )=f$, then $\eta$ is determined in ${\mathcal{F}}_T$ up to unique isomorphism. One then says that the inverse image of $\xi$ by $f$ exists, and an object $\eta$ of ${\mathcal{F}}_T$ endowed with a cartesian $f$-morphism $\alpha :\eta \to \xi$ is called an inverse image of $\xi$ by $f$.

Often, once $\mathcal{F}$ is fixed, one assumes such an inverse image chosen whenever it exists. The inverse image will then be denoted by symbols such as $f{\ast }_{\mathcal{F}}(\xi )$, or simply $f\ast (\xi )$, or $\xi {\times }_{S}T$ when these notations cause no confusion. In what follows, the canonical morphism $\alpha :\eta \to \xi$ will then be denoted ${\alpha }_f(\xi )$.

If for every $\xi \in Ob({\mathcal{F}}_S)$ the inverse image of $\xi$ by $f$ exists, one also says that the inverse-image functor by $f$ in $\mathcal{F}$ exists, and $f\ast (\xi )$ then becomes a covariant functor in $\xi$, from ${\mathcal{F}}_S$ to ${\mathcal{F}}_T$. This comes from the fact that the second member in (i) depends covariantly on $\xi$, or more precisely denotes a functor from ${\mathcal{F}}_T^{\circ }\times {\mathcal{F}}_S$ to Set.

This functorial dependence of $f\ast (\xi )$ is made explicit as follows. Consider cartesian $f$-morphisms

α: η → ξ,    α′: η′ → ξ′

and an $S$-morphism $\lambda :\xi \to {\xi }^{\prime }$. Then there exists a unique $T$-morphism $\mu :\eta \to {\eta }^{\prime }$ such that

α′ μ = λ α,

as follows from the fact that ${\alpha }^{\prime }$ is cartesian.

Also note the following immediate fact. Consider a commutative diagram in $\mathcal{F}$

η  --α-->  ξ
|         |
μ         λ
↓         ↓
η′ --α′-> ξ′

where $\alpha$ and ${\alpha }^{\prime }$ are $f$-morphisms, $\lambda$ is an $S$-isomorphism, and $\mu$ is a $T$-isomorphism. Then $\alpha$ is cartesian if and only if ${\alpha }^{\prime }$ is cartesian.

Definition.

An $\mathcal{E}$-functor $F:\mathcal{F}\to \mathcal{G}$ is called a cartesian functor if it transforms cartesian morphisms into cartesian morphisms. We denote by Hom̲_cart(ℱ,𝒢) the full subcategory of Hom̲_{ℰ/-}(ℱ,𝒢) formed by the cartesian functors.

For example, regarding $\mathcal{E}$ as a category over $\mathcal{E}$ by means of the identity functor, every morphism of $\mathcal{E}$ is cartesian. Thus a cartesian functor from $\mathcal{E}$ to $\mathcal{F}$ is a section functor $F:\mathcal{E}\to \mathcal{F}$ that transforms every morphism of $\mathcal{E}$ into a cartesian morphism. Such a functor is called a cartesian section of $\mathcal{F}$ over $\mathcal{E}$.

Proposition.

1. A functor $F:\mathcal{F}\to \mathcal{G}$ that is an $\mathcal{E}$-equivalence is a cartesian functor.
2. Let $F$, $G$ be two isomorphic $\mathcal{E}$-functors $\mathcal{F}\to \mathcal{G}$. If one is cartesian, then so is the other.
3. The composite of two cartesian functors $\mathcal{F}\to \mathcal{G}$ and $\mathcal{G}\to \mathcal{H}$ is a cartesian functor.

Assertion (3) is trivial from the definition; (2) follows from the remark preceding VI.5.2; (1) follows easily from the definition and criterion VI.4.2 (3). More precisely, a morphism $\alpha$ in $\mathcal{F}$ is cartesian if and only if $F(\alpha )$ is cartesian.

Corollary.

Let $F:\mathcal{F}\to \mathcal{G}$ be an $\mathcal{E}$-equivalence. Then for every category $\mathcal{H}$ over $\mathcal{E}$, the corresponding functors $G\mapsto G\circ F$ and $G\mapsto F\circ G$ induce equivalences of categories:

Hom̲_cart(𝒢,ℋ) ≃ Hom̲_cart(ℱ,ℋ),
Hom̲_cart(ℋ,ℱ) ≃ Hom̲_cart(ℋ,𝒢).

This follows in the usual way from criterion VI.4.2 (1) and from VI.5.3 (1), (2), (3).

One can specify that the $\mathcal{E}$-functor $G:\mathcal{G}\to \mathcal{H}$ is cartesian if and only if $G\circ F$ is cartesian, and likewise an $\mathcal{E}$-functor $G:\mathcal{H}\to \mathcal{F}$ is cartesian if and only if $F\circ G$ is cartesian.

It follows from VI.5.4 (3) that, if one considers the subcategory $Ca{t}_{/}^{cart}\mathcal{E}$ of $Ca{t}_{/}\mathcal{E}$ whose objects are the same as those of $Ca{t}_{/}\mathcal{E}$ and whose morphisms are the cartesian functors, then, as in VI.2, one has pairings

Hom̲_cart(ℱ,𝒢) × Hom̲_cart(𝒢,ℋ) → Hom̲_cart(ℱ,ℋ)

induced by those of VI.2. These pairings allow one to regard Hom̲_cart(ℱ,𝒢) as a functor in $\mathcal{F}$ and $\mathcal{G}$, from the category (Cat^cart_/ℰ)° × Cat^cart_/ℰ to Cat. We shall need this remark chiefly in the case $\mathcal{F}=\mathcal{G}$.

Definition.

Let $\mathcal{F}$ be a category over $\mathcal{E}$. We denote by

$$Lim\leftarrow (\mathcal{F}/\mathcal{E})$$

the category of cartesian $\mathcal{E}$-functors $\mathcal{E}\to \mathcal{F}$, i.e. the cartesian sections of $\mathcal{F}$ over $\mathcal{E}$.

By what has just been said, $Lim\leftarrow (\mathcal{F}/\mathcal{E})$ is a functor in $\mathcal{F}$, from the category $Ca{t}_{/}^{cart}\mathcal{E}$ to the category Cat.

We shall see below the relations between this operation $Lim\leftarrow$ and the notion of projective limit of categories, as well as numerous examples.

6. Fibered Categories and Prefibered Categories. Products and Base Change in Them

Definition.

A category $\mathcal{F}$ over $\mathcal{E}$ is called a fibered category, and the functor $\mathcal{F}\to \mathcal{E}$ is then said to be fibrant, if it satisfies the two following axioms:

Fib I. For every morphism $f:T\to S$ in $\mathcal{E}$, the inverse-image functor by $f$ in $\mathcal{F}$ exists.

Fib II. The composite of two cartesian morphisms is cartesian.

A category $\mathcal{F}$ over $\mathcal{E}$ satisfying condition Fib I is called a prefibered category over $\mathcal{E}$.

If $\mathcal{F}$ is a fibered, respectively prefibered, category over $\mathcal{E}$, a subcategory $\mathcal{G}$ of $\mathcal{F}$ is called a fibered subcategory, respectively a prefibered subcategory, if it is a fibered, respectively prefibered, category over $\mathcal{E}$ and, moreover, the inclusion functor is cartesian. If, for example, $\mathcal{G}$ is a full subcategory of $\mathcal{F}$, one sees that this means that, for every morphism $f:T\to S$ in $\mathcal{E}$ and every $\xi \in Ob({\mathcal{G}}_S)$, $f{\ast }_{\mathcal{F}}(\xi )$ is $T$-isomorphic to an object of ${\mathcal{G}}_T$.

Another interesting case is the following. Let $\mathcal{F}$ be a fibered category over $\mathcal{E}$, and consider the subcategory $\mathcal{G}$ of $\mathcal{F}$ with the same objects and whose morphisms are the cartesian morphisms of $\mathcal{F}$; in particular the morphisms of ${\mathcal{G}}_S$ are the isomorphisms of ${\mathcal{F}}_S$. One sees at once that this is indeed a fibered subcategory of $\mathcal{F}$, because in the bijection

$${\mathrm{Hom}}_T({\eta }^{\prime },\eta )\simeq {\mathrm{Hom}}_f({\eta }^{\prime },\xi )$$

relative to a cartesian $f$-morphism $\alpha$ in $\mathcal{F}$, the $T$-isomorphisms of the first member correspond to the cartesian morphisms of the second. By definition, the cartesian sections $\mathcal{E}\to \mathcal{F}$ then correspond bijectively to arbitrary $\mathcal{E}$-functors $\mathcal{E}\to \mathcal{G}$. However, note that the natural functor

Hom̲_{ℰ/-}(ℰ,𝒢) → Hom̲_cart(ℰ,ℱ) = Lim←(ℱ/ℰ)

is faithful, but in general is not fully faithful, i.e. is not an isomorphism.

Remarks. Let $\mathcal{F}$ be a category over $\mathcal{E}$. The following conditions are equivalent:

1. All morphisms of $\mathcal{F}$ are cartesian.
2. $\mathcal{F}$ is a fibered category over $\mathcal{E}$, and the ${\mathcal{F}}_S$ are groupoids, i.e. every morphism in ${\mathcal{F}}_S$ is an isomorphism.

One then says that $\mathcal{F}$ is a category fibered in groupoids over $\mathcal{E}$.

These are the ones encountered especially in “theory of moduli”. If $\mathcal{E}$ is a groupoid, one shows that conditions (1) and (2) are also equivalent to the following:

1. $\mathcal{F}$ is a groupoid, and the projection functor $p:\mathcal{F}\to \mathcal{E}$ is transportable; cf. VI.4.4.

For example, if $\mathcal{E}$ and $\mathcal{F}$ are groupoids such that $Ob(\mathcal{E})$ and $Ob(\mathcal{F})$ are reduced to a point, so that $\mathcal{E}$ and $\mathcal{F}$ are defined, up to isomorphism, by groups $E$ and $F$, and the functor $p:\mathcal{F}\to \mathcal{E}$ is defined by a group homomorphism $p:F\to E$, then $\mathcal{F}$ is fibered over $\mathcal{E}$ if and only if $p$ is surjective, i.e. if $p$ defines an extension of the group $E$ by the group $G=Kerp$.

Proposition.

Let $F:\mathcal{F}\to \mathcal{G}$ be an $\mathcal{E}$-equivalence. In order that $\mathcal{F}$ be a fibered, respectively prefibered, category over $\mathcal{E}$, it is necessary and sufficient that $\mathcal{G}$ be so.

This follows easily from the definitions and from the remark made above that a morphism $\alpha$ in $\mathcal{F}$ is cartesian if and only if $F(\alpha )$ is.

Proposition.

Let ${\mathcal{F}}_1$, ${\mathcal{F}}_2$ be two categories over $\mathcal{E}$, and let $\alpha =({\alpha }_1,{\alpha }_2)$ be a morphism in $\mathcal{F}={\mathcal{F}}_1{\times }_{\mathcal{E}}{\mathcal{F}}_2$. Then $\alpha$ is cartesian if and only if its components are cartesian.

Indeed, let ${\xi }_i$ be the target and ${\eta }_i$ the source of ${\alpha }_i$, and let $f:T\to S$ be the morphism of $\mathcal{E}$ such that ${\alpha }_1$ and ${\alpha }_2$ are $f$-morphisms. For every ${\eta }^{\prime }=({\eta }_1^{\prime },{\eta }_2^{\prime })$ in ${\mathcal{F}}_T$, one has a commutative diagram

$${\mathrm{Hom}}_T({\eta }^{\prime },\eta )\to {\mathrm{Hom}}_f({\eta }^{\prime },\xi )\downarrow \downarrow {\mathrm{Hom}}_T({\eta }_1^{\prime },{\eta }_1)\times {\mathrm{Hom}}_T({\eta }_2^{\prime },{\eta }_2)\to {\mathrm{Hom}}_f({\eta }_1^{\prime },{\xi }_1)\times {\mathrm{Hom}}_f({\eta }_2^{\prime },{\xi }_2),$$

where the vertical arrows are bijections. Thus if one of the horizontal arrows is a bijection, the same is true of the other. This already shows that if ${\alpha }_1$ and ${\alpha }_2$ are cartesian, hence the second horizontal arrow is bijective, then $\alpha$ is cartesian. The converse is seen by taking, in the diagram above, ${\eta }_i^{\prime }={\eta }_i$, whence ${\mathrm{Hom}}_T({\eta }_i^{\prime },{\eta }_i)\ne \varnothing$: first for $i=2$, which proves that ${\alpha }_1$ is cartesian, then for $i=1$, which proves that ${\alpha }_2$ is cartesian.

Corollary.

Let $\mathcal{F}={\mathcal{F}}_1{\times }_{\mathcal{E}}{\mathcal{F}}_2$, and let $F=(F_1,F_2)$ be an $\mathcal{E}$-functor $\mathcal{G}\to \mathcal{F}$. Then $F$ is cartesian if and only if $F_1$ and $F_2$ are cartesian. One obtains in this way an isomorphism of categories

Hom̲_cart(𝒢, ℱ₁ ×_ℰ ℱ₂) ≃ Hom̲_cart(𝒢,ℱ₁) × Hom̲_cart(𝒢,ℱ₂),

and in particular, taking $\mathcal{G}=\mathcal{E}$, an isomorphism of categories

Lim←((ℱ₁ ×_ℰ ℱ₂)/ℰ) ≃ Lim←(ℱ₁/ℰ) × Lim←(ℱ₂/ℰ).

Corollary.

Let ${\mathcal{F}}_1$ and ${\mathcal{F}}_2$ be two fibered, respectively prefibered, categories over $\mathcal{E}$. Then their fiber product $\mathcal{F}={\mathcal{F}}_1{\times }_{\mathcal{E}}{\mathcal{F}}_2$ is a fibered, respectively prefibered, category over $\mathcal{E}$.

These results moreover extend to the case of the fiber product of an arbitrary family of categories over $\mathcal{E}$.

Proposition.

Let $\mathcal{F}$ be a category over $\mathcal{E}$, with projection functor $p$, and let $\lambda :{\mathcal{E}}^{\prime }\to \mathcal{E}$ be a functor. Regard ${\mathcal{F}}^{\prime }=\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }$ as a category over ${\mathcal{E}}^{\prime }$ by the projection functor $p^{\prime }=p{\times }_{\mathcal{E}}i{d}_{\mathcal{E}}^{\prime }$. Let ${\alpha }^{\prime }$ be a morphism of ${\mathcal{F}}^{\prime }$. Then ${\alpha }^{\prime }$ is a cartesian morphism if and only if its image $\alpha$ in $\mathcal{F}$ is cartesian.

The proof is immediate and is left to the reader.

Corollary.

For every cartesian functor $F:\mathcal{F}\to \mathcal{G}$ of categories over $\mathcal{E}$, the functor

F′ = F ×_ℰ ℰ′

from ${\mathcal{F}}^{\prime }=\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }$ to ${\mathcal{G}}^{\prime }=\mathcal{G}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }$ is cartesian.

Consequently, the functor Hom̲_ℰ(ℱ,𝒢) → Hom̲_ℰ′(ℱ′,𝒢′) considered in VI.3 induces a functor

Hom̲_cart(ℱ,𝒢) → Hom̲_cart(ℱ′,𝒢′).

In other words, for fixed $\mathcal{F}$ and $\mathcal{G}$, one may regard

Hom̲_cart(ℱ ×_ℰ ℰ′, 𝒢 ×_ℰ ℰ′)

as a functor in ${\mathcal{E}}^{\prime }$, from the category $Ca{t}_{/}{\mathcal{E}}^{\circ }$ to Cat. If $\mathcal{F}$ and $\mathcal{G}$ are also allowed to vary, one finds a functor from the category

Cat_/ℰ° × (Cat^cart_/ℰ)° × Cat^cart_/ℰ

to Cat.

When one takes into account the isomorphism

Hom_ℰ′(ℱ′,𝒢′) ≃ Hom_ℰ(ℱ ×_ℰ ℰ′,𝒢)

considered in VI.3, the cartesian ${\mathcal{E}}^{\prime }$-functors from ${\mathcal{F}}^{\prime }$ to ${\mathcal{G}}^{\prime }$ correspond to the $\mathcal{E}$-functors $\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }\to \mathcal{G}$ that transform every morphism whose first projection is a cartesian morphism of $\mathcal{F}$ into a cartesian morphism of $\mathcal{G}$. Taking $\mathcal{F}=\mathcal{E}$, one finds, after a change of notation:

Corollary.

$Lim\leftarrow ({\mathcal{F}}^{\prime }/{\mathcal{E}}^{\prime })$ is isomorphic to the full subcategory of Hom̲_{ℰ/-}(ℰ′,ℱ) formed by the $\mathcal{E}$-functors ${\mathcal{E}}^{\prime }\to \mathcal{F}$ that transform arbitrary morphisms into cartesian morphisms. In particular, if $\mathcal{F}$ is a fibered category and if $\tilde{\mathcal{F}}$ is the subcategory of $\mathcal{F}$ whose morphisms are the cartesian morphisms of $\mathcal{F}$, then one has a bijection

$$ObLim\leftarrow ({\mathcal{F}}^{\prime }/{\mathcal{E}}^{\prime })\simeq {\mathrm{Hom}}_{\mathcal{E}/-}({\mathcal{E}}^{\prime },\tilde{\mathcal{F}}).$$

This makes precise the way in which the expression

$$Lim\leftarrow ((\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime })/{\mathcal{E}}^{\prime })$$

must be regarded as a functor in ${\mathcal{E}}^{\prime }$ and $\mathcal{F}$, from the category Cat_/ℰ° × Cat^cart_/ℰ to the category Cat. Later we shall see a more complete functorial dependence with respect to ${\mathcal{E}}^{\prime }$ when $\mathcal{F}$ is required to be a fibered category.

Corollary.

If $\mathcal{F}$ is a fibered, respectively prefibered, category over $\mathcal{E}$, then ${\mathcal{F}}^{\prime }=\mathcal{F}{\times }_{\mathcal{E}}{\mathcal{E}}^{\prime }$ is a fibered, respectively prefibered, category over ${\mathcal{E}}^{\prime }$.

Proposition.

Let $\mathcal{F}$ and $\mathcal{G}$ be prefibered categories over $\mathcal{E}$, and let $F$ be a cartesian $\mathcal{E}$-functor from $\mathcal{F}$ to $\mathcal{G}$. In order that $F$ be faithful, respectively fully faithful, respectively an $\mathcal{E}$-equivalence, it is necessary and sufficient that for every $S\in Ob(\mathcal{E})$, the induced functor

$$F_S:{\mathcal{F}}_S\to {\mathcal{G}}_S$$

be faithful, respectively fully faithful, respectively an equivalence.

The proof is immediate from the definitions.

To finish this number, we give a few properties of fibered categories using axiom Fib II.

Proposition.

Let $\mathcal{F}$ be a prefibered category over $\mathcal{E}$. In order that $\mathcal{F}$ be fibered, it is necessary and sufficient that it satisfy the following condition:

Fib II′. Let $\alpha :\eta \to \xi$ be a cartesian morphism in $\mathcal{F}$ over the morphism $f:T\to S$ of $\mathcal{E}$. For every morphism $g:U\to T$ in $\mathcal{E}$, and every $\zeta \in Ob({\mathcal{F}}_U)$, the map $u\mapsto \alpha \circ u$

$${\mathrm{Hom}}_g(\zeta ,\eta )\to {\mathrm{Hom}}_{fg}(\zeta ,\xi )$$

is bijective.

In other words, in a category fibered over $\mathcal{E}$, cartesian diagrams are characterized by a property a priori stronger than the one in the definition, which is recovered by taking $g=i{d}_T$ in the preceding statement.

Corollary.

Let $\mathcal{F}$ be a category over $\mathcal{E}$ and let $\alpha$ be a morphism in $\mathcal{F}$. In order that $\alpha$ be an isomorphism, it is necessary that $p(\alpha )=f$ be an isomorphism and that $\alpha$ be cartesian. The converse is true if $\mathcal{F}$ is fibered over $\mathcal{E}$.

Indeed, if $\alpha$ is an isomorphism then evidently so is $f=p(\alpha )$. For every ${\eta }^{\prime }\in Ob({\mathcal{F}}_T)$, the map $u\mapsto \alpha \circ u$

$$\mathrm{Hom}({\eta }^{\prime },\eta )\to \mathrm{Hom}({\eta }^{\prime },\xi )$$

is bijective. Since $f$ is an isomorphism, one sees at once that an element of the first member is a $T$-morphism if and only if its image in the second is an $f$-morphism. Thus one obtains a bijection

$${\mathrm{Hom}}_T({\eta }^{\prime },\eta )\to {\mathrm{Hom}}_f({\eta }^{\prime },\xi ),$$

which proves the first assertion. Conversely, suppose that $f$ is an isomorphism and that $\alpha$ satisfies the condition stated in Fib II′, which means, when $\mathcal{F}$ is fibered over $\mathcal{E}$, that $\alpha$ is cartesian. Then one sees at once that for every $\zeta \in Ob(\mathcal{F})$, the map $u\mapsto \alpha \circ u$ from $\mathrm{Hom}(\zeta ,\eta )$ to $\mathrm{Hom}(\zeta ,\xi )$ is bijective, and hence $\alpha$ is an isomorphism.

Corollary.

Let $\alpha :\eta \to \xi$ and $\beta :\zeta \to \eta$ be two composable morphisms in the category $\mathcal{F}$ fibered over $\mathcal{E}$. If $\alpha$ is cartesian, then $\beta$ is cartesian if and only if $\alpha \beta$ is cartesian.

One uses the definition of cartesian morphisms in the strengthened form VI.6.11.

7. Categories Cloven over ℰ

Definition.

Let $\mathcal{F}$ be a category over $\mathcal{E}$. A cleavage of $\mathcal{F}$ over $\mathcal{E}$ means a function that attaches to every $f\in Fl(\mathcal{E})$ an inverse-image functor for $f$ in $\mathcal{F}$, denoted $f\ast$. The cleavage is said to be normalized if $f=i{d}_S$ implies $f\ast =i{d}_{{\mathcal{F}}_S}$. A cloven category, respectively a normalized cloven category, means a category $\mathcal{F}$ over $\mathcal{E}$ endowed with a cleavage, respectively with a normalized cleavage.

It is evident that $\mathcal{F}$ admits a cleavage if and only if $\mathcal{F}$ is prefibered over $\mathcal{E}$, and then $\mathcal{F}$ admits a normalized cleavage. The set of cleavages on $\mathcal{F}$ is in bijective correspondence with the set of subsets $K$ of $Fl(\mathcal{F})$ satisfying the following conditions:

1. The $\alpha \in K$ are cartesian morphisms.
2. For every morphism $f:T\to S$ in $\mathcal{E}$ and every $\xi \in Ob({\mathcal{F}}_S)$, there exists a unique $f$-morphism in $K$ with target $\xi$.

For the cleavage defined by $K$ to be normalized, it is necessary and sufficient that $K$ also satisfy the condition:

1. The identity morphisms in $\mathcal{F}$ belong to $K$.

The morphisms that are elements of $K$ may be called the “transport morphisms” for the cleavage in question.

The notion of isomorphism of cloven categories over $\mathcal{E}$ is clear. More generally, one can define morphisms of cloven $\mathcal{E}$-categories as functors of $\mathcal{E}$-categories $\mathcal{F}\to \mathcal{G}$ that send transport morphisms to transport morphisms. These are, in particular, cartesian functors. In this way the cloven categories over $\mathcal{E}$ are the objects of a category, the category of cloven categories over $\mathcal{E}$. The reader may spell out the existence of products, tied to the fact that if a category over $\mathcal{E}$ is the product of categories ${\mathcal{F}}_i$ over $\mathcal{E}$, each endowed with a cleavage, then $\mathcal{F}$ is endowed with the corresponding natural cleavage. We also leave to the reader the task of spelling out the notion of base change in cloven categories.

We shall denote by ${\alpha }_f(\xi )$ the canonical morphism

$${\alpha }_f(\xi ):f\ast (\xi )\to \xi .$$

As was said, it is functorial in $\xi$, i.e. one has a functorial homomorphism

α_f: i_T f* → i_S,

where for every $S\in Ob(\mathcal{E})$, $i_S$ denotes the inclusion functor

$$i_S:{\mathcal{F}}_S\to \mathcal{F}.$$

Now consider morphisms

f: T → S    and    g: U → T

in $\mathcal{E}$, and let $\xi \in Ob({\mathcal{F}}_S)$. There then exists a unique $U$-morphism

$$c_{f,g}(\xi ):g\ast f\ast (\xi )\to (fg)\ast (\xi )$$

making commutative

the diagram

$$g\ast f\ast (\xi )--{\alpha }_g(f\ast (\xi ))-->f\ast (\xi )|c_{f,g}(\xi )|{\alpha }_f(\xi )\downarrow \downarrow (fg)\ast (\xi )--{\alpha }_{fg}(\xi )-->\xi ,$$

by the definition of $(fg)\ast (\xi )$. For variable $\xi$, this homomorphism is functorial; that is, one has a homomorphism

$$c_{f,g}:g\ast f\ast \to (fg)\ast$$

of functors ${\mathcal{F}}_S\to {\mathcal{F}}_U$. Note at once:

Proposition.

In order that the cloven category $\mathcal{F}$ over $\mathcal{E}$ be fibered, it is necessary and sufficient that the $c_{f,g}$ be isomorphisms.

It follows, taking $f$ to be an isomorphism and $g$ its inverse, and considering the isomorphisms $c_{f,g}$ and $c_{g,f}$:

Corollary.

If $\mathcal{F}$ is a fibered cloven category over $\mathcal{E}$, then for every isomorphism $f:T\to S$ in $\mathcal{E}$, $f\ast$ is an equivalence of categories ${\mathcal{F}}_S\to {\mathcal{F}}_T$.

Proposition.

Let $\mathcal{F}$ be a cloven category over $\mathcal{E}$. One has:

A)
  c_{f,id_T}(ξ) = α_{id_T}(f*(ξ)),
  c_{id_S,f}(ξ) = f*(α_{id_S}(ξ)).

and

B)  c_{f,gh}(ξ) · c_{g,h}(f*(ξ))
      = c_{fg,h}(ξ) · h*(c_{f,g}(ξ)).

In these formulas, $f$, $g$, $h$ denote morphisms

V → U → T → S

and $\xi$ is an object of ${\mathcal{F}}_S$.

In the case of a normalized cleavage, the first and second relations take the simpler form

A′) c_{f,id_T} = id_{f*},    c_{id_S,f} = id_{f*}.

As for the third, it is visualized by the commutativity of the diagram

$$h\ast g\ast f\ast (\xi )--c_{g,h}(f\ast (\xi ))-->(gh)\ast f\ast (\xi )|h\ast (c_{f,g}(\xi ))|c_{f,gh}(\xi )\downarrow \downarrow h\ast (fg)\ast (\xi )--c_{fg,h}(\xi )-->(fgh)\ast (\xi ).$$

In the case of fibered categories, where the $c_{f,g}$ are isomorphisms, this commutativity may be expressed intuitively by saying that the successive use of isomorphisms of the form $c_{f,g}$ does not lead to “contradictory identifications.” One may also write this formula without the argument $\xi$, using the convolution product of homomorphisms of functors:

c_{fg,h} ∘ (h* ∗ c_{f,g}) = c_{f,gh} ∘ (c_{g,h} ∗ f*).

The proof of the first two formulas in VI.7.4 is trivial; let us sketch that of the third. For this, consider, in addition to the square (D), the square of homomorphisms

$$g\ast f\ast (\xi )--{\alpha }_g(f\ast (\xi ))-->f\ast (\xi )|c_{f,g}(\xi )|{\alpha }_f(\xi )\downarrow \downarrow (fg)\ast (\xi )--{\alpha }_{fg}(\xi )-->\xi ,$$

which is commutative by definition of $c_{f,g}(\xi )$. Consider the diagram obtained by joining the vertices of (D) to the corresponding vertices of this square by homomorphisms of the form $\alpha$:

$${\alpha }_h(g\ast f\ast (\xi )),{\alpha }_{gh}(f\ast (\xi )),{\alpha }_h((fg)\ast (\xi )),{\alpha }_{fgh}(\xi ).$$

The four lateral faces of the cube so obtained are also commutative. For the left face, this comes from the fact that the left column of (D) is obtained from the left column of the preceding square by applying $h$, and that ${\alpha }_h$ is a functorial homomorphism. For the other three faces, this is nothing other than the definition of the operations $c$ on the remaining three sides of (D). Thus the five faces of the cube other than the upper face are commutative. It follows that the two (fgh)-morphisms $h\ast g\ast f\ast (\xi )\to (fgh)\ast (\xi )$ defined by (D) have the same composite with ${\alpha }_{fgh}(\xi ):(fgh)\ast (\xi )\to \xi$. Hence they are equal by the definition of $(fgh)\ast$.

Let us confine ourselves, in what follows, to normalized cloven categories. Such a category gives rise to the following objects:

1. A map $S\mapsto {\mathcal{F}}_S$ from $Ob(\mathcal{E})$ to Cat.
2. A map $f\mapsto f\ast$, associating to every $f\in Fl(\mathcal{E})$, with source $T$ and target $S$, a functor $f\ast :{\mathcal{F}}_S\to {\mathcal{F}}_T$.
3. A map $(f,g)\mapsto c_{f,g}$, associating to every pair of arrows $(f,g)$ of $\mathcal{E}$ a functorial homomorphism $c_{f,g}:g\ast f\ast \to (fg)\ast$.

Moreover, these data satisfy the conditions expressed in formulas A′) and B) above. N.B. If one had not confined oneself to the case of a normalized cleavage, one would have had to introduce an additional object, namely a function $S\mapsto {\alpha }_S$ associating to every object $S$ of $\mathcal{E}$ a functorial homomorphism ${\alpha }_S:(i{d}_S)\ast \to i{d}_{{\mathcal{F}}_S}$; condition A′) would then be replaced by condition A).

We shall now show how one can reconstruct, up to unique isomorphism, the normalized cloven category $\mathcal{F}$ over $\mathcal{E}$ from the preceding objects.

8. Cloven Category Defined by a Pseudofunctor ℰ° → Cat

For short, call a pseudofunctor from ${\mathcal{E}}^{\circ }$ to Cat, one should say a normalized pseudofunctor, a set of data a), b), c) as above, satisfying conditions A′) and B). In the preceding number we associated to a normalized cloven category over $\mathcal{E}$ a pseudofunctor ${\mathcal{E}}^{\circ }\to Cat$. Here we indicate the inverse construction. We shall leave to the reader the verification of most of the details, as well as of the fact that these constructions are indeed “inverse” to one another. More precisely, one should regard the pseudofunctors ${\mathcal{E}}^{\circ }\to Cat$ as the objects of a new category, and show that our constructions provide equivalences, quasi-inverse to one another, between this latter category and the category of cloven categories over $\mathcal{E}$ defined in the preceding number.

Put

$${\mathcal{F}}^{\circ }=\coprod _{S\in Ob(\mathcal{E})}Ob(\mathcal{F}(S)),$$

the sum set of the sets $Ob(\mathcal{F}(S))$. N.B. here we write $\mathcal{F}(S)$, and not ${\mathcal{F}}_S$, for the value at the object $S$ of $\mathcal{E}$ of the given pseudofunctor, to avoid notational confusion later. We therefore have an evident map

$$p^{\circ }:{\mathcal{F}}^{\circ }\to Ob(\mathcal{E}).$$

Let

ξ̄ = (S,ξ),    η̄ = (T,η),    with ξ ∈ Ob(ℱ(S)), η ∈ Ob(ℱ(T)),

be two elements of ${\mathcal{F}}^{\circ }$, and let $f\in \mathrm{Hom}(T,S)$. Put

$$h_f(\bar{\eta },\bar{\xi })={\mathrm{Hom}}_{\mathcal{F}(T)}(\eta ,f\ast (\xi )).$$

If in addition one has a morphism $g:U\to T$ in $\mathcal{E}$ and $\zeta \in Ob(\mathcal{F}(U))$, one defines a map, denoted $(u,v)\mapsto u\circ v$,

$$h_f(\bar{\eta },\bar{\xi })\times h_g(\bar{\zeta },\bar{\eta })\to h_{fg}(\bar{\zeta },\bar{\xi }),$$

i.e. a map

Hom_{ℱ(T)}(η, f*(ξ)) × Hom_{ℱ(U)}(ζ, g*(η))
  → Hom_{ℱ(U)}(ζ, (fg)*(ξ)),

by the formula

u ∘ v = c_{f,g}(ξ) · g*(u) · v.

That is, $u\circ v$ is the composite of the sequence

ζ --v--> g*(η) --g*(u)--> g*f*(ξ) --c_{f,g}(ξ)--> (fg)*(ξ).

On the other hand, put

$$h(\bar{\eta },\bar{\xi })=\coprod _{f\in \mathrm{Hom}(T,S)}{h}_f(\bar{\eta },\bar{\xi }).$$

The preceding pairings define pairings

$$h(\bar{\eta },\bar{\xi })\times h(\bar{\zeta },\bar{\eta })\to h(\bar{\zeta },\bar{\xi }),$$

while the definition of the $h(\bar{\eta },\bar{\xi })$ gives an evident map

$$p_{\bar{\eta },\bar{\xi }}:h(\bar{\eta },\bar{\xi })\to \mathrm{Hom}(T,S).$$

This being said, one verifies the following points:

1. Composition between elements of the $h(\bar{\eta },\bar{\xi })$ is associative.

2. For every $\bar{\xi }=(S,\xi )$ in ${\mathcal{F}}^{\circ }$, consider the identity element of

$$h_{i{d}_S}(\bar{\xi },\bar{\xi })={\mathrm{Hom}}_{\mathcal{F}(S)}(i{d}_S\ast (\xi ),\xi )={\mathrm{Hom}}_{\mathcal{F}(S)}(\xi ,\xi ),$$

and its image in $h(\bar{\xi },\bar{\xi })$. This object is a left and right unit for composition between elements of the $h(\bar{\eta },\bar{\xi })$.

This already shows that one obtains a category $\mathcal{F}$ by putting

$$Ob(\mathcal{F})={\mathcal{F}}^{\circ },Fl(\mathcal{F})=\coprod _{\bar{\xi },\bar{\eta }\in {\mathcal{F}}^{\circ }}h(\bar{\eta },\bar{\xi }).$$

N.B. one cannot simply take $Fl(\mathcal{F})$ to be the union of the sets $h(\bar{\eta },\bar{\xi })$, since these latter sets are not necessarily disjoint. Moreover:

1. The maps $p^{\circ }:Ob(\mathcal{F})\to Ob(\mathcal{E})$ and $p_1=(p_{\bar{\eta },\bar{\xi }}):Fl(\mathcal{F})\to Fl(\mathcal{E})$ define a functor $p:\mathcal{F}\to \mathcal{E}$. In this way $\mathcal{F}$ becomes a category over $\mathcal{E}$; moreover, the evident map $h_f(\bar{\eta },\bar{\xi })\to \mathrm{Hom}(\bar{\eta },\bar{\xi })$ induces a bijection

$$h_f(\bar{\eta },\bar{\xi })\simeq {\mathrm{Hom}}_f(\bar{\eta },\bar{\xi }).$$

1. The evident maps

Ob(ℱ(S)) → ℱ° = Ob(ℱ),    Fl(ℱ(S)) → Fl(ℱ),

where the second is defined by the evident maps

$${\mathrm{Hom}}_{\mathcal{F}(S)}(\xi ,{\xi }^{\prime })=h_{i{d}_S}(\bar{\xi },{\bar{\xi }}^{\prime })\to \mathrm{Hom}(\bar{\xi },{\bar{\xi }}^{\prime }),$$

define an isomorphism

$$i_S:\mathcal{F}(S)\simeq {\mathcal{F}}_S.$$

1. For every object $\bar{\xi }=(S,\xi )$ of $\mathcal{F}$, and every morphism $f:T\to S$ of $\mathcal{E}$, consider

the element $\bar{\eta }=(T,\eta )$ of ${\mathcal{F}}_T$, with $\eta =f\ast (\xi )$, and the element ${\alpha }_f(\xi )$ of $\mathrm{Hom}(\bar{\eta },\bar{\xi })$, image of $i{d}_{f\ast (\xi )}$ by the morphism

$${\mathrm{Hom}}_{\mathcal{F}(T)}(f\ast (\xi ),f\ast (\xi ))=h_f(\bar{\eta },\bar{\xi })\to {\mathrm{Hom}}_f(\bar{\eta },\bar{\xi }).$$

This element is cartesian, and it is the identity of $\bar{\xi }$ if $f=i{d}_S$. In other words, the set of the ${\alpha }_f(\xi )$ defines a normalized cleavage of $\mathcal{F}$ over $\mathcal{E}$. Moreover, by construction, one has commutativity in the diagram of functors

$$\mathcal{F}(S)--f\ast -->\mathcal{F}(T)|i_S|i_T\downarrow \downarrow {\mathcal{F}}_S--f{\ast }_{\mathcal{F}}->{\mathcal{F}}_T,$$

where $f{\ast }_{\mathcal{F}}$ is the inverse-image functor by $f$, relative to the cleavage considered on $\mathcal{F}$. Finally:

1. The homomorphisms $c_{f,g}$ given with the pseudofunctor are transformed, by the isomorphisms $i_S$, into the functorial homomorphisms $c_{f,g}$ associated with the cleavage of $\mathcal{F}$.

We restrict ourselves to giving the verification of 1), which is, if anything, less trivial than the others. It suffices to prove associativity of composition between objects of sets of the form $h_f(\bar{\eta },\bar{\xi })$. Thus consider in $\mathcal{E}$ morphisms

S ←^f T ←^g U ←^h V

and objects

ξ, η, ζ, τ

in $\mathcal{F}(S)$, $\mathcal{F}(T)$, $\mathcal{F}(U)$, $\mathcal{F}(V)$, and finally elements

u ∈ h_f(η̄,ξ̄) = Hom_{ℱ(T)}(η, f*(ξ)),
v ∈ h_g(ζ̄,η̄) = Hom_{ℱ(U)}(ζ, g*(η)),
w ∈ h_h(τ̄,ζ̄) = Hom_{ℱ(V)}(τ, h*(ζ)).

We want to prove the formula

(u ∘ v) ∘ w = u ∘ (v ∘ w),

which is an equality in ${\mathrm{Hom}}_{\mathcal{F}(V)}(\tau ,(fgh)\ast (\xi ))$. By the definitions, the two members of this equality are obtained by composition along the upper and lower contours of the diagram below:

τ --w--> h*(ζ) --h*(v)--> h*g*(η) --h*g*(u)--> h*g*f*(ξ) --h*(c_{f,g}(ξ))--> h*(fg)*(ξ)
 \____________________ v∘w ____________________/       | c_{g,h}(f*(ξ))              | c_{fg,h}(ξ)
                         ↓                              ↓                             ↓
                  (gh)*(η) --(gh)*(u)-->        (gh)*f*(ξ) --c_{f,gh}(ξ)-->       (fgh)*(ξ).

The middle square is commutative because $c_{g,h}$ is a functorial homomorphism, and the square on the right is commutative by condition B) for a pseudofunctor. This gives the asserted result.

Of course, it remains to specify, when the pseudofunctor considered already comes from a normalized cloven category ${\mathcal{F}}^{\prime }$ over $\mathcal{E}$, how one obtains a natural isomorphism between ${\mathcal{F}}^{\prime }$ and $\mathcal{F}$. We leave the details to the reader.

We likewise leave to the reader the interpretation, in terms of pseudofunctors, of the notion of inverse image of a cloven category $\mathcal{F}$ over $\mathcal{E}$ by a base-change functor ${\mathcal{E}}^{\prime }\to \mathcal{E}$.

9. Example: Cloven Category Defined by a Functor ℰ° → Cat; Split Categories over ℰ

Suppose one has a functor

$$\varphi :{\mathcal{E}}^{\circ }\to Cat.$$

It then defines a pseudofunctor by putting

ℱ(S) = φ(S),    f* = φ(f),    c_{f,g} = id_{(fg)*}.

Thus the construction of the preceding number gives us a category $\mathcal{F}$ cloven over $\mathcal{E}$, said to be associated with the functor $\varphi$. For a cloven category over $\mathcal{E}$ to be isomorphic to a cloven category defined by a functor $\varphi :{\mathcal{E}}^{\circ }\to Cat$, it is manifestly necessary and sufficient that it satisfy the conditions

(fg)* = g*f*,    c_{f,g} = id_{(fg)*}.

In terms of the set $K$ of transport morphisms, this also simply means that the composite of two transport morphisms is a transport morphism. A cleavage of a category $\mathcal{F}$ over $\mathcal{E}$ satisfying the preceding condition is called a splitting of $\mathcal{F}$ over $\mathcal{E}$, and a category $\mathcal{F}$ over $\mathcal{E}$ endowed with a splitting is called a split category over $\mathcal{E}$. It is therefore a special case of the notion of cloven category. The category of split categories over $\mathcal{E}$ is therefore equivalent to Hom̲(ℰ°,Cat). Note that a split category over $\mathcal{E}$ is a fortiori a cloven category over $\mathcal{E}$.

If $\mathcal{F}$ is a fibered category over $\mathcal{E}$, there does not always exist a splitting on $\mathcal{F}$. Suppose for example that $Ob(\mathcal{E})$ and $Ob(\mathcal{F})$ are reduced to one element, and that the set of endomorphisms of that element is a group $E$, respectively $F$, so that the projection functor $p$ is given by a group homomorphism $p:F\to E$, surjective since $p$ is fibrant. One verifies at once that the set of cleavages of $\mathcal{F}$ over $\mathcal{E}$ is in bijective correspondence with the set of maps $s:E\to F$ such that $ps=i{d}_E$, i.e. the set of “systems of representatives” for the classes modulo the subgroup $G$ that is the kernel of the surjective homomorphism $p:F\to E$. A cleavage is a splitting if and only if $s$ is a group homomorphism. To say that a splitting exists therefore means that the group extension $F$ of $E$ by $G$ is trivial, which is expressed, when $G$ is commutative, by the vanishing of a certain cohomology class in $H^2(E,G)$, where $G$ is regarded as a group on which $E$ operates.

Suppose, however, that $\mathcal{F}$ is a fibered category over $\mathcal{E}$ such that the

${\mathcal{F}}_S$ are rigid categories, i.e. the automorphism group of every object of ${\mathcal{F}}_S$ is reduced to the identity. It is then easy to prove that $\mathcal{F}$ admits a splitting over $\mathcal{E}$. Indeed, one first observes that the question of existence of a splitting is not changed if $\mathcal{F}$ is replaced by an $\mathcal{E}$-equivalent category. This reduces us in the present case to the case where the ${\mathcal{F}}_S$ are rigid and reduced categories, i.e. two isomorphic objects in ${\mathcal{F}}_S$ are identical. But if $G$ is a rigid and reduced category, every isomorphism between two functors $H\to G$, where $H$ is any category, is an identity. It follows that if $\mathcal{F}$ is a fibered category over $\mathcal{E}$ whose fiber categories are rigid and reduced, then there exists a unique cleavage of $\mathcal{F}$ over $\mathcal{E}$, which is necessarily a splitting. Thus $\mathcal{F}$ is isomorphic to the category defined by a functor $\varphi :{\mathcal{E}}^{\circ }\to Cat$ such that the $\varphi (S)$ are rigid and discrete categories, and the functor $\varphi$ is defined up to isomorphism.

10. Co-Fibered Categories, Bi-Fibered Categories

Consider a category $\mathcal{F}$ over $\mathcal{E}$, with projection functor

$$p:\mathcal{F}\to \mathcal{E}.$$

It defines a category ${\mathcal{F}}^{\circ }$ over ${\mathcal{E}}^{\circ }$ by the projection functor

$$p^{\circ }:{\mathcal{F}}^{\circ }\to {\mathcal{E}}^{\circ }.$$

A morphism $\alpha :\eta \to \xi$ in $\mathcal{F}$ is said to be co-cartesian if it is a cartesian morphism for ${\mathcal{F}}^{\circ }$ over ${\mathcal{E}}^{\circ }$. Spelling this out, one sees that it means that for every object ${\xi }^{\prime }$ of ${\mathcal{F}}_S$, the map $u\mapsto u\circ \alpha$

$${\mathrm{Hom}}_S(\xi ,{\xi }^{\prime })\to {\mathrm{Hom}}_f(\eta ,{\xi }^{\prime })$$

is bijective. One then also says that $(\xi ,\alpha )$ is a direct image of $\eta$ by $f$, in the category $\mathcal{F}$ over $\mathcal{E}$. If it exists for every $\eta$ in ${\mathcal{F}}_T$, one says that the direct-image functor by $f$ exists; once it has been chosen, this functor is denoted

$$f{\ast }_{\mathcal{F}}or{f}_{\ast }.$$

It is therefore defined by an isomorphism of bifunctors on ${\mathcal{F}}_T^{\circ }\times {\mathcal{F}}_S$:

$${\mathrm{Hom}}_S(f_{\ast }(\eta ),\xi )\simeq {\mathrm{Hom}}_f(\eta ,\xi ).$$

Thus, if $f_{\ast }$ exists, then for $f\ast$ to exist it is necessary and sufficient that $f_{\ast }$ admit an adjoint functor, i.e. that there exist a functor $f\ast :{\mathcal{F}}_S\to {\mathcal{F}}_T$ and an isomorphism of bifunctors

$${\mathrm{Hom}}_S(f_{\ast }(\eta ),\xi )\simeq {\mathrm{Hom}}_T(\eta ,f\ast (\xi )).$$

Let $g:U\to T$ be another morphism in $\mathcal{E}$, and suppose that the inverse and direct images by $f$, $g$, and fg exist. Consider then the functorial homomorphisms

c^{f,g}: f_* g_* ← (fg)_*,
c_{f,g}: g* f* → (fg)*.

One observes that, if $f_{\ast }{g}_{\ast }$ and $g\ast f\ast$ are regarded as a pair of adjoint functors, and likewise $(fg{)}_{\ast }$ and $(fg)\ast$, then the two preceding homomorphisms are adjoint to one another. Thus one is an isomorphism if and only if the other is. In particular:

Proposition.

Suppose that the category $\mathcal{F}$ over $\mathcal{E}$ is prefibered and co-prefibered. In order that it be fibered, it is necessary and sufficient that it be co-fibered.

Of course, $\mathcal{F}$ is said to be co-prefibered, respectively co-fibered, over $\mathcal{E}$ if ${\mathcal{F}}^{\circ }$ is prefibered, respectively fibered, over $\mathcal{E}$. We shall say that $\mathcal{F}$ is bi-fibered over $\mathcal{E}$ if it is both fibered and co-fibered over $\mathcal{E}$.

11. Various Examples

a) Categories of Arrows of ℰ

Let $\mathcal{E}$ be a category. Denote by ${\Delta }^1$ the category associated with the totally ordered set with two elements [0,1]. It therefore has two objects 0 and 1, and besides the two identity morphisms one arrow (0,1) with source 0 and target 1. Let

Ar(ℰ) = Hom̲(Δ¹,ℰ).

This is called the category of arrows of $\mathcal{E}$. The object 1 of ${\Delta }^1$ defines a canonical functor, called the target functor

$$Ar(\mathcal{E})\to \mathcal{E}$$

the functor defined by the object 0 of ${\Delta }^1$ being called the source functor. For every object $S$ of $\mathcal{E}$, the fiber category $Ar(\mathcal{E}{)}_S$ is canonically isomorphic to the category ${\mathcal{E}}_{/}S$ of objects of $\mathcal{E}$ over $S$.

Consider a morphism $f:T\to S$ in $\mathcal{E}$. To it there corresponds a canonical functor

f_*: ℰ_/T = ℱ_T → ℰ_/S = ℱ_S

and a functorial isomorphism

$${\mathrm{Hom}}_S(f_{\ast }(\eta ),\xi )\simeq {\mathrm{Hom}}_f(\eta ,\xi ),$$

which therefore makes $f_{\ast }$ a direct-image functor for $f$ in $\mathcal{F}$. Moreover, here

(id_S)_* = id_{ℱ_S},    (fg)_* = f_* g_*,    c^{f,g} = id_{(fg)},

i.e. $\mathcal{F}$ is endowed with a co-splitting over $\mathcal{E}$. A fortiori, $\mathcal{F}$ is co-fibered over $\mathcal{E}$. Note now that the set of morphisms in $\mathcal{F}$ is in bijective correspondence with the set of commutative square diagrams in $\mathcal{E}$:

Y --f′--> X
| v      | u
↓        ↓
T --f--> S.

By definition, the morphism in question is cartesian if the square is cartesian in $\mathcal{E}$, i.e. if it makes $Y$ a fiber product of $X$ and $T$ over $S$. The inverse-image functor $f\ast$ therefore exists if and only if, for every object $X$ over $S$, the fiber product $X{\times }_{S}T$ exists. It follows from VI.10.1 that if the product of two objects over a third always exists in $\mathcal{E}$, i.e. if $\mathcal{F}$ is prefibered over $\mathcal{E}$, then $\mathcal{F}$ is even fibered over $\mathcal{E}$.

b) Category of Presheaves or Sheaves on Variable Spaces

Let $\mathcal{E}=Top$ be the category of topological spaces. If $T$ is a topological space, we denote by $\mathcal{U}(T)$ the category of open subsets of $T$, whose morphisms are inclusion maps. If $\mathcal{C}$ is a category, a functor $\mathcal{U}(T{)}^{\circ }\to \mathcal{C}$ is called a presheaf on $T$ with values in $\mathcal{C}$, and a sheaf if it satisfies a left-exactness condition that we shall not repeat here.

The category $\mathcal{P}(T)$ of presheaves on $T$ with values in $\mathcal{C}$ is, by definition, the category Hom̲(𝒰(T)°,𝒞), and the category $\mathcal{F}(T)$ of sheaves on $T$ with values in $\mathcal{C}$ is the full subcategory whose objects are the objects of Hom̲(𝒰(T)°,𝒞) that are sheaves. If $f:T\to S$ is a morphism in $\mathcal{E}$, i.e. a continuous map of topological spaces, then by the increasing map $U\mapsto f^{-1}(U)$ there corresponds to it a functor $\mathcal{U}(S)\to \mathcal{U}(T)$, whence a functor

f_*: Hom̲(𝒰(T)°,𝒞) → Hom̲(𝒰(S)°,𝒞)

called the direct-image functor of presheaves by $f$. One sees at once that the direct image of a sheaf is a sheaf. Thus the functor $f_{\ast }:\mathcal{P}(T)\to \mathcal{P}(S)$ induces a functor, also denoted

$$f_{\ast }:\mathcal{F}(T)\to \mathcal{F}(S).$$