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SGA 4-I — Expose III: Functoriality of Categories of Sheaves

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Expose III — Functoriality of Categories of Sheaves

By J.-L. Verdier.

In I 5, the behavior of categories of presheaves with respect to functors between the categories of arguments was studied.

In this expose, this study is extended to the case of sites and categories of sheaves (§§ 1 and 2). After introducing the induced topology (§ 3), § 4 takes up the comparison lemma, which will play an important role in Expose IV. In § 5, for the patient reader, we study certain commutative diagrams connected with localized categories of the type $C/X$.

In Theorem I 5.1, smallness hypotheses are imposed on the categories under consideration. These hypotheses are not, in general, satisfied by the sites encountered in practice, that is, by $U$-sites. This forces a few contortions (4.2, 4.3, 4.4).

1. Continuous Functors

Definition 1.1. Let $C$ and $C^{\prime }$ be two $U$-sites, and let $u:C\to C^{\prime }$ be a functor between the underlying categories. We say that $u$ is continuous if, for every sheaf of sets $F$ on $C^{\prime }$, the presheaf $X\mapsto F(u(X))$ on $C$ is a sheaf.

This notion of continuity depends a priori on the universe $U$ for which the two sites are $U$-sites. Proposition 1.5 shows that it does not depend on it.

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1.1.1. Denote by $i:\tilde{C}\to \hat{C}$ the canonical inclusion functor from sheaves into presheaves, and by $i^{\prime }:\widetilde{C^{\prime }}\to \widehat{C^{\prime }}$ the analogous functor for $C^{\prime }$. By Definition 1.1, the functor $u$ is continuous if and only if there exists a functor $u_s:\widetilde{C^{\prime }}\to \tilde{C}$ such that $i{u}_s=u^{\ast }{i}^{\prime }$, where $u^{\ast }F=F\circ u$ (I 5.0).

Proposition 1.2. Let $C$ be a small site, $C^{\prime }$ a $U$-site, and $u:C\to C^{\prime }$ a functor between the underlying categories. The following properties are equivalent:

1. The functor $u$ is continuous.
2. For every object $X$ of $C$ and every sieve $R\to X$ covering $X$, the morphism $u_{!}R\to u_{!}X$ is bicovering in $\widehat{C^{\prime }}$ (II 5.2 and I 5.1). (*)
3. For every bicovering family $H_i\to K$, $i\in I$, of $\hat{C}$, the family $u_{!}{H}_i\to u_{!}K$, $i\in I$, is bicovering in $\widehat{C^{\prime }}$.
4. There exists a functor $u^s:\tilde{C}\to \widetilde{C^{\prime }}$, commuting with inductive limits and “extending $u$”, that is, making commutative, up to isomorphism, the canonical diagram between $C$, $C^{\prime }$, and their categories of sheaves.

Moreover, when $C$ is no longer necessarily a small site, but a $U$-site, one still has $1⟺4$. The functor of 4 is necessarily a left adjoint of the functor $u_s:\widetilde{C^{\prime }}\to \tilde{C}$; it is therefore determined up to unique isomorphism.

(*) “Covering” instead of “bicovering” does not necessarily suffice; cf. Example 1.9.3 below.

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Proof. The proof of $1⟺4$ when $C$ is a $U$-site will be given in 4.2. Suppose $C$ is small. It is clear that $3\Rightarrow 2$.

$1\Rightarrow 3$. Let $H_i\to K$, $i\in I$, be a bicovering family of $\hat{C}$. For every sheaf $F$ on $C^{\prime }$, the presheaf $u^{\ast }F$ (I 5.0) is a sheaf on $C$. Therefore, by II 5.3, the canonical map

$$\mathrm{Hom}(K,u^{\ast }F)\to \prod _i\mathrm{Hom}(H_i,u^{\ast }F)$$

is bijective. By adjunction (I 5.1), one deduces that the canonical map

$$\mathrm{Hom}(u_{!}K,F)\to \prod _i\mathrm{Hom}(u_{!}{H}_i,F)$$

is bijective. Consequently, by II 5.3, the family $u_{!}{H}_i\to u_{!}K$, $i\in I$, is bicovering.

$2\Rightarrow 1$. Let $X$ be an object of $C$, let $R\to X$ be a covering sieve, and let $F$ be a sheaf on $C^{\prime }$. By II 5.3, the map

$$\mathrm{Hom}(u_{!}X,F)\to \mathrm{Hom}(u_{!}R,F)$$

is bijective. By adjunction (I 5.1), one deduces that the map

$$\mathrm{Hom}(X,u^{\ast }F)\to \mathrm{Hom}(R,u^{\ast }F)$$

is bijective. Consequently $u^{\ast }F$ is a sheaf.

$1\Rightarrow 4$. Use the usual notation $a$ and $i$, respectively $a^{\prime }$ and $i^{\prime }$, for the “associated sheaf” and “inclusion into presheaves” functors. For every sheaf $G$ on $C$, put $u^{s}G=a^{\prime }{u}_{!}iG$. For every sheaf $F$ on $C^{\prime }$, one has the sequence of natural isomorphisms

$$\begin{array}{rl}\mathrm{Hom}(G,u_{s}F)& =\mathrm{Hom}(iG,i{u}_{s}F)\\ & =\mathrm{Hom}(iG,u^{\ast }{i}^{\prime }F)\\ & =\mathrm{Hom}(u_{!}iG,i^{\prime }F)\\ & =\mathrm{Hom}(a^{\prime }{u}_{!}iG,F).\end{array}$$

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The first isomorphism comes from the fact that $i$ is fully faithful; the second comes from the definition of $u_s$; the third and fourth are obtained by adjunction. Consequently $u^s$ is left adjoint to $u_s$. In particular, $u^s$ commutes with inductive limits.

For every presheaf $K$ on $C$ and every sheaf $F$ on $C^{\prime }$, one has the sequence of natural isomorphisms

$$\begin{array}{rl}\mathrm{Hom}(u^{s}aK,F)& =\mathrm{Hom}(aK,u_{s}F)\\ & =\mathrm{Hom}(K,i{u}_{s}F)\\ & =\mathrm{Hom}(K,u^{\ast }{i}^{\prime }F)\\ & =\mathrm{Hom}(u_{!}K,i^{\prime }F)\\ & =\mathrm{Hom}(a^{\prime }{u}_{!}K,F).\end{array}$$

The third isomorphism comes from the definition of $u_s$; the others are obtained by adjunction. We thus obtain a functorial isomorphism $u^{s}aK\simeq a^{\prime }{u}_{!}K$. In particular, when $K$ is representable, taking I 5.4.3 into account, one obtains a diagram commutative up to isomorphism:

C --u--> C'
|        |
epsilon_C      epsilon_C'
v        v
C~ --u^s--> C'~

$4\Rightarrow 2$. Let $h:C\to \hat{C}$, respectively $h^{\prime }:C^{\prime }\to \widehat{C^{\prime }}$, be the functor that associates to an object the presheaf it represents. Consider the canonical diagram corresponding to the functors $u$, $u_{!}$, $u^s$, $a$, $a^{\prime }$, $h$, and $h^{\prime }$.

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The upper square is commutative (I 5.4.3), and one has $ah={\epsilon }_C$ and $a^{\prime }{h}^{\prime }={\epsilon }_{C^{\prime }}$. For every object $K$ of $\hat{C}$, one has a canonical isomorphism (I 3.4)

$${\mathrm{colim}}_{(h(X)\to K)\in \mathrm{ob}(C/K)}h(X)\to K.$$

Since the functors $u_{!}$, $u^s$, $a$, and $a^{\prime }$ commute with inductive limits, one deduces isomorphisms

$$\mathrm{colim}{a}^{\prime }{h}^{\prime }u(X)\to u^{s}aK,\mathrm{colim}{a}^{\prime }{u}_{!}h(X)\to a^{\prime }{u}_{!}K.$$

We have $u_{!}h=h^{\prime }u$ and, by hypothesis, an isomorphism $a^{\prime }{h}^{\prime }u\simeq u^{s}ah$. It follows that there is an isomorphism

$$u^{s}aK\simeq a^{\prime }{u}_{!}K,$$

functorial in $K$. The diagram is therefore commutative up to isomorphism. Since $ai$ is isomorphic to the identity functor, one has an isomorphism $u^s\simeq a^{\prime }{u}_{!}i$, whence the uniqueness of $u^s$.

Let $v:H\to K$ be a bicovering morphism of $\hat{C}$. Then $a(v)$ is an isomorphism (II 5.3), and consequently $a^{\prime }{u}_{!}(v)$, isomorphic to $u^{s}a(v)$, is an isomorphism. Thus $u_{!}(v)$ is transformed by $a^{\prime }$ into an isomorphism, and consequently $u_{!}(v)$ is bicovering (II 5.3). This proves $2$.

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The additional assertion, in the case where $C$ is small, was proved during the proof.

1.2.1. The functor $u^s:\tilde{C}\to \widetilde{C^{\prime }}$ of 1.2(iv) can often be interpreted as an “inverse image” functor for a “morphism of topoi” $\widetilde{C^{\prime }}\to \tilde{C}$; cf. IV 4.9. Its properties are summarized in the following proposition.

Proposition 1.3. Let $u:C\to C^{\prime }$ be a continuous functor between a small site $C$ and a $U$-site $C^{\prime }$.

1. The functor $u^s$ is left adjoint to the functor $u_s$.
2. There is a canonical isomorphism $a^{\prime }{u}_{!}i\simeq u^s$.
3. There is a canonical isomorphism $u^{s}a\simeq a^{\prime }{u}_{!}$.
4. The functor $u^s$ commutes with inductive limits.
5. When the functor $u_{!}$ is left exact (cf. I 5.4.4), the functor $u^s$ is left exact as well. More generally, the functor $u^s$ commutes with the types of finite projective limits with which the functor $u_{!}$ commutes.

Proof. Assertions 1, 2, 3, and 4 were proved during the proof of 1.2. Assertion 5 follows from the isomorphism 2, taking into account that $i$ commutes with projective limits and that $a^{\prime }$ commutes with finite projective limits (II 4.1).

Remark 1.4. Combining I 5.4.3 and 1.3.3, one obtains a diagram:

C --u--> C'
|        |
h        h'
v        v
C^ --u_!--> C'^
|          |
a          a'
v          v
C~ --u^s--> C'~

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where the upper square is commutative and the lower square is commutative up to isomorphism. But the functors $a$, $a^{\prime }$, $u_{!}$, and $u^s$ are defined only up to isomorphism. The reader may verify that one can choose the functors $a$, $a^{\prime }$, and $u^s$ so that:

1. the composite functors $ah$ and $a^{\prime }{h}^{\prime }$ are injective on sets of objects;
2. the diagram indicated in the scan is commutative.

More precisely, one can choose the functors $a$ and $a^{\prime }$ so as to satisfy condition 1. Once the functors $a$ and $a^{\prime }$ have been chosen, one can choose the functor $u^s$ so as to satisfy condition 2.

Proposition 1.5. Let $U\in V$ be two universes, let $C$ and $C^{\prime }$ be two $U$-sites, and let $u:C\to C^{\prime }$ be a functor between the underlying categories. Then $u$ is continuous relative to $U$ if and only if it is continuous relative to $V$.

Moreover, when $u$ is continuous, denote by $u_U^s$, respectively $u_V^s$, the functor between categories of $U$-sheaves, respectively of $V$-sheaves, introduced in 1.2(iv). Then the diagram

C~_U  --u^s_U-->  C'~_U
 |                  |
 v                  v
C~_V  --u^s_V-->  C'~_V

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where the vertical functors are the canonical inclusion functors, is commutative up to canonical isomorphism.

Proof. We give the proof only in the case where $C$ is $U$-small. The general case will be proved in 4.3. It is clear that if the functor $u^{\ast }$ transforms every $V$-sheaf on $C^{\prime }$ into a $V$-sheaf on $C$, then it transforms every $U$-sheaf on $C^{\prime }$ into a $U$-sheaf on $C$.

Suppose then that $u^{\ast }$ transforms every $U$-sheaf on $C^{\prime }$ into a $U$-sheaf on $C$. Denote by ${\hat{C}}_U$, ${\widehat{C^{\prime }}}_U$, respectively ${\hat{C}}_V$, ${\widehat{C^{\prime }}}_V$, the categories of $U$-presheaves and $V$-presheaves, and denote by $u_{!}^U$ and $u_{!}^V$ the “inverse image” functors for $U$-presheaves and $V$-presheaves respectively. One has a commutative diagram (cf. the explicit construction of $u_{!}$ in I 5.1):

C^_U   --u_!^U-->   C'^_U
 |                   |
 v                   v
C^_V   --u_!^V-->   C'^_V

The inclusion functors from $U$-presheaves into $V$-presheaves are fully faithful and commute with projective limits. It then follows from II 5.1(i) that a bicovering morphism of $U$-presheaves is a bicovering morphism of $V$-presheaves.

Let $R\to X$ be a sieve covering an object $X$ of $C$. By 1.2(ii), the morphism $u_{!}^{U}R\to u_{!}^{U}X$ is bicovering. Using the commutativity of the preceding diagram and what precedes, one deduces that the morphism $u_{!}^{V}R\to u_{!}^{V}X$ is bicovering. Consequently, by 1.2, the functor $u^{\ast }$ transforms $V$-sheaves on $C^{\prime }$ into $V$-sheaves on $C$.

The commutativity of the diagram of 1.5 then follows from the commutativity of the preceding diagram, from II 3.6, and from 1.3.2.

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Proposition 1.6. Let $C$ and $C^{\prime }$ be two $U$-sites, and let $u:C\to C^{\prime }$ be a functor between the underlying categories. Consider the following conditions:

1. $u$ is continuous (cf. 1.1 and 1.2).
2. For every covering family $(X_i\to X)$, $i\in I$, of $C$, the family $(u{X}_i\to uX)$, $i\in I$, of $C^{\prime }$, is covering.

One has the implication $1\Rightarrow 2$.

Suppose that the topology of $C$ can be defined by a pretopology $T$ (II 1.3) and that the functor $u$ commutes with fiber products. Then conditions 1 and 2 are equivalent and are equivalent to the following condition:

1. The functor $u$ transforms the covering families of $T$ into covering families of $C^{\prime }$.

In particular, suppose that, in $C$, fiber products are representable and that the functor $u$ commutes with fiber products. Then conditions 1 and 2 are equivalent.

Note: it in fact suffices that $u$ commute with the fiber products occurring in base changes for morphisms coming from covering families for $T$.

Proof. We show that $1\Rightarrow 2$. Let $X_i\to X$, $i\in I$, be a covering family of $C$. For every sheaf $F$ on $C^{\prime }$, $u^{\ast }F$ is a sheaf. Consequently the map

$$\mathrm{Hom}(X,u^{\ast }F)\to \prod _i\mathrm{Hom}(X_i,u^{\ast }F)$$

is injective (II 5.1). Hence an injective map

$$\mathrm{Hom}(uX,F)\to \prod _i\mathrm{Hom}(u{X}_i,F).$$

Thus the family $u{X}_i\to uX$, $i\in I$, is covering in $C^{\prime }$ (II 5.1).

Every covering family of the pretopology $T$ is a covering family of $C$, whence $2\Rightarrow 3$. We show that $3\Rightarrow 1$. Let $X$ be an object of $C$ and let $X_i\to X$, $i\in I$, be a covering family of $\mathrm{Cov}(X)$ (II 1.3).

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The morphisms $X_i\to X$ are squarable, and the functor $u$ commutes with fiber products. Consequently the fiber products $u(X_i){\times }_{u(X)}u(X_j)$ are representable and canonically isomorphic to $u(X_i{\times }_X{X}_j)$. Let $F$ be a sheaf on $C^{\prime }$. Then the diagram of sets

$$F(uX)\to \prod _{i}F(u{X}_i)\rightrightarrows \prod _{i,j}F(u(X_i{\times }_X{X}_j))$$

is exact (II 2.1, I 3.5 and I 2.12). Consequently the diagram of sets

$$u^{\ast }F(X)\to \prod _i{u}^{\ast }F(X_i)\rightrightarrows \prod _{i,j}{u}^{\ast }F(X_i{\times }_X{X}_j)$$

is exact. Thus $u^{\ast }F$ is a sheaf (II 2.4), whence 1.

The last assertion of 1.6 follows from what precedes and from the fact that, when fiber products are representable in $C$, all covering families of $C$ define on $C$ a pretopology whose associated topology is the topology of $C$.

Proposition 1.7. Let $C$ be a small site, $C^{\prime }$ a $U$-site, and $u:C\to C^{\prime }$ a continuous functor. Let $\gamma$ be an algebraic structure defined by finite projective limits, such that, in the category of $\gamma$-objects of $U$-Ens, $U$-inductive limits are representable. We use the notation of Proposition II 6.4.

The functor $u_s$ commutes with projective limits and consequently defines a functor $u_{s,\gamma }$ between the categories of sheaves with values in $\gamma$-objects. The functor $u_{s,\gamma }$ admits a left adjoint $u_{\gamma }^s$, which has the following properties.

Note: the inductive limits are indexed by sets belonging to a suitable universe. One can in fact show that this condition is always satisfied.

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1. There is a canonical isomorphism $u_{\gamma }^s\simeq a_{\gamma }^{\prime }{u}_{!}{i}_{\gamma }$, and $u_{\gamma }^s$ commutes with the finite projective limits with which $u_s$ commutes.

2. There is a canonical isomorphism $u_{\gamma }^s{a}_{\gamma }\simeq a_{\gamma }^{\prime }{u}_{!}$.

3. The functor $u_{\gamma }^s$ commutes with inductive limits.

4. If $u^s$ is left exact, the diagram below, in the notation of II 6.3.0, is commutative up to isomorphism and the functor $u_{\gamma }^s$ is exact.

   C~_gamma --u^s_gamma--> C'~_gamma
    |                       |
   ens                     ens'
    v                       v
   C~        --u^s-->       C'~
   

5. Suppose that the functor $\widetilde{\mathrm{ens}}$, respectively ${\widetilde{\mathrm{ens}}}^{\prime }$, has a left adjoint $\widetilde{\mathrm{Lib}}$, respectively ${\widetilde{\mathrm{Lib}}}^{\prime }$ (II 6.5). There is a canonical isomorphism

   $$u_{\gamma }^s\widetilde{\mathrm{Lib}}\simeq {\widetilde{\mathrm{Lib}}}^{\prime }{u}^s.$$

Moreover, when $C$ is not necessarily a small site but a $U$-site, the functor $u_{s,\gamma }$ admits a left adjoint $u_{\gamma }^s$. Finally, if $V$ is a universe containing $U$, and if $u_{\gamma ,U}^s$, respectively $u_{\gamma ,V}^s$, denotes the left adjoint of $u_{s,\gamma }$ relative to the universe $U$, respectively $V$, the following diagram, analogous to diagram 1.5.1,

C~_{gamma,U} --u^s_{gamma,U}--> C'~_{gamma,U}
   |                                 |
   v                                 v
C~_{gamma,V} --u^s_{gamma,V}--> C'~_{gamma,V}

is commutative up to canonical isomorphism.

Proof. The proof, in the case where $C$ is a small site, is left to the reader. It will be completed in 4.3 in the case where $C$ is a $U$-site.

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Notation 1.8. From now on we shall use the notation $u_s$ to denote the functor $u_{s,\gamma }$. This notation risks no confusion, because the functor $u_s$, defined on sheaves of sets, commutes with finite projective limits.

Similarly, when $u^s$ is left exact, we shall use the notation $u^s$ to denote the functor $u_{\gamma }^s$. This abuse of notation is justified by 1.7.4.

Example 1.9.1. Let $X$ be a small topological space. Denote by $\mathrm{Ouv}(X)$ the category of opens of $X$: its objects are the open subsets of $X$, and its morphisms are inclusions. We equip $\mathrm{Ouv}(X)$ with the canonical topology. A family $(U_i\to U)$, $i\in I$, is covering if and only if the opens $U_i$ cover $U$ (II 2.6).

Sheaves of sets on $\mathrm{Ouv}(X)$ are therefore sheaves of sets in the sense of [TF].

Let $f:X\to Y$ be a continuous map. One deduces a functor $f^{-1}:\mathrm{Ouv}(Y)\to \mathrm{Ouv}(X)$: for every open $U$ of $Y$, $f^{-1}(U)$ is the inverse image of $U$ by the map $f$. The functor $f^{-1}$ is continuous (1.6). The functor

$$f_{\ast }:\widetilde{\mathrm{Ouv}(X)}\to \widetilde{\mathrm{Ouv}(Y)}$$

is the direct image functor for sheaves, in the sense of [TF]. The functor

$$f^{\ast }:\widetilde{\mathrm{Ouv}(Y)}\to \widetilde{\mathrm{Ouv}(X)}$$

is the inverse image functor for sheaves, in the sense of [TF]. The functor $(f^{-1}{)}^s$ commutes with finite projective limits, thanks to 1.3.5.

Example 1.9.2. Let $X$ be a topological space and let $Y\to X$ be an open of $X$. Every open of $Y$ is an open of $X$, whence a functor $j:\mathrm{Ouv}(Y)\to \mathrm{Ouv}(X)$. The functor $j$ is continuous (1.6). The functor $j_s:\widetilde{\mathrm{Ouv}(X)}\to \widetilde{\mathrm{Ouv}(Y)}$ is the “restriction to $Y$” functor. The functor $\widetilde{\mathrm{Lib}}:\widetilde{\mathrm{Ouv}(Y)}\to \widetilde{\mathrm{Ouv}(X)}$ (II 6.3.3, I 1.7) is the “

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extension by zero outside $Y$“ functor introduced in [TF].

The functor $j^{\ast }:\widetilde{\mathrm{Ouv}(Y)}\to \widetilde{\mathrm{Ouv}(X)}$ (1.3) is analogous to the “extension by zero” functor: for every sheaf $H$ on $Y$ and every point $y\in Y$, the fiber at $y$ of $j^{\ast }H$ is canonically isomorphic to the fiber at $y$ of $H$; for every point $x\in X$, $x\notin Y$, the fiber at $x$ of $j^{\ast }H$ is empty. The functor $j^{\ast }$ is called the “extension by the empty object outside $Y$” functor.

The functor $j^{\ast }$ commutes with fiber products and finite products over a nonempty set of indices, but it does not transform the final object of $\widetilde{\mathrm{Ouv}(Y)}$ into the final object of $\widetilde{\mathrm{Ouv}(X)}$.

Example 1.9.3. Let $C$ and $C^{\prime }$ be two small sites, and let $u:C\to C^{\prime }$ be a functor between the underlying categories that transforms every covering family of $C$ into a covering family of $C^{\prime }$. The functor $u$ is not continuous in general (cf. however 1.6). Here is a counterexample.

Let $S$ be a set of infinite cardinality in the universe $U$, and let $C$ be the full subcategory of the category of sets whose objects are the nonempty finite subsets of $S$. We equip $C$ with the canonical topology. The covering families of $C$ are the surjective families of maps, and covering families are not empty. Moreover, up to isomorphism, there are only two constant sheaves on $C$: the sheaf with value the empty set and the sheaf with value the one-element set.

Put $C=C^{\prime }$ and let $u:C\to C^{\prime }$ be a constant functor. The functor $u$ transforms covering families of $C$ into covering families of $C^{\prime }$. But the functor $u$ is not continuous. Indeed, since the representable presheaves of $C^{\prime }$ are sheaves, for every object $X$ of $C^{\prime }$, there exists a

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sheaf $F$ on $C^{\prime }$ such that the cardinality of the set $F(X)$ is $\ge 2$, and consequently the presheaf $u^{\ast }F$ is not a sheaf.

2. Cocontinuous Functors

Definition 2.1. Let $C$ and $C^{\prime }$ be two $U$-sites, and let $u:C\to C^{\prime }$ be a functor between the underlying categories. We say that $u$ is cocontinuous if it has the following property:

(COC) For every object $Y$ of $C$ and every sieve $R\to u(Y)$ covering $u(Y)$, the sieve of $Y$ generated by the arrows $Z\to Y$ such that $u(Z)\to u(Y)$ factors through $R$, covers $Y$.

Proposition 2.2. Let $C$ be a small site, $C^{\prime }$ a $U$-site, and $u:C\to C^{\prime }$ a functor between the underlying categories. Denote by $u_{\ast }:\hat{C}\to \widehat{C^{\prime }}$ the functor right adjoint to the functor $F\mapsto F\circ u=u^{\ast }F$ (I 5.1). The functor $u$ is cocontinuous if and only if, for every sheaf $G$ on $C$, $u_{\ast }G$ is a sheaf on $C^{\prime }$.

Proof. The condition is sufficient. Suppose that, for every sheaf $G$ on $C$, the presheaf $u_{\ast }G$ is a sheaf. Let $Y$ be an object of $C$ and $R\to u(Y)$ a covering sieve. We have an isomorphism

$$\mathrm{Hom}(u(Y),u_{\ast }G)\to \mathrm{Hom}(R,u_{\ast }G).$$

By adjunction (I 5.1), one deduces an isomorphism

$$\mathrm{Hom}(u^{\ast }u(Y),G)\to \mathrm{Hom}(u^{\ast }R,G).$$

It follows (II 5.3) that the morphism $u^{\ast }R\to u^{\ast }u(Y)$ is bicovering. But the functor $u^{\ast }$ commutes with projective limits; consequently the morphism $u^{\ast }R\to u^{\ast }u(Y)$ is a monomorphism, hence a covering monomorphism. On the other hand, we have a canonical morphism $Y\to u^{\ast }u(Y)$ deduced from the adjunction morphism

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$\mathrm{id}\to u^{\ast }{u}_{\ast }$ (I 5.4.3). Make the base change $Y\to u^{\ast }u(Y)$. We obtain a sieve covering $Y$:

$$u^{\ast }R{\times }_{u^{\ast }u(Y)}Y\to Y.$$

The arrows $Z\to Y$ that factor through this sieve are exactly those of the sieve described by property (COC).

The condition is necessary. Let $S$ be an object of $C^{\prime }$ and $R\to S$ a sieve covering $S$. We must prove that, for every sheaf $G$ on $C$, one has an isomorphism

$$\mathrm{Hom}(S,u_{\ast }G)\to \mathrm{Hom}(R,u_{\ast }G).$$

Using Proposition II 5.3 and the adjunction isomorphisms, it suffices to prove that the monomorphism $u^{\ast }R\to u^{\ast }S$ is covering. For this, it suffices to prove (II 5.1) that, for every base change $Y\to u^{\ast }S$, with $Y$ an object of $C$, the sieve

$$u^{\ast }R{\times }_{u^{\ast }S}Y\to Y$$

is covering.

By the properties of adjoint functors and I 5.4.3, the morphism $Y\to u^{\ast }S$ factors through

$$Y\to u^{\ast }u(Y)\to u^{\ast }S.$$

Consequently the sieve $u^{\ast }R{\times }_{u^{\ast }S}Y\to Y$ is the transform, by the base change $Y\to u^{\ast }u(Y)$, of the monomorphism

$$u^{\ast }R{\times }_{u^{\ast }S}{u}^{\ast }u(Y)\to u^{\ast }u(Y).$$

Now the functor $u^{\ast }$ commutes with projective limits; this monomorphism is therefore the transform by $u^{\ast }$ of the sieve $R{\times }_{S}u(Y)\to u(Y)$, which is covering. Hypothesis (COC) then says that the sieve $u^{\ast }R{\times }_{u^{\ast }S}Y\to Y$ is covering.

Proposition 2.3. Let $C$ and $C^{\prime }$ be two $U$-sites, and let $u:C\to C^{\prime }$ be a cocontinuous functor. Denote by $u^{\ast }:\widetilde{C^{\prime }}\to \tilde{C}$ the functor $u^{\ast }=a{u}^{\ast }{i}^{\prime }$, where $a$ denotes the “associated sheaf” functor for $C$, where, in the right-hand member, $u^{\ast }$ denotes the functor $F\mapsto F\circ u$, and where $i^{\prime }$ denotes the

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inclusion functor $\widetilde{C^{\prime }}\to \widehat{C^{\prime }}$.

1. The functor $u^{\ast }$ commutes with inductive limits and is exact.

2. There is a canonical isomorphism $u^{\ast }{a}^{\prime }\simeq a{u}^{\ast }$, where $a^{\prime }$ denotes the “associated sheaf” functor for $\widehat{C^{\prime }}$.

3. The functor $u^{\ast }$ admits a right adjoint $u_{\ast }:\tilde{C}\to \widetilde{C^{\prime }}$. When $C$ is small, the diagram

   C~ --u_*--> C'~
   |           |
   i           i'
   v           v
   C^ --u_*--> C'^
   

   where the lower functor is right adjoint to the functor $u^{\ast }$ (I 5.1), is commutative up to canonical isomorphism.

4. Let $V\supset U$ be a universe. Denote by $u_{\ast ,U}$ (resp. $u_{\ast ,V}$) the right adjoint functor relative to the universe $U$ (resp. $V$), whose existence is asserted in 3). The diagram

   C~_U --u_{*,U}--> C'~_U
    |                  |
    v                  v
   C~_V --u_{*,V}--> C'~_V
   

   is commutative up to isomorphism.

Proof. Assertions 1) and 2) follow immediately from II 3.4. Assertion 3), when $C$ is small, follows from 2.2. When $C$ is a $U$-site, it will be proved in 4.4. Assertion 4), when $C$ is small, follows from the analogous commutativity assertion when the topologies on $C$ and $C^{\prime }$ are chaotic, and from I 3.5. When the topologies on $C$ and $C^{\prime }$ are

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chaotic, the commutativity of (2.3.2) is seen immediately from the explicit description of $u_{\ast }$ (I 5.1).

2.4. We shall not develop here the considerations concerning $\gamma$-objects of categories of sheaves. It will suffice to observe that the functors $u^{\ast }$ and $u_{\ast }$ introduced in this section always commute with finite projective limits (contrary to what happened in the preceding section for the functor $u_s$). Consequently they extend naturally to functors defined on $\gamma$-objects, which are adjoint to one another and which commute with the “underlying sheaf of sets” functors. We shall make the abuses of notation indicated in 1.8, consisting in writing $u^{\ast }$ and $u_{\ast }$ for the extensions to $\gamma$-objects.

Proposition 2.5. Let $C$ and $C^{\prime }$ be two $U$-sites, and let $v:C^{\prime }\to C$ and $u:C\to C^{\prime }$ be two functors, where $v$ is left adjoint to $u$. The following properties are equivalent:

1. The functor $u$ is continuous.
2. The functor $v$ is cocontinuous.

Moreover, under these equivalent conditions, one has canonical isomorphisms

$$v_{\ast }\simeq u_s,v^{\ast }\simeq u^s.$$

In particular, the functor $u^s$ commutes with finite projective limits.

Proof. The property, for a functor, of being continuous or cocontinuous does not depend on the universe $U$ for which $C$ and $C^{\prime }$ are $U$-sites: this is immediate in the case of a cocontinuous functor, and follows from 1.5 in the case of a continuous functor. We may therefore, after increasing the universe if necessary, suppose that $C$ and $C^{\prime }$ are small. The proposition then follows from 2.2 and I 5.6.

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Proposition 2.6. Let $u:C\to C^{\prime }$ be a continuous and cocontinuous functor between two $U$-sites. Then the functor $u^{\ast }:\widetilde{C^{\prime }}\to \tilde{C}$ (2.3) commutes with $U$-inductive and projective limits. Denote by $u_{!}:\tilde{C}\to \widetilde{C^{\prime }}$ and $u_{\ast }:\tilde{C}\to \widetilde{C^{\prime }}$ the functors adjoint to $u^{\ast }$ on the left and on the right respectively. The functor $u_{!}$ is fully faithful if and only if the functor $u_{\ast }$ is fully faithful. When $u$ is fully faithful, $u_{!}$ is fully faithful, and the converse is true when the topologies of $C$ and $C^{\prime }$ are less fine than the canonical topology.

The functor $u^{\ast }$ admits a right adjoint $u_{\ast }$ (2.3) and a left adjoint $u_{!}$ (1.2). The functor $u^{\ast }$ therefore commutes with inductive and projective limits (I 2.11). The fact that $u_{!}$ is fully faithful if and only if $u_{\ast }$ is fully faithful is a general property of adjoint functors (I 5.7.1).

When $u$ is fully faithful, the functor $u^{\ast }:{\widehat{C^{\prime }}}_V\to {\hat{C}}_V$, relative to $V$-presheaves, for a sufficiently large universe $V$, is fully faithful (I 5.7). Consequently the functor $u_{\ast }:\tilde{C}\to \widetilde{C^{\prime }}$ is fully faithful by virtue of the commutativity of (2.3.1) and (2.3.2), hence $u_{!}$ is fully faithful by what precedes. The converse is deduced from the existence of the commutative diagram 1.2 iv), taking into account the fact that the functors ${\epsilon }_C$ and ${\epsilon }_{C^{\prime }}$ are fully faithful when the topologies are less fine than the canonical topology. QED.

Example 2.7. With the notation of 1.9.2, the functor

$$\phi :\mathrm{Ouv}(Y)\to \mathrm{Ouv}(X)$$

is continuous (1.9.2) and cocontinuous (2.2). Moreover, let $i:Y\to X$

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be the canonical injection and $i^{-1}:\mathrm{Ouv}(X)\to \mathrm{Ouv}(Y)$ the functor it makes possible to define (1.9.1). The functor $i$ is right adjoint to the functor $\phi$. We therefore have a sequence of three adjoint functors:

$${\phi }^s,{\phi }_s\simeq i^{-1},{\phi }_{\ast },$$

and canonical isomorphisms

$${\phi }^{\ast }\simeq (i^{-1}{)}^s,{\phi }_{\ast }\simeq i_s^{-1}.$$

3. Induced Topology

3.1. Let $C^{\prime }$ be a site, $C$ a category, and $u:C\to C^{\prime }$ a functor. For every universe $U$ such that $C^{\prime }$ is a $U$-site and $C$ is a $U$-small category, denote by ${\mathcal{T}}_U$ the finest among the topologies $T$ on $C$ that make $u$ continuous (1.1). Such a topology exists thanks to II 2.2.

The topology ${\mathcal{T}}_U$ does not depend on the universe $U$. Indeed, if $V\supset U$ is a universe, then ${\mathcal{T}}_U={\mathcal{T}}_V$ (1.1 and 1.5). The topology ${\mathcal{T}}_U$ is called the topology induced on $C$ by the topology of $C^{\prime }$ by means of the functor $u$.¹

Proposition 3.2. Let $C$ be a small category, $C^{\prime }$ a $U$-site, $u:C\to C^{\prime }$ a functor, and $\mathcal{T}$ the topology on $C$ induced by $u$. Let $X$ be an object of $C$ and $R\to X$ a sieve of $X$. The following properties are equivalent:

1. The sieve $R\to X$ is covering for $\mathcal{T}$.
2. For every base change $Y\to X$, where $Y$ is an object of $C$, the morphism $u_{!}(R{\times }_{X}Y)\to u(Y)$ is bicovering in $\widehat{C^{\prime }}$ (II 5.2).

Proof. $i)\Rightarrow ii)$ follows from axiom (T1) for topologies and from 1.2.

$ii)\Rightarrow i)$: For every sheaf $F$ on $C^{\prime }$, the map

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$$\mathrm{Hom}(Y,u^{\ast }F)\to \mathrm{Hom}(R{\times }_{X}Y,u^{\ast }F)$$

is isomorphic, by adjunction (I 5.1), to the map

$$\mathrm{Hom}(u(Y),F)\to \mathrm{Hom}(u_{!}(R{\times }_{X}Y),F),$$

which is bijective (II 5.3). Consequently (II 2.2), the sieve $R\to X$ is covering for $\mathcal{T}$.

Corollary 3.3. Let $C^{\prime }$ be a site, $C$ a category, $u:C\to C^{\prime }$ a functor, and $\mathcal{T}$ the topology on $C$ induced by the topology of $C^{\prime }$. Let $(X_i\to X{)}_{i\in I}$ be a family of squarable morphisms of $C$, and suppose that $u$ commutes with fiber products.² The following properties are equivalent:

1. The family $X_i\to X$, $i\in I$, is covering for $\mathcal{T}$.
2. The family $(u(X_i)\to u(X))$, $i\in I$, is covering.

Proof. $i)\Rightarrow ii)$ follows from 1.6.

$ii)\Rightarrow i)$: Let $U$ be a universe such that $C$ is $U$-small and $C^{\prime }$ is a $U$-site. Let $R\to X$ be the sieve generated by the $X_i\to X$. The presheaf $R$ is the cokernel of the pair of arrows (I 2.12 and I 3.5)

$$\coprod _{i,j}{X}_i{\times }_X{X}_j\rightrightarrows \coprod _i{X}_i$$

where the direct sum is taken here in $\hat{C}$. Since the functor $u_{!}$ commutes with inductive limits (I 5.4), the presheaf $u_{!}R$ is the cokernel of the pair of arrows

$$\coprod _{i,j}u(X_i{\times }_X{X}_j)\rightrightarrows \coprod _{i}u(X_i).$$

Since the functor $u$ commutes with fiber products, the presheaf $u_{!}R$ is the cokernel of the pair of arrows

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$$\coprod _{i,j}u(X_i){\times }_{u(X)}u(X_j)\rightrightarrows \coprod _{i}u(X_i),$$

and consequently $u_{!}R\to u(X)$ is a sieve of $u(X)$ generated by the $u(X_i)\to u(X)$, $i\in I$. Again using the fact that $u$ commutes with fiber products, one shows that, for every base change $Y\to X$, where $Y$ is an object of $C$, $u_{!}(R{\times }_{X}Y)\to u(Y)$ is a sieve of $u(Y)$ generated by the $u(X_i){\times }_{u(X)}u(Y)\to u(Y)$, $i\in I$. Since the family $u(X_i)\to u(X)$, $i\in I$, is covering, the family $u(X_i){\times }_{u(X)}u(Y)\to u(Y)$, $i\in I$, is covering. Thus $u_{!}(R{\times }_{X}Y)\to u(Y)$ is a covering sieve, and consequently (3.2) $R\to X$ is covering for $\mathcal{T}$. QED.

Corollary 3.4. Let $C^{\prime }$ be a $U$-site, let $C$ be a full subcategory of $C^{\prime }$, and let $u:C\to C^{\prime }$ be the inclusion functor. Suppose that fiber products are representable in $C$ and that $u$ commutes with fiber products. The following conditions are equivalent:

1. The following two conditions hold:

   a. For every object $X$ of $C$, every covering family $(Y_j\to X)$, $j\in J$, of $C^{\prime }$ is dominated by a covering family $X_i\to X$, $i\in I$, where the $X_i$ are objects of $C$.

   b. There exists a small set $G$ of objects of $C$ such that every object of $C$ is the target of a covering family in $C^{\prime }$ of morphisms whose sources lie in $G$.

2. The topology induced by the topology of $C^{\prime }$ is a $U$-topology (I 3.0.2), and the functor $u$ is continuous and cocontinuous for this topology.

Proof. This follows immediately from 3.3 and 2.1.

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Proposition 3.5. Let $C$ be a $U$-site and let ${\epsilon }_C:C\to \tilde{C}$ be the canonical functor (II 4.4.0). Equip $\tilde{C}$ with the canonical topology. Then the topology of the site $C$ is the topology induced by the topology of $\tilde{C}$.

Proof. The covering families of $\tilde{C}$ for the canonical topology are the universally effective epimorphic families (II 2.5), that is (II 4.3), the epimorphic families. Let $T$ be the topology of the site $C$ and ${\mathcal{T}}_U$ the finest topology among the topologies $T^{\prime }$ on $C$ such that, for every sheaf $F$ on $\tilde{C}$, $F\circ {\epsilon }_C$ is a sheaf for $T^{\prime }$. It suffices to show that $T={\mathcal{T}}_U$, because then ${\mathcal{T}}_U$ is a $U$-topology and consequently ${\mathcal{T}}_U$ is the induced topology.

A. The topology $T$ is finer than ${\mathcal{T}}_U$. Let $(X_i\to X{)}_{i\in I}$ be a covering family of ${\mathcal{T}}_U$. Then, for every sheaf $F$ on $\tilde{C}$,

$$F({\epsilon }_C(X))\to \prod _{i}F({\epsilon }_C(X_i))$$

is injective. Consequently (II 5.2), $({\epsilon }_C(X_i)\to {\epsilon }_C(X){)}_{i\in I}$ is covering for the canonical topology of $\tilde{C}$, hence (II 5.2) $(X_i\to X{)}_{i\in I}$ is covering for $T$.

B. The topology $T$ is less fine than ${\mathcal{T}}_U$. It suffices to show that ${\epsilon }_C:C\to \tilde{C}$ is continuous (1.1). But it will be proved in IV no. 1 that every sheaf on $\tilde{C}$ is representable. Consequently, for every sheaf $F$ on $\tilde{C}$, $F\circ {\epsilon }_C$ is a sheaf on $C$. We shall not use 3.5 until IV 1.

Let us note two results that will be used in VI 7.

Proposition 3.6. Let $(C_i{)}_{i\in I}$ be a family of sites, $C$ a category, and, for every $i\in I$, a functor $u_i:C_i\to C$. Let $U$ be a universe such that the categories $C_i$ and $C$ are $U$-small. There exists a topology ${\mathcal{T}}_U$ on $C$ which is the least fine of the topologies for which the $u_i$ are continuous. The topology ${\mathcal{T}}_U$ does not depend on the universe $U$ for which the categories considered are small.

The last assertion of 3.6 follows from 1.5. Let $T$ be a topology on $C$. Then the functors $u_i$ are continuous if and only if, for every $i\in I$ and every sieve covering $R\to X$ of an object $X$ of

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$C_i$, the morphism

$$u_{i!}(R)\to u_i(X)$$

of $\hat{C}$ is bicovering (1.2 ii). Thus 3.6 is a consequence of Lemma 3.6.1.

Lemma 3.6.1. Let $C$ be a small category and let $(u_i:F_i\to G_i{)}_{i\in I}$ be a family of arrows of $\hat{C}$. Then there exists on $C$ a topology least fine among those that make the morphisms $u_i$ covering, respectively bicovering (II 5.2).

To say that $u:F\to G$ is covering for a given topology $T$ means that, for every arrow $X\to G$, with $X\in \mathrm{Ob}C$, the corresponding arrow $F{\times }_{G}X\to X$ is covering, or equivalently that the family of arrows $X^{\prime }\to X$ of $C$ that factor through the preceding arrow is covering. The fact that there exists a least fine topology among those for which the $u_i:F_i\to G_i$ are covering therefore follows from I 1.1.6; whence the covering assertion of 3.6.1.

The bicovering assertion is deduced from it by remembering that a morphism $u:F\to G$ is bicovering if and only if the morphisms $u:F\to G$ and ${\mathrm{diag}}_u:F\to F{\times }_{G}F$ are covering.

Proposition 3.7. Let $(C_i{)}_{i\in I}$ be a family of sites, $C$ a category, and, for every $i\in I$, a functor $u_i:C_i\to C$. Let $U$ be a universe such that the categories $C_i$ and $C$ are $U$-small. There exists a topology ${\mathcal{T}}_U$ which is the finest for which the $u_i$ are cocontinuous. The topology ${\mathcal{T}}_U$ does not depend on the universe $U$ for which the categories considered are small.

Let $U$ be a universe for which the categories considered are small. The functors $u_i$ are cocontinuous for a topology $\mathcal{T}$ of $C$ if and only if, for every $i\in I$ and every sheaf $F$ on $C_i$, the presheaf $u_{i\ast }F$ is a sheaf for $\mathcal{T}$ (2.2). It follows that the topology ${\mathcal{T}}_U$ is the finest topology for which the presheaves $u_{i\ast }F$, $i\in I$, $F\in \mathrm{Ob}\widetilde{C_i}$, are sheaves (II 2.2). The last assertion follows from 2.2.

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4. Comparison Lemma

Theorem 4.1 (comparison lemma). Let $C$ be a small category, let $C^{\prime }$ be a site whose underlying category is a $U$-category, and let $u:C\to C^{\prime }$ be a fully faithful functor. Equip $C$ with the topology induced by $u$ (3.1). Consider the properties:

1. Every object of $C^{\prime }$ can be covered by objects coming from $C$.
2. The functor $F\mapsto F\circ u$ induces an equivalence of categories from the category of sheaves on $C^{\prime }$ to the category of sheaves on $C$.

One always has $i)\Rightarrow ii)$. When $C^{\prime }$ is a $U$-site and when the topology of $C^{\prime }$ is less fine than the canonical topology, one has $ii)\Rightarrow i)$.

4.1.1. We first prove $i)\Rightarrow ii)$. The proof proceeds in two steps.

First step. For every presheaf $H$ of $\widehat{C^{\prime }}$, the adjunction morphism

$$\phi :u_{!}{u}^{\ast }H\to H$$

(I 5.1) is bicovering (II 5.3), and the functor $u_s:\widetilde{C^{\prime }}\to \tilde{C}$ (1.1.1) is fully faithful.

Let $C/H$ be the small category whose objects are the objects $Y$ of $C$ equipped with a morphism $u(Y)\to H$, and whose morphisms are the commutative diagrams

u(Y) --u(m)--> u(Y')
  \             /
   \           /
        H

We have

$$u^{\ast }H={\mathrm{colim}}_{Y\in \mathrm{Ob}(C/H)}Y$$

(I 3.4), and consequently (I 5.4)

$$u_{!}{u}^{\ast }H={\mathrm{colim}}_{Y\in \mathrm{Ob}(C/H)}u(Y).$$

The adjunction morphism is the evident morphism resulting from the description of $u_{!}{u}^{\ast }H$ as an inductive limit. The morphism $\phi$ has the

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following property:

(*) For every object $Y$ of $C$, every morphism $m:u(Y)\to H$ factors uniquely as the canonical morphism $\alpha (m):u(Y)\to u_{!}{u}^{\ast }H$ followed by the morphism $\phi$.

It follows immediately from i) that the morphism $\phi$ is covering (II 5.1). We show that it is bicovering. Let $p,q:Z\to u_{!}{u}^{\ast }H$ be two morphisms from an object $Z$ of $C^{\prime }$ such that $\phi p=\phi q$. For every object $Y$ of $C$ and every morphism $n:u(Y)\to Z$, we have $\phi pn=\phi qn$. Property (*) then implies that $pn=qn$, and since the $u(Y)$ cover $Z$, the kernel of $(p,q)$ is a sieve covering $Z$. The morphism $\phi$ is therefore bicovering (II 5.3).

For every sheaf $H$ on $C^{\prime }$, $u^{\ast }H$ is a sheaf on $C$, denoted $u_{s}H$ (1.1.1). Moreover $u^s{u}_{s}H=a^{\prime }{u}_{!}{u}_{s}H$ (1.3), and the adjunction morphism $u^s{u}_{s}H\to H$ is obtained by applying the “associated sheaf” functor to the morphism $\phi :u_{!}{u}^{\ast }H\to H$. Consequently (II 5.3), the adjunction morphism $u^s{u}_{s}H\to H$ is an isomorphism. Thus $u_s:\widetilde{C^{\prime }}\to \tilde{C}$ is fully faithful.

Second step. The functor $u$ is cocontinuous and the functor $u^s:\tilde{C}\to \widetilde{C^{\prime }}$ (1.2) is fully faithful. Consequently $u_s:\widetilde{C^{\prime }}\to \tilde{C}$ is an equivalence.

Let $Y$ be an object of $C$ and $i:R\to u(Y)$ a covering sieve. Since the functor $u^{\ast }$ commutes with projective limits (I 5.5), the morphism $u^{\ast }(i):u^{\ast }(R)\to u^{\ast }u(Y)$ is a monomorphism. Since $u$ is fully faithful, we have $u^{\ast }u(Y)\simeq Y$, whence a sieve of $Y$, $u^{\ast }(i):u^{\ast }(R)\to Y$. To show that $u$ is cocontinuous, it suffices to show that $u^{\ast }(i):u^{\ast }(R)\to Y$ is a sieve covering for the induced topology on $C$ (2.1). For this, it suffices to show (3.2) that, for every base change $m:X\to Y$, the morphism

$$u_{!}(u^{\ast }(R){\times }_{Y}X)\to u(X)$$

is bicovering. But since $u^{\ast }$ commutes

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with projective limits, we have

$$u^{\ast }(R){\times }_{Y}X=u^{\ast }(R{\times }_{u(Y)}u(X)),$$

and $R{\times }_{u(Y)}u(X)$ is a sieve covering $u(X)$. It therefore suffices to show that, for every object $Y$ of $C$ and every covering sieve $i:R\to u(Y)$, the morphism of $\widehat{C^{\prime }}$

$$u_{!}{u}^{\ast }(R)\to u(Y)$$

is bicovering. But this morphism factors as the adjunction morphism $u_{!}{u}^{\ast }(R)\to R$, which is bicovering by the first step, followed by the monomorphism $R\to u(Y)$, which is covering. It is therefore bicovering (II 5.3). This shows that $u$ is cocontinuous.

Since $u$ is continuous and fully faithful, it follows from 2.6, used in the case where $C$ is small, that $u^s$ (denoted $u_{!}$ in 2.6) is fully faithful. Since $u^s$ and $u_s$ are adjoint to one another and are fully faithful, they are quasi-inverse functors, and consequently $u_s$ is an equivalence.

4.1.2. We now prove $ii)\Rightarrow i)$. The objects $Y$ of $C$ form a generating family of $\tilde{C}$ (II 4.10). Consequently, for every object $X$ of $C^{\prime }$, there exists an epimorphic family in $\tilde{C}$, $v_i:Y_i\to u_{s}X$, $i\in I$. We have $u_{s}u(Y_i)=Y_i$, and, since the functor $u_s$ is an equivalence of categories, we have $v_i=u_s(w_i)$, where the $w_i:u(Y_i)\to X$ form an epimorphic family of $\widetilde{C^{\prime }}$. It then follows from II 5.1 that the family $w_i:u(Y_i)\to X$, $i\in I$, is covering.

4.2. End of the proof of 1.2. Let $G$ be a full subcategory of $C$ whose objects form a small topologically generating family of $C$ (II 3.0.1). Equip $G$ with the topology induced by the inclusion functor $i:G\to C$. The functor $(u\circ i{)}_s=i_s\circ u_s$ (1.1.1) admits a left adjoint by the first part of the proof of 1.2. Since $i_s$ is an equivalence

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of categories (4.1), the functor $u_s$ admits a left adjoint $u^s$, and one has a functorial isomorphism

$$u^s\simeq (u\circ i{)}^s\circ i_s.$$

We show that the diagram

C  --u-->   C'
|          |
epsilon_C        epsilon_C'
v          v
C~ --u^s-> C'~

is commutative up to isomorphism. For every sheaf $H$ on $C^{\prime }$, one has

$${\mathrm{Hom}}_{\widetilde{C^{\prime }}}({\epsilon }_{C^{\prime }}uX,H)\simeq u_{s}H(X)$$

for every $X\in \mathrm{Ob}C$. Moreover one has

$${\mathrm{Hom}}_{\widetilde{C^{\prime }}}(u^s{\epsilon }_{C}X,H)\simeq {\mathrm{Hom}}_{\tilde{C}}({\epsilon }_{C}X,u_{s}H)$$

by adjunction, and then

$${\mathrm{Hom}}_{\tilde{C}}({\epsilon }_{C}X,u_{s}H)\simeq u_{s}H(X)$$

by definition of ${\epsilon }_C$. Hence an isomorphism

$${\epsilon }_{C^{\prime }}uX\simeq u^s{\epsilon }_{C}X$$

for every $X\in \mathrm{Ob}C$. This proves $i)\Rightarrow iv)$.

We show that $iv)\Rightarrow i)$. We have a commutative diagram

G --i--> C --u--> C'
|        |        |
epsilon_G      epsilon_C      epsilon_C'
v        v        v
G~--i^s->C~--u^s->C'~

and consequently, by the first part of the proof, the functor $u\circ i:G\to C^{\prime }$ is continuous. It follows at once from 4.1 and 1.1 that $u$ is continuous, whence $iv)\Rightarrow i)$. One moreover obtains the uniqueness of $u^s$, knowing, by the first part of the proof, the uniqueness when $C$ is small.

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4.3. End of the proof of 1.5 and 1.7. Let $u:C\to C^{\prime }$ be a functor between two $U$-sites and let $G$ be a full subcategory of $C$ whose set of objects is a small topologically generating family of $C$ (II 3.0.1). Equip $G$ with the topology induced by the inclusion functor $i:G\to C$. It follows immediately from 4.1 and 1.1 that $u$ is continuous if and only if $u\circ i$ is continuous. From this remark and from the first part of the proof of 1.5, the general case follows. This remark also makes it possible to reduce the proof of 1.7 to the case where the category $C$ is small.

4.4. End of the proof of 2.3. Let $u:C\to C^{\prime }$ be a functor between two $U$-sites, and let $G$ be a full subcategory of $C$ whose set of objects is a small topologically generating family of $C$ (II 3.0.1), equipped with the topology induced by the inclusion functor $i:G\to C$. It follows from the proof of 4.1 (4.1.1, second step) that $i$ is cocontinuous.

Consequently, when $u$ is cocontinuous, the functor $u\circ i:G\to C^{\prime }$ is cocontinuous. Moreover, $i^{\ast }:\tilde{C}\to \tilde{G}$ is an equivalence of categories (4.1). Thus the functor $u^{\ast }:\widetilde{C^{\prime }}\to \tilde{C}$ admits a right adjoint. This proves 3). To prove 4), it suffices to observe that $(u\circ i{)}_{\ast }\simeq u_{\ast }\circ i_{\ast }$ and that $i_{\ast }$ is an equivalence (4.1). The commutativity of (2.3.2) then follows from the commutativity of these diagrams when $C$ is small.

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5. Localization

5.1. Let now $C$ be a $U$-site and let $X$ be an object of $\hat{C}$. Unless expressly stated otherwise, the category $C/X$ will be equipped with the topology $\mathcal{T}$ induced by the functor $j_X:C/X\to C$ (3.1). The notation $C/X$ will denote the category $C/X$ equipped with the topology $\mathcal{T}$.

Proposition I 5.11 shows that the functor $j_X$ commutes with fiber products and consequently transforms every monomorphism into a monomorphism. In particular, for every object $(Y\to X)$ of $C/X$, the functor $j_X$ establishes a one-to-one correspondence between the sieves, in the category $C/X$, of the object $(Y\to X)$ and the sieves, in $C$, of the object $Y$.

Proposition 5.2. Let $C$ be a $U$-site, let $X$ be an object of $\hat{C}$, and let $j_X:C/X\to C$ be the corresponding continuous functor.

1. A sieve $R$ of an object $(Z\to X)$ is covering in $C/X$ if and only if the sieve $j_X(R)\to Z$ is covering in $C$.

2. The functor $j_X:C/X\to C$ is cocontinuous and continuous.

3. Let $m:Y\to X$ be a morphism of $\hat{C}$. We then have the commutative diagram

   C/Y --j_m--> C/X
    \           /
   j_Y        j_X
      \       /
          C
   

   The topology induced by the functor $j_Y$ on $C/Y$ is equal to the topology induced by $j_m$ on $C/Y$.

4. The topology induced by $j_X:C/X\to C$ is a $U$-topology (II 3.0.2).

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Proof.

1. If the sieve $R$ of $(Z\to X)$ is covering, the sieve $j_X(R)\to Z$ is covering (1.6). Conversely, if the sieve $j_X(R)\to Z$ is covering in $C$, one sees that the same is true for every sieve obtained by making a base change in $C/X$. The sieve $R$ is therefore covering (3.2).
2. This is deduced immediately from 1) by applying 2.1.
3. This is deduced immediately from the description of covering sieves given by 1).
4. Let $(G_i{)}_{i\in I}$ be a small topologically generating family of $C$. One verifies immediately that the small family $(u:G_i\to X)$, where $u\in {\bigcup }_{i\in I}{\mathrm{Hom}}_{\hat{C}}(G_i,X)$, is topologically generating in $C/X$.

Terminology and Notation 5.3. By the preceding proposition, the functor $j_X$ is both continuous and cocontinuous. It therefore defines a sequence of three adjoint functors (4.3.2) between the categories of sheaves of sets (1.3 and 2.3):

$$j_X^s:\widetilde{(C/X)}\to \tilde{C}$$

$$j_X^{\ast }=j_{X,s}:\tilde{C}\to \widetilde{(C/X)}$$

$$j_{X\ast }:\widetilde{(C/X)}\to \tilde{C}$$

In the particular situation of Proposition 5.2, we shall use the following terminology and notation:

1. The functor $j_{X\ast }$ will be called the direct image functor.
2. The functor $j_X^{\ast }=j_{X,s}$ will be denoted $j_X^{\ast }$ and will be called the restriction functor to $C/X$.

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3. The functor $j_X^s$ on sheaves of sets will be called the extension by the empty object functor to the category $\tilde{C}$, and will be denoted $j_{X!}$ (cf. 2.9.2).

We therefore have a sequence of three adjoint functors between $\widetilde{(C/X)}$ and $\tilde{C}$:

$$j_{X!},j_X^{\ast },j_{X\ast }.$$

Proposition 5.4. The functor $j_{X!}:\widetilde{(C/X)}\to \tilde{C}$ factors through the category $\tilde{C}/aX$, where $a$ is the “associated sheaf” functor:

(C/X)~ --e_X~--> C~/aX --> C~

The functor

$$\widetilde{e_X}:\widetilde{(C/X)}\to \tilde{C}/aX$$

is an equivalence of categories. The restriction functor to $C/X$, composed with the equivalence $\widetilde{e_X}$, is isomorphic to the “base change by $aX\to e$” functor, where $e$ is the final object of $\tilde{C}$:

$$F\mapsto (F\times aX\to aX).$$

Proof. The image by $j_{X!}$ of the final object of $\widetilde{(C/X)}$ is the object $aX$; hence the factorization. To show that the functor $\widetilde{e_X}$ is an equivalence, we shall confine ourselves to a few indications. By I 5.11, a presheaf on $C/X$ is defined by a presheaf $F$ on $C$ equipped with a morphism $F\to X$. One then proves that the following two properties are equivalent:

1. The presheaf on $C/X$ defined by $F\to X$ is a sheaf.

2. The following diagram is cartesian, where $i$ denotes the injection into presheaves and $a$ the associated sheaf functor:

   F -> iaF
   |    |
   v    v
   X -> iaX
   

[PDF p. 311]

One then immediately deduces that $\widetilde{e_X}$ is an equivalence. The last assertion is trivial.

Proposition 5.5.

1. Let $C$ be a $U$-site and let $X$ be an object of $\hat{C}$. The diagram below of categories and functors is commutative up to canonical isomorphisms:

   Diagram (5.5.1) of the scan: comparison between the categories of
   presheaves/sheaves on C/X and the categories of arrows over X,
   aX, and iaX. The horizontal arrows are h_X, a_X, i_X above,
   then h/X, a/X, i/aX, and pi below; the vertical arrows include
   e_X~ and e~/X.
   

   The notation $a_X$ and $i_X$ denotes the “associated sheaf” and “injection into presheaves” functors for the site $C/X$. The notation $a/X$ denotes the natural extension of the functor $a$, associated sheaf for the site $C$, to the category of arrows with target $X$; similarly for the notation $i/aX$. Finally, the notation $\pi$ denotes the base-change functor by the canonical arrow $X\to iaX$.

2. Let moreover $m:Y\to X$ be a morphism of $\hat{C}$. The diagram below is commutative up to canonical isomorphism:

   Diagram (5.5.2) of the scan: comparison between C~/aX/aY,
   (C/X)~/a_XY, (C/X/Y)~, C~/aY, and (C/Y)~, with the arrows
   f, g, e_Y~, and e_m~.
   

[PDF p. 312]

The arrow $f$ is the natural extension of the equivalence $\tilde{e}/X$ to the category of arrows with target $a_{X}Y$, and is therefore an equivalence of categories. The arrow $g$ is none other than the equivalence of 5.4 applied to the situation $(C/X)/[m]$.