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SGA 4-I — Expose II: Topologies and Sheaves

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Expose II — Topologies and Sheaves

By J.-L. Verdier.

After the definition of topologies and pretopologies (§1) and of sheaves of sets (§2), §3 takes up the central theorem of this expose: the existence of the sheaf associated to a presheaf (3.4). In order to cover all cases encountered in practice, the existence of this functor is proved for presheaves on a $U$-site (3.0.2).

Sections 4 and 5 draw the consequences of this theorem for the behavior of inductive and projective limits in categories of sheaves. In §6, one defines and studies sheaves with values in arbitrary categories, immediately turning attention to sheaves of rings, groups, modules, and so on.

1. Topologies. Covering Families. Pretopologies

Definition 1.1. A topology on a category $C$ is the datum, for every object $X$ of $C$, of a set $J(X)$ of sieves of $X$, subject to the following axioms:

• (T1) Stability under base change. For every object $X$ of $C$, every sieve $R\in J(X)$, and every morphism $f:Y\to X$, with $Y\in \mathrm{ob}(C)$, the sieve $R{\times }_{X}Y$ of $Y$ belongs to $J(Y)$.
• (T2) Local character. If $R$ and $R^{\prime }$ are two sieves of $X$, if $R\in J(X)$, and if, for every $Y\in \mathrm{ob}(C)$ and every morphism $Y\to R$, the sieve $R^{\prime }{\times }_{X}Y$ of $Y$ belongs to $J(Y)$, then $R^{\prime }$ belongs to $J(X)$.

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• (T3) For every object $X$ of $C$, $X$ belongs to $J(X)$.

1.1.1. The sieves belonging to $J(X)$ will be called the sieves covering $X$, or also the refinements of $X$. From axioms (T1), (T2), and (T3), one immediately deduces that the set of sieves covering $X$ is stable under finite intersections, and that every sieve containing a covering sieve is a covering sieve. The set $J(X)$, ordered by inclusion, is therefore cofiltered (I 8).

1.1.2. Let $C$ be a category, and let $T$ and $T^{\prime }$ be two topologies on $C$. The topology $T$ is said to be finer than the topology $T^{\prime }$ if, for every object $X$ of $C$, every refinement of $X$ for the topology $T^{\prime }$ is a refinement of $X$ for the topology $T$. In this way one defines an order structure on the set of topologies.

1.1.3. Let $(T_i)$ be a family of topologies on $C$. For every object $X$ of $C$, the sieves of $X$ that are covering for all the topologies $T_i$ verify axioms (T1), (T2), and (T3), and therefore define a topology: the intersection topology of the $T_i$, that is, the lower bound of the $T_i$. It is the finest of the topologies that is less fine than all the topologies $T_i$. Consequently the family $(T_i)$ admits an upper bound: the intersection topology of the topologies finer than each of the $T_i$.

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1.1.4. In particular, the datum $J(X)$ equal to the set of all sieves of $X$ is a topology finer than every topology on $C$; we shall call it the discrete topology.

There exists a topology less fine than every topology on $C$: the topology for which $J(X)=\{X\}$ for every $X$ of $C$. This topology is called the coarse or chaotic topology.

1.1.5. A category $C$ equipped with a topology is called a site. The category $C$ is called the category underlying the site.

Definition 1.2. Let $C$ be a site and let $X$ be an object of $C$. A family of morphisms $(f_i:X_i\to X{)}_{i\in I}$ is said to be covering if the sieve generated by the family $f_i$ (I 4.3.3) is a sieve covering $X$.

1.1.6. Let $C$ be a category. Suppose we are given, for each object $X$ of $C$, a set of families of morphisms with target $X$. Then there exists a topology $T$ that is the least fine of the topologies for which the given families are covering, namely the intersection (1.1.3) of all the topologies in question. This topology is called the topology generated by the given sets of families of morphisms. It is difficult, in general, to describe all covering sieves of this topology. However the situation is more pleasant in the following case.

Definition 1.3. Let $C$ be a category. A pretopology on $C$ is the datum, for each object $X$ of $C$, of a set $\mathrm{Cov}(X)$ of families of morphisms with target $X$, subject to the following axioms:

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• (PT0) For every object $X$ of $C$, the morphisms in the families of morphisms of $\mathrm{Cov}(X)$ are squarable. Recall that a morphism $Y\to X$ is said to be squarable if, for every morphism $Z\to X$, the fiber product $Z{\times }_{X}Y$ is representable.
• (PT1) Stability under base change. For every object $X$ of $C$, every family $(X_i\to X{)}_{i\in I}$ belonging to $\mathrm{Cov}(X)$, and every morphism $Y\to X$, with $Y\in \mathrm{ob}(C)$, the family $(X_i{\times }_{X}Y\to Y{)}_{i\in I}$ belongs to $\mathrm{Cov}(Y)$.
• (PT2) Stability under composition. If $(X_i\to X{)}_{i\in I}$ belongs to $\mathrm{Cov}(X)$, and if for each $i\in I$ a family $(X_{ij}\to X_i{)}_{j\in J_i}$ belongs to $\mathrm{Cov}(X_i)$, then the composite family $(X_{ij}\to X)$ belongs to $\mathrm{Cov}(X)$.
• (PT3) The family reduced to the identity morphism $X\to X$ belongs to $\mathrm{Cov}(X)$.

1.3.1. Definition 1.1.6 allows us, for a given pretopology on $C$, to consider the topology on $C$ generated by this pretopology. Note that if $C$ is a category in which fiber products are representable, then every topology $T$ on $C$ can be defined by a pretopology, namely the one for which $\mathrm{Cov}(X)$ is formed by all families covering $X$ for the topology $T$.

Proposition 1.4. Let $C$ be a category, $E$ a pretopology on $C$, $T$ the topology defined by the pretopology $E$ (1.1.6), and $X$ an object of $C$. Denote by $J_E(X)$ the set of sieves of $X$ generated by families of the pretopology, and by $J_T(X)$ the set of refinements of $X$ for

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the topology $T$. Then $J_E(X)$ is cofinal in $J_T(X)$. In other words, for a sieve $R$ of $X$ to belong to $J_T(X)$, it is necessary and sufficient that there exist a sieve $R^{\prime }\in J_E(X)$ such that $R^{\prime }\subset R$.

Proof. Let $J^{\prime }(X)$, for every object $X$ of $C$, be the set of sieves of $X$ that contain a sieve of $J_E(X)$. Evidently $J^{\prime }(X)\subset J_T(X)$. To show that $J^{\prime }(X)=J_T(X)$, it suffices to show that the datum of the $J^{\prime }(X)$ defines a topology on $C$. Now the $J^{\prime }(X)$ evidently verify axioms (T1) and (T3). It remains to verify (T2).

For this it suffices to prove that, if $R^{\prime }$ is a subsieve of $R\in J_E(X)$ such that for every $Y\to R$ the sieve $R^{\prime }{\times }_{X}Y$ of $Y$ belongs to $J^{\prime }(Y)$, then the sieve $R^{\prime }$ belongs to $J^{\prime }(X)$. But the sieve $R$ is generated by a family $(X_i\to X)$ belonging to $\mathrm{Cov}(X)$, and for every $i$ the sieve $R^{\prime }{\times }_X{X}_i$ contains a sieve generated by a family $(X_{ij}\to X_i)$ belonging to $\mathrm{Cov}(X_i)$. We deduce that the sieve $R^{\prime }$ contains a sieve generated by the composite family $(X_{ij}\to X)$. Therefore, by axiom (PT2), $R^{\prime }$ contains a sieve of $J_E(X)$, and consequently belongs to $J^{\prime }(X)$. QED.

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2. Sheaves of Sets

Definition 2.1. Let $C$ be a site whose underlying category is a $U$-category. A presheaf $F$ with values in $U$-Ens is said to be separated (respectively is a sheaf) if for every sieve $R$ covering $X$, an object of $C$, the map

$${\mathrm{Hom}}_{\hat{C}}(X,F)\to {\mathrm{Hom}}_{\hat{C}}(R,F)$$

is injective (respectively bijective). The full subcategory of $\hat{C}$

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whose objects are the sheaves is called the category of sheaves of sets on $C$, and is most often denoted $\tilde{C}$ (1). When no ambiguity results, we shall simply say category of sheaves on $C$; by contrast, we shall sometimes be more precise and say category of sheaves with values in $U$-Ens, or category of $U$-sheaves.

Proposition 2.2. Let $C$ be a $U$-category, and let $\mathcal{F}=(F_i{)}_{i\in I}$ be a family of $U$-presheaves on $C$. For each object $X$ of $C$, denote by $J_{\mathcal{F}}(X)$ the set of sieves $R\to X$ such that for every morphism $Y\to X$ of $C$ with target $X$, the sieve $R{\times }_{X}Y$ has the following property: the map

$${\mathrm{Hom}}_{\hat{C}}(Y,F_i)\to {\mathrm{Hom}}_{\hat{C}}(R{\times }_{X}Y,F_i)$$

is bijective (respectively injective) for every $i\in I$. Then the sets $J_{\mathcal{F}}(X)$ define a topology on $C$, which is the finest of the topologies for which each $F_i$ is a sheaf (respectively a separated presheaf).

Proof. The $J_{\mathcal{F}}(X)$ evidently verify axioms (T1) and (T3). It remains to show that they verify (T2). For this it suffices to show that:

1. If $R^{\prime }\to R\to X$ are two sieves of $X$, with $R\in J_{\mathcal{F}}(X)$ and such that for every $Y\to R$ (where $Y$ is an object of $C$) the sieve $R^{\prime }{\times }_{X}Y$ belongs to $J_{\mathcal{F}}(Y)$, then the sieve $R^{\prime }$ belongs to $J_{\mathcal{F}}(X)$.

2. If $R^{\prime }\to R\to X$ are two sieves of $X$ such that $R^{\prime }\in J_{\mathcal{F}}(X)$, then the sieve $R$ belongs to $J_{\mathcal{F}}(X)$.

Note that, in case 2), for every $Y\to R$ (where $Y$ is an object of $C$) the sieve $R^{\prime }{\times }_{X}Y$ belongs to $J_{\mathcal{F}}(Y)$. Now, in cases 1) and 2), $R$ is the inductive limit of the $Y$, objects of $C$, over $R$ (I 3.4).

(1) or $\hat{C}$, depending on the mood of the typewriter.

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Inductive limits in $\hat{C}$ are universal (I 3.3). Consequently, in cases 1) and 2), $R^{\prime }$ is the inductive limit of the $R^{\prime }{\times }_{X}Y$. Now, in cases 1) and 2), $R^{\prime }{\times }_{X}Y$ belongs to $J_{\mathcal{F}}(Y)$. Passing to the inductive limit over the objects $Y$ of $C$ over $R$, one deduces that the map

$${\mathrm{Hom}}_{\hat{C}}(R,F_i)\to {\mathrm{Hom}}_{\hat{C}}(R^{\prime },F_i)$$

is a bijection (respectively an injection) for every $i\in I$. Consequently the maps

$${\mathrm{Hom}}_{\hat{C}}(X,F_i)\to {\mathrm{Hom}}_{\hat{C}}(R^{\prime },F_i)$$

$${\mathrm{Hom}}_{\hat{C}}(X,F_i)\to {\mathrm{Hom}}_{\hat{C}}(R,F_i)$$

are, in cases 1) and 2), bijections (respectively injections).

Moreover, hypotheses 1) and 2) are visibly stable under arbitrary base change $Y\to X$ ($Y$ an object of $C$). It follows that for every object $Y$ of $C$ over $X$ and every $i\in I$, the maps

$${\mathrm{Hom}}_{\hat{C}}(Y,F_i)\to {\mathrm{Hom}}_{\hat{C}}(R{\times }_{X}Y,F_i)$$

$${\mathrm{Hom}}_{\hat{C}}(Y,F_i)\to {\mathrm{Hom}}_{\hat{C}}(R^{\prime }{\times }_{X}Y,F_i)$$

are, in both cases, bijections (respectively injections); this shows that $R^{\prime }$ and $R$ belong to $J_{\mathcal{F}}(X)$. QED.

Corollary 2.3. Let $C$ be a $U$-category, and for every $X$ of $C$ let $K(X)$ be a set of sieves of $X$. Suppose that the $K(X)$ verify axiom (T1) of (1.1). For a presheaf $F$ to be a sheaf (respectively a separated presheaf) for the topology generated (1.1.6) by the $K(X)$, it is necessary and sufficient that for every object $X$ of $C$ and every sieve $R\in K(X)$, the map

$${\mathrm{Hom}}_{\hat{C}}(X,F)\to {\mathrm{Hom}}_{\hat{C}}(R,F)$$

be a bijection (respectively an injection).

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Corollary 2.4. In particular, let $C$ be a $U$-category equipped with a pretopology. For a presheaf $F$ to be a sheaf (respectively a separated presheaf), it is necessary and sufficient that for every object $X$ of $C$ and every family $(X_{\alpha }\to X{)}_{\alpha \in A}$ belonging to $\mathrm{Cov}(X)$, the diagram of sets

$$F(X)\to \prod _{\alpha \in A}F(X_{\alpha })\rightrightarrows \prod _{(\alpha ,\beta )\in A\times A}F(X_{\alpha }{\times }_X{X}_{\beta })$$

be exact (respectively the map

$$F(X)\to \prod _{\alpha \in A}F(X_{\alpha })$$

be injective).

Proof. Apply 2.3, then (I 3.5) and (I 2.12).

With Corollary 2.4 one recovers the definition given in [1].

Definition 2.5. Let $C$ be a $U$-category. The canonical topology of $C$ is the finest topology for which the representable functors are sheaves (2.2). A sieve covering $X$ for the canonical topology will be called a universally strict epimorphic sieve. A family covering for the canonical topology will be called a universally strict epimorphic family. When, moreover, the morphisms of the covering family are squarable, the family will be said to be universally effective epimorphic [2].

Remark 2.5.1. For almost all sites that have had to be used up to now, the topology is less fine than the canonical topology; in other words, the representable functors on $C$ are sheaves, i.e. the covering families of $C$ are universally strict epimorphic.

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The only exception to this rule is the site $\tilde{C}$ studied in no. 5 (whose topology is finer, and most often strictly finer, than the canonical topology).

Proposition 2.6. For a sieve $R\to X$ to be universally strict epimorphic, it is necessary and sufficient that, for every object $Y\to X$ of $C$ over $X$ and every object $Z$ of $C$, the map

$${\mathrm{Hom}}_C(Y,Z)\to \underset{C/(Y{\times }_{X}R)}{\underleftarrow{\lim }}{\mathrm{Hom}}_C(-,Z)$$

be a bijection.

Proof. Immediate by applying 2.2 and then (I 5.3).

Remarks 2.7.

1. Proposition 2.6 gives a characterization of the universally strict epimorphic sieves of a category $C$, independent of the universe in which the presheaves take their values, with the sole condition that the morphism sets $\mathrm{Hom}(X,Y)$ of $C$ belong to this universe. It thus allows one to define the canonical topology for every category $C$, without needing to specify universes.

2. Let $C$ be a site whose underlying category is a $U$-category, let $F$ be a $U$-presheaf, and let $V$ be a universe containing $U$. The category underlying $C$ is a $V$-category, and $F$ may be considered as a $V$-presheaf. The $U$-presheaf $F$ is a sheaf (respectively a separated presheaf) if and only if the $V$-presheaf $F$ is a sheaf (respectively a separated presheaf). In other words, the condition that $F$ be a sheaf (respectively a separated presheaf) does not depend (in the sense just

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specified) on the universe in which the presheaf $F$ takes its values. In particular, let $C$ be a site and $F$ a $U$-presheaf; one says that $F$ is a sheaf (respectively a separated presheaf) if there exists a universe $V$ containing $U$ such that the category $C$ is a $V$-category and such that $F$ is a $V$-sheaf (respectively a separated $V$-presheaf). This property does not depend on the universe $V$.

3. Sheaf Associated to a Presheaf

Definition 3.0.1. Let $C$ be a site. A topologically generating family (or, when no confusion results, generating family) of $C$ is a set $G$ of objects of $C$ such that every object of $C$ is the target of a covering family of morphisms of $C$ (1.2) whose sources are elements of $G$.

Definition 3.0.2. Let $U$ be a universe. A $U$-site is a site $C$ whose underlying category is a $U$-category (I 1.1), which possesses a small topologically generating family. Let $C$ be a $U$-category; a $U$-topology on $C$ is a topology on $C$ making $C$ into a $U$-site. A site $C$ is said to be $U$-small, or by abuse of language small, if the category underlying $C$ is small (I 1.0).

3.0.3. It follows immediately from the definitions that every topology finer than a $U$-topology is a $U$-topology, and that a small site is a $U$-site.

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Proposition 3.0.4. Let $C$ be a $U$-site, and let $G$ be a small topologically generating family of $C$. For every $X\in \mathrm{ob}C$, denote by $J_G(X)$ the set of sieves covering $X$ generated by a family of morphisms

$$(Y_{\alpha }\to X),\alpha \in A,\text{where }{Y}_{\alpha }\in G.$$

1. The set $J_G(X)$ is small.
2. The set $J_G(X)$ is cofinal in the set $J(X)$ of all sieves covering $X$, ordered by inclusion.
3. For every $R\in J_G(X)$, there exists a small epimorphic family (I 10.2)

$$(u_{\alpha }:Y_{\alpha }\to R),\alpha \in A,\text{with }{Y}_{\alpha }\in G.$$

Proof. 1) Put $A(X)={\coprod }_{Y\in G}\mathrm{Hom}(Y,X)$. The set $A(X)$ is small (I 0) and $\mathrm{card}(J_G(X))\le 2^{\mathrm{card}(A(X))}$. Consequently $J_G(X)$ is small.

2. Let $R\in J(X)$. Put $A(R)={\coprod }_{Y\in G}\mathrm{Hom}(Y,R)$ and let $R^{\prime }$ be the sieve of $X$ generated by the family $(Y\to R\to X)$, $u\in A(R)$. We have $R^{\prime }\subset R$, and it suffices to show that $R^{\prime }$ is covering. By axiom (T2) of 1.1, it suffices to show that for every morphism $Z\to R$, $Z\in \mathrm{ob}C$, the sieve $R^{\prime }{\times }_{X}Z$ of $Z$ is covering. But the sieve $R^{\prime }{\times }_{X}Z$ contains the sieve generated by the family of all morphisms $Y\to Z$ with $Y\in G$, a family which is covering by hypothesis. The sieve $R^{\prime }{\times }_{X}Z$ of $Z$ is therefore covering (axiom T2 of 1.1).

3. Let $R\in J_G(X)$. The family $(Y_u\to R)$, $u\in A(R)={\coprod }_{Y\in G}\mathrm{Hom}(Y,R)$, is epimorphic by hypothesis. Now, for every $Y\in \mathrm{ob}C$, $\mathrm{Hom}(Y,R)$ is contained in $\mathrm{Hom}(Y,X)$, which is small. Consequently $A(R)$ is small.

3.0.5. Let $C$ be a $U$-site, let $V$ be a universe such that $C\in V$ and $U\subset V$, and let $G$ be a $U$-small topologically generating family of $C$. The category $\hat{C}$ of presheaves of $U$-sets on $C$ is a $V$-category (I 1.1.1). Let

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$X$ be an object of $C$. The set $J(X)$ of sieves covering $X$ is $V$-small and, ordered by inclusion, is cofiltered (1.1.1). For every $U$-presheaf $F$, the inductive limit

$$\underset{J(X)}{\underrightarrow{\lim }}{\mathrm{Hom}}_{\hat{C}}(-,F)$$

is therefore representable by an element of $V$ (I 2.4.1). Moreover, it follows from (3.0.4 3)) that, for every $R\in J_G(X)$, ${\mathrm{Hom}}_{\hat{C}}(R,F)$ is $U$-small; and since $J_G(X)$ is a $U$-small set cofinal in $J(X)$ (loc. cit.), it follows from (I 2.4.2) that ${\underrightarrow{\lim }}_{J(X)}{\mathrm{Hom}}_{\hat{C}}(-,F)$ is $U$-small. Choose then, for every $F$ and every $X$, an element of $U$ representing this inductive limit and put

$$LF(X)=\underset{J(X)}{\underrightarrow{\lim }}{\mathrm{Hom}}_{\hat{C}}(-,F).$$

Let $g:Y\to X$ be a morphism of $C$. The base-change functor $g^{\ast }:J(X)\to J(Y)$ defines a map

$$LF(g):LF(X)\to LF(Y)$$

which makes $X\mapsto LF(X)$ into a presheaf on $C$.

Since the morphism ${\mathrm{id}}_X:X\to X$ is an element of $J(X)$, for every object $X$ of $C$ we have a map

$$\ell (F)(X):F(X)\to LF(X),$$

which visibly defines a morphism of functors

$$\ell (F):F\to LF.$$

It is moreover clear that $F\mapsto LF$ is a functor in $F$ and that the morphisms $\ell (F)$ define a morphism

$$\ell :\mathrm{Id}\to L.$$

Finally let $R\to X$ be a refinement of $X$. The definition of $LF(X)$ and (I 1.4) provide a map

$$Z_R:{\mathrm{Hom}}_{\hat{C}}(R,F)\to {\mathrm{Hom}}_{\hat{C}}(X,LF),$$

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and for every morphism of $C$, $g:Y\to X$, the definition of the functor $LF$ shows that the diagram below is commutative:

Z_R:       Hom_{C^}(R, F)          -> Hom_{C^}(X, LF)
              |                         |
              v                         v
Z_{R x_X Y}: Hom_{C^}(R x_X Y, F)  -> Hom_{C^}(Y, LF)

where the vertical arrows are the evident arrows. We now copy [2].

Lemma 3.1.

1. For every refinement $i_R:R\to X$ and every $u:R\to F$, the diagram

   F --ell(F)--> LF
   ^             ^
   | u           | Z_R(u)
   R --i_R---->  X
   

   is commutative.

2. For every morphism $v:X\to LF$, there exists a refinement $R$ of $X$ and a morphism $u:R\to F$ such that $Z_R(u)=v$.

3. Let $Y$ be an object of $C$ and let $u,v:Y\to F$ be two morphisms such that $\ell (F)\circ u=\ell (F)\circ v$. Then the kernel of the pair $(u,v)$ is a refinement of $Y$.

4. Let $R$ and $R^{\prime }$ be two refinements of $X$, and let $u:R\to F$ and $u^{\prime }:R^{\prime }\to F$ be two morphisms. For $Z_R(u)=Z_{R^{\prime }}(u^{\prime })$, it is necessary and sufficient that $u$ and $u^{\prime }$ coincide on a refinement $R^{\prime \prime }\to R{\times }_X{R}^{\prime }$.

Proof. The only nontrivial assertion is assertion 1). We must show that $Z_R(u)\circ i_R=\ell (F)\circ u$. For this it suffices to show (I 3.4) that the composites of these morphisms with every morphism $g:Y\to R$ ($Y$ an object

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of $C$) are equal. Now consider $f=i_R\circ g$ and the fiber product $R{\times }_{X}Y$:

F  --ell(F)--> LF
^              ^
| u            | Z_R(u)
R  --i_R---->  X
^              ^
| g'           | f
R x_X Y --i'-> Y

In the above diagram, $i^{\prime }$ is a monomorphism that admits a section. Consequently (I 3.3) $i^{\prime }$ is an isomorphism. Now by definition of the morphism $\ell (F)$, the morphism $Z_{R{\times }_{X}Y}(u\circ g^{\prime })$ is equal to $\ell (F)\circ u\circ g^{\prime }$. On the other hand, the commutativity of diagram (*) gives the equality $Z_{R{\times }_{X}Y}(u\circ g^{\prime })=Z_R(u)\circ f$, and consequently we indeed have the equality $\ell (F)\circ u\circ g=Z_R(u)\circ i_R\circ g$.

Proposition 3.2.

1. The functor $L$ is left exact (I 2.3.2).
2. For every presheaf $F$, $LF$ is a separated presheaf.
3. The presheaf $F$ is separated if and only if the morphism $\ell (F):F\to LF$ is a monomorphism. The presheaf $LF$ is then a sheaf.
4. The following properties are equivalent:

i) $\ell (F):F\to LF$ is an isomorphism.

ii) $F$ is a sheaf.

Proof. 1) It suffices to show (I 3.1) that, for every object $X$ of $C$, the functor $F\mapsto LF(X)$ commutes with finite projective limits. But, by definition of

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the projective limit, the functor

$$F\mapsto {\mathrm{Hom}}_{\hat{C}}(R,F),R\to X\in \mathrm{ob}(J(X)),$$

commutes with projective limits, and the inductive limit ${\underrightarrow{\lim }}_{J(X)}$ commutes with finite projective limits because $J(X)$ is a cofiltered set (I 2.7).

2. Let $X$ be an object of $C$, and let $f,g:X\to LF$ be two morphisms that coincide on a refinement $R\to X$ of $X$. By virtue of 3.1 2), there exists a covering sieve $R^{\prime }\to X$, which one may always suppose contained in $R$, and two morphisms $u,v:R^{\prime }\to F$ such that $Z_{R^{\prime }}(u)=f$ and $Z_{R^{\prime }}(v)=g$. By virtue of 3.1 1), we then have $\ell (F)\circ u=\ell (F)\circ v$. Consequently (3.1 4)) $u$ and $v$ coincide on a refinement $R^{\prime \prime }\to R^{\prime }$. Let $w$ be the restriction of $u$ to $R^{\prime \prime }$. We have $Z_{R^{\prime \prime }}(w)=Z_{R^{\prime }}(u)=Z_{R^{\prime }}(v)$, and consequently $f=g$. The presheaf $LF$ is therefore separated.

3. If $F$ is separated, the morphism $\ell (F)$ is a monomorphism because a filtered inductive limit of monomorphisms is a monomorphism. If $\ell (F)$ is a monomorphism, the presheaf $F$ is a subpresheaf of a separated presheaf, hence is separated. We show that $LF$ is then a sheaf. Let $i:R\to X$ be a sieve covering an object $X$ of $C$, and let $u:R\to LF$ be a morphism. It suffices to show that $u$ factors through $X$. Put $R^{\prime }=F{\times }_{LF}R$ and $v=Z_{R^{\prime }}({\mathrm{pr}}_1)$:

F  --ell(F)--> LF
^              ^ \ v
| pr_1         | u
R' --pr_2---> R --i--> X
^              ^
| p_1          | m
R'' --p_2--->  Y

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To show that $u=v\circ i$, it suffices to show (I 3.4) that for every morphism $m:Y\to R$ ($Y$ an object of $C$), $v\circ i\circ m=u\circ m$. Put $R^{\prime \prime }=Y{\times }_R{R}^{\prime }$ and let $p_1$ and $p_2$ be the projections. The projection $R^{\prime \prime }\to Y$ is a monomorphism and makes $R^{\prime \prime }$ into a sieve covering $Y$ (3.1 2)). We have $v\circ i\circ {\mathrm{pr}}_2=\ell (F)\circ {\mathrm{pr}}_1$ (3.1 1)), and therefore $v\circ i\circ {\mathrm{pr}}_2=u\circ {\mathrm{pr}}_2$. We deduce $v\circ i\circ {\mathrm{pr}}_2\circ p_1=u\circ {\mathrm{pr}}_2\circ p_1$, i.e. $v\circ i\circ m\circ p_2=u\circ m\circ p_2$. Since the presheaf $LF$ is separated, we have $v\circ i\circ m=u\circ m$, QED.

4. Clear.

Remark 3.3. Let $J^{\prime }(X)$ be a cofinal subset of $J(X)$. We have

$$LF(X)=\underset{J^{\prime }(X)}{\underrightarrow{\lim }}{\mathrm{Hom}}_{\hat{C}}(-,F).$$

In particular, if the topology of $C$ is defined by a pretopology $X\mapsto \mathrm{Cov}(X)$ (1.1.3), the functor $L$ can be described using the covering families of $\mathrm{Cov}(X)$ (I 2.12 and I 3.5). By spelling out the formulas one recovers the construction of [2].

Theorem 3.4. Let $C$ be a $U$-site. The inclusion functor $i:\tilde{C}\to \hat{C}$ from sheaves into presheaves admits a left adjoint $a$, left exact (I 2.3.2):

C^  --a-->  C~
C~  --i-->  C^

The functor $i\circ a$ is canonically isomorphic to the functor $L\circ L$ (cf. 3.0.5). For every presheaf $F$, the adjunction morphism $F\to i\circ a(F)$ is obtained, by the preceding isomorphism, from the morphism $\ell (LF)\circ \ell (F):F\to L\circ L(F)$.

[PDF p. 250 duplicates PDF p. 249 in the source scan.]

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Definition 3.5. The sheaf $aF$ is called the sheaf associated to the presheaf $F$.

Theorem 3.4 follows immediately from Proposition 3.2.

Proposition 3.6. Let $C$ be a $U$-site and $V\supset U$ a universe. Denote by ${\hat{C}}_U$ and ${\tilde{C}}_U$ (respectively ${\hat{C}}_V$ and ${\tilde{C}}_V$) the categories of $U$-presheaves and $U$-sheaves (respectively of $V$-presheaves and $V$-sheaves), and by $a_U:{\hat{C}}_U\to {\tilde{C}}_U$ (respectively $a_V:{\hat{C}}_V\to {\tilde{C}}_V$) the corresponding “associated sheaf” functors. The diagram

C^_U --a_U--> C~_U
 |             |
 v             v
C^_V --a_V--> C~_V

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

Proof. This follows from the construction of the functors $a_U$ and $a_V$ (3.4 and 3.0.5).

4. Exactness Properties of the Category of Sheaves

The exactness properties of the category of sheaves are deduced from the exactness properties of the category of presheaves via Theorem 3.4. The present section makes this philosophy explicit in typical statements among the most useful ones.

Theorem 4.1. Let $C$ be a $U$-site, $\tilde{C}$ the category of sheaves, $a:\hat{C}\to \tilde{C}$ the associated sheaf functor, and $i:\tilde{C}\to \hat{C}$ the inclusion functor.

1. The functor $a$ commutes with inductive limits and is exact.

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2. $U$-inductive limits in $\tilde{C}$ are representable. For every small category $I$ and every functor $E:I\to \tilde{C}$, the canonical morphism

   $$\underset{I}{\underrightarrow{\lim }}E\to a(\underset{I}{\underrightarrow{\lim }}iE)$$

   is an isomorphism.

3. $U$-projective limits in $\tilde{C}$ are representable. For every object $X$ of $C$, the functor on $\tilde{C}$

   $$F\mapsto F(X)$$

   commutes with projective limits, i.e. the inclusion functor $i:\tilde{C}\to \hat{C}$ commutes with projective limits.

Proof. These properties result essentially from Theorem 3.4 and from (I 2.11).

Thus, in the category of sheaves, products indexed by an element of $U$, fiber products, sums indexed by an element of $U$, amalgamated sums, kernels, cokernels, images, and coimages are representable.

Corollary 4.1.1. Let $C$ be a $U$-site and $F$ a sheaf of sets on $C$. The canonical homomorphism

$$\underset{C/F}{\underrightarrow{\lim }}aX\to F$$

is an isomorphism.

Proof. This follows from (I 3.4) and from the fact that $a$ commutes with inductive limits.

Proposition 4.2. Every morphism of $\tilde{C}$ that is both an epimorphism and a monomorphism is an isomorphism.

Proof. Let $u:G\to H$ be a morphism of $\tilde{C}$ that is an epimorphism and a monomorphism. First note that the morphism $u$ is a monomorphism

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of presheaves (4.1 1)). Construct the amalgamated sum $H{⨿}_{G}H=K$ and the two canonical morphisms $i_1,i_2:H\to K$ in the category of presheaves. Since $u$ is a monomorphism of presheaves, one verifies immediately that the following diagram is cartesian and cocartesian (I 10.1):

G --u--> H
|        |
u        i_2
v        v
H --i_1-> K

Applying the “associated sheaf” functor, we therefore obtain a cartesian and cocartesian diagram of the category of sheaves (4.1 1)):

G --u-->  H
|         |
u         a i_2
v         v
H --a i_1-> aK

Since $u$ is an epimorphism of sheaves, the morphism $a{i}_1$ is an isomorphism; and since the diagram (*) is cartesian, the morphism $u$ is an isomorphism.

Proposition 4.3.

1. Inductive limits in $\tilde{C}$ that are representable are universal (I 2.5).
2. Every epimorphic family (I 10.2) of morphisms is universally effective epimorphic (2.6).
3. Every equivalence relation is universally effective (I 10.6).
4. Filtered $U$-inductive limits commute with finite projective limits (I 2.6).

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Proof. Assertions 1) to 4) are true in the category of sets, hence in the category of presheaves (I 3.1). Assertions

1. and 4) then follow immediately from the corresponding assertions for presheaves and from 4.1. Let us prove 3). Let $p_1,p_2:R\to X$ be an equivalence relation of $\tilde{C}$. The diagram

$$R\begin{array}{c}\stackrel{p_1}{\to }\\ \underset{p_2}{\to }\end{array}X$$

is then an equivalence relation of presheaves (4.1 1)). There therefore exists a morphism of presheaves $u:X\to Y$ such that the diagram

R --p_2--> X
|          |
p_1        u
v          v
X --u----> Y

is cartesian and cocartesian in the category of presheaves. Applying the “associated sheaf” functor, one obtains a cartesian and cocartesian diagram in the category of sheaves (4.1 1)). Consequently $R\to X\times X$ is an effective equivalence relation. Since every equivalence relation is effective by what precedes, every equivalence relation is universally effective.

Let us prove 2). Let $(u_i:X_i\to X{)}_{i\in I}$ be an epimorphic family of $\tilde{C}$. By (II 2.6) and (I 2.12), it suffices to prove the following assertions:

a) For every morphism of sheaves $v:Y\to X$, the family of morphisms ${\mathrm{pr}}_{2,i}:X_i{\times }_{X}Y\to Y$, $i\in I$, is an epimorphic family of $\tilde{C}$.

b) For every sheaf $Z$, the diagram of sets

$$\mathrm{Hom}(X,Z)\to \prod _{i\in I}\mathrm{Hom}(X_i,Z)\rightrightarrows \prod _{(i,k)\in I\times I}\mathrm{Hom}(X_i{\times }_X{X}_k,Z)$$

is exact.

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Let $X^{\prime }$ be the union of the images of the morphisms $u_i$ in the sense of presheaves, and let $j:X^{\prime }\to X$ be the canonical injection. For every $i\in I$, the morphism $u_i:X_i\to X$ factors through a morphism $u_i^{\prime }:X_i\to X^{\prime }$ and the morphism $j$, and the family $u_i^{\prime }:X_i\to X^{\prime }$, $i\in I$, is an epimorphic family of $\hat{C}$. Since $j$ is a monomorphism, the morphism $aj:a{X}^{\prime }\to X$ is a monomorphism of sheaves (4.1). Since, for every $i\in I$, we have $u_i=aj\circ a{u}_i^{\prime }$, $aj$ is an epimorphism of sheaves. Consequently $aj$ is an isomorphism (4.2). Let $v:Y\to X$ be a morphism of sheaves. We deduce by base change a morphism $j_v:X^{\prime }{\times }_{X}Y\to Y$ and morphisms $u_{i,v}^{\prime }:X_i{\times }_{X}Y\to X^{\prime }{\times }_{X}Y$. Since $a$ commutes with fiber products, $a{j}_v$ is an isomorphism. Since epimorphic families of $\hat{C}$ remain epimorphic after base change, the family $u_{i,v}^{\prime }$, $i\in I$, is epimorphic in $\hat{C}$. Since $a$ commutes with inductive limits, the family $a{u}_{i,v}^{\prime }:X_i{\times }_{X}Y\to a(X^{\prime }{\times }_{X}Y)$, $i\in I$, is epimorphic in $\tilde{C}$, which proves a).

Let us prove b). Let $Z$ be a sheaf. Since $aj$ is an isomorphism, the map

$$\mathrm{Hom}(X,Z)\to \mathrm{Hom}(X^{\prime },Z)$$

is a bijection. Since the $u_i^{\prime }$, $i\in I$, form an epimorphic family of $\hat{C}$, the diagram of sets

$$\mathrm{Hom}(X^{\prime },Z)\to \prod _{i\in I}\mathrm{Hom}(X_i,Z)\rightrightarrows \prod _{(i,k)\in I\times I}\mathrm{Hom}(X_i{\times }_{X^{\prime }}{X}_k,Z)$$

is exact. Finally, since $X^{\prime }$ is a subpresheaf of $X$, the presheaf $X_i{\times }_{X^{\prime }}{X}_k$, $(i,k)\in I\times I$, is canonically isomorphic to $X_i{\times }_X{X}_k$, which proves b).

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Remarks 4.3.1.

1. Proposition 4.3 2) formally gives a second proof of 4.2: it is clear that, in any category, a morphism that is both a universally effective epimorphism and a monomorphism is in fact an isomorphism.

2. One deduces from the preceding propositions that every morphism in the category of sheaves factors uniquely as an effective epimorphism followed by an effective monomorphism. This property naturally generalizes, for nonadditive categories, axiom (AB2) of abelian categories ($\mathrm{coim}\simeq \mathrm{im}$) [Tohoku].

4.4.0. Let $C$ be a $U$-site. The functor $h:C\to \hat{C}$ (I 1.3.1), composed with the associated sheaf functor, provides a functor

$${\epsilon }_C:C\to \tilde{C},$$

called the canonical functor from $C$ to $\tilde{C}$, which will be used constantly in what follows. The functor ${\epsilon }_C$ commutes with finite projective limits. When the topology of $C$ is less fine than the canonical topology, ${\epsilon }_C$ is fully faithful and commutes with projective limits; it is then, moreover, defined when $C$ does not necessarily possess a small topologically generating family. We shall not study in detail the behavior of ${\epsilon }_C$ with respect to inductive limits (cf. [SGA 3 IV]). We shall however need the propositions below.

Theorem 4.4. Let $C$ be a $U$-site and let $(u_i:X_i\to X{)}_{i\in I}$ be a family of morphisms of $C$ with target $X$. The following conditions are equivalent (*):

i) The family $({\epsilon }_C(u_i):{\epsilon }_C(X_i)\to {\epsilon }_C(X){)}_{i\in I}$ is an epimorphic family of $\tilde{C}$ (I 10.2).

(*) When the site $C$ does not necessarily possess a small topologically generating family and when the topology of $C$ is less fine than the canonical topology, one has ii) $\Rightarrow$ i) (cf. proof of 4.4).

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ii) The family $(u_i:X_i\to X{)}_{i\in I}$ is a covering family of $C$ (1.2).

Proof. ii) $\Rightarrow$ i). Let $R\to X$ be the sieve generated by the $u_i$, $i\in I$ (I 4.3.3), and let $u_i^{\prime }:X_i\to R$ be the morphisms induced by the $u_i$. The family of the $u_i^{\prime }$, $i\in I$, is an epimorphic family of $\hat{C}$, and $R$ is a sieve covering $X$. Consequently, for every sheaf $F$, the map

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

is bijective, and the map

$$\mathrm{Hom}(R,F)\to \prod _i\mathrm{Hom}(X_i,F)$$

is injective. Therefore the map

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

is injective, and consequently the map

$$\mathrm{Hom}(aX,F)\to \prod _i\mathrm{Hom}(a{X}_i,F)$$

is injective, whence i).

i) $\Rightarrow$ ii). With the notation introduced above, let $J_R:R\to X$ be the canonical injection into $X$ of the sieve generated by the $u_i$, $i\in I$. It follows from i) that the morphism of sheaves $a{J}_R:aR\to aX$ is both an epimorphism of sheaves and a monomorphism of sheaves. It is therefore an isomorphism of sheaves (4.2). With the notation of no. 3, we have a commutative diagram:

X --ell(X)--> LX --ell(LX)--> aX
^             ^               ^
J_R           LJ_R            aJ_R
R --ell(R)--> LR --ell(LR)--> aR

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First, by 3.1 2), there exists a sieve covering $J_{S_1}:S_1\to X$ and a morphism $u_1:S_1\to LR$ such that

$$\begin{array}{rl}(\mathrm{4.4.1})\ell (LX)\circ \ell (X)\circ J_{S_1}& =a{J}_R\circ \ell (LR)\circ u_1\\ & =\ell (LX)\circ L{J}_R\circ u_1.\end{array}$$

Put $m=\ell (X)\circ J_{S_1}$ and $n=L{J}_R\circ u_1$. The morphisms $m$ and $n$ are morphisms from $S_1$ to $LX$. Let $S_2\to S_1$ be the kernel of the pair of arrows $(m,n)$. For every object $Y$ of $C$ and every morphism $\alpha :Y\to S_1$, we have, by virtue of (4.4.1), $\ell (LX)\circ m\circ \alpha =\ell (LX)\circ n\circ \alpha$. It follows from (3.1 3)) that the kernel $S_2{\times }_{S_1}Y$ of the pair of arrows $(m\circ \alpha ,n\circ \alpha )$ is a sieve covering $Y$. Consequently, by axiom (T2) for topologies, $J_{S_2}:S_2\to X$ is a sieve covering $X$. We therefore have a morphism $u_2:S_2\to LR$ and a commutative diagram:

(4.4.2)
S_2 --J_{S_2}--> X
| \              |
u_2 \            | ell(X)
v    \ J_R       v
LR <--ell(R)-- R --LJ_R--> LX

Let $Y$ be an object of $C$ and $\beta :Y\to S_2$ a morphism. By 3.1 1), there exists a sieve covering $J_{Q_1}:Q_1\to Y$ and a morphism $v_1:Q_1\to R$ such that $u_2\circ \beta \circ J_{Q_1}=\ell (R)\circ v_1$. Put $f=J_{S_2}\circ \beta \circ J_{Q_1}$ and $g=J_R\circ v_1$. Let $Q_2$ be the kernel of the pair $(f,g):Q_1\rightrightarrows X$ and let $J_{Q_2}:Q_2\to Y$ be the canonical injection. For every object $Z$ of $C$ and every morphism $\gamma :Z\to Q_1$, we have, by the commutativity of (4.4.2):

$$\begin{array}{rl}\ell (X)\circ f\circ \gamma & =\ell (X)\circ J_{S_2}\circ \beta \circ J_{Q_1}\circ \gamma \\ & =L{J}_R\circ u_2\circ \beta \circ J_{Q_1}\circ \gamma \\ & =L{J}_R\circ \ell (R)\circ v_1\circ \gamma \\ & =\ell (X)\circ J_R\circ v_1\circ \gamma \\ & =\ell (X)\circ g\circ \gamma .\end{array}$$

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It follows from 3.1 2) that the kernel $Q_2{\times }_{Q_1}Z$ of the pair $(f\circ \gamma ,g\circ \gamma )$ is a sieve covering $Z$. By axiom (T2) for topologies, $J_{Q_2}:Q_2\to Y$ is a sieve covering $Y$. For every morphism $\beta :Y\to S_2$, there therefore exists a sieve covering $J_{Q_2}:Q_2\to Y$ and a morphism $v_2:Q_2\to R$ such that the following diagram is commutative:

Y --beta--> S_2 --J_{S_2}--> X
^                         /
| J_{Q_2}                / J_R
Q_2 --v_2-------------> R

Now denote by $S_2\cap R$ the fiber product of $S_2$ and $R$ over $X$. The sieve of $Y$ $(S_2\cap R){\times }_{S_2}Y$ contains the sieve $Q_2$, and consequently $(S_2\cap R){\times }_{S_2}Y$ is a sieve covering $Y$. It then follows from axiom (T2) for topologies that $S_2\cap R$ is a sieve covering $X$, and consequently the sieve $R$, which contains $S_2\cap R$, is a sieve covering $X$, QED.

Corollary 4.4.4. Let $T$ be the topology of the $U$-site $C$, and let $T^{\prime }$ (respectively $T^{\prime \prime }$) be the finest topology on $C$ among those for which the representable functors are sheaves (respectively separated presheaves) (2.2). Then $T=T^{\prime }=T^{\prime \prime }$.

Indeed, we trivially have $T\subset T^{\prime }\subset T^{\prime \prime }$, and 4.4 and 2.2 imply that $T^{\prime \prime }\subset T$, QED.

Definition 4.5. An initial object of a category $A$ is an object ${\varnothing }_A$ representing the empty inductive limit, i.e. such that, for every $X\in \mathrm{ob}A$, there exists one and only one arrow ${\varnothing }_A\to X$. An object ${\varnothing }_A$ is said to be strictly initial if it is initial and if every morphism with target ${\varnothing }_A$ is an isomorphism. Let $(S_i{)}_{i\in I}$ be a family of objects of a category $A$. Suppose that the sum $S={\coprod }_{i\in I}{S}_i$ is representable. The sum $S$ is said to be disjoint if the structural morphisms $S_i\to S$ are squarable, if they are monomorphisms, and if, for every pair $i,j\in I$, $i\ne j$, the product $S_i{\times }_S{S}_j$ is an initial object of $A$. The sum $S$ is said to be disjoint

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universal if it is disjoint and if it remains a disjoint sum after every base change $T\to S$. It follows that for every pair $i,j$, $i\ne j$, the objects $S_i{\times }_S{S}_j$ are strictly initial objects of $A$.

Example 4.5.1. In the category of sets, direct sums are disjoint and universal. The same is therefore true in every category of presheaves of sets (I 3.1), and consequently in every category of sheaves of sets on a $U$-site (4.1); in particular, the initial object of $\tilde{C}$ is strict.

Proposition 4.6. Let $C$ be a $U$-category and let $(s_i:X_i\to X{)}_{i\in I}$ be a family of morphisms of $C$. For every $U$-topology $\mathcal{G}$ on $C$ (3.0.2), denote by ${\tilde{C}}_{\mathcal{G}}$ the corresponding category of sheaves and by ${\epsilon }_{\mathcal{G}}:C\to {\tilde{C}}_{\mathcal{G}}$ the corresponding functor.

1. Let $\mathcal{G}$ be a $U$-topology such that

   $$\coprod _{i\in I}{\epsilon }_{\mathcal{G}}(X_i)\stackrel{({\epsilon }_{\mathcal{G}}(s_i))}{\to }{\epsilon }_{\mathcal{G}}(X)$$

   is an isomorphism. Then for every topology ${\mathcal{G}}^{\prime }$ finer than $\mathcal{G}$, the morphism

   $$\coprod _{i\in I}{\epsilon }_{{\mathcal{G}}^{\prime }}(X_i)\stackrel{({\epsilon }_{{\mathcal{G}}^{\prime }}(s_i))}{\to }{\epsilon }_{{\mathcal{G}}^{\prime }}(X)$$

   is an isomorphism.

2. Let $\mathcal{G}$ be a $U$-topology on $C$. The following properties (i) and (ii) are equivalent:

i) a) The family $(s_i:X_i\to X{)}_{i\in I}$ is covering for $\mathcal{G}$.

b) For every $i\in I$, the diagonal morphism of presheaves

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$${\Delta }_i:X_i\to X_i{\times }_X{X}_i$$

is transformed by the “associated sheaf” functor (for $\mathcal{G}$) into an isomorphism, which is the case if the $s_i$ are monomorphisms.

c) For every pair $(i,j)$, $i\ne j$, of elements of $I$, the presheaf $X_i{\times }_X{X}_j$ is transformed by the “associated sheaf” functor (for $\mathcal{G}$) into the initial object of ${\tilde{C}}_{\mathcal{G}}$.

ii) ${\epsilon }_{\mathcal{G}}(X)$ is the sum of the ${\epsilon }_{\mathcal{G}}(X_i)$.

Proof. 1) A sheaf $F$ for ${\mathcal{G}}^{\prime }$ is a sheaf for $\mathcal{G}$. Consequently, for every sheaf $F$ for ${\mathcal{G}}^{\prime }$, the morphism

$$\mathrm{Hom}(X,F)\to \prod _{i\in I}\mathrm{Hom}(X_i,F)$$

is an isomorphism, which implies the assertion.

2. By definition, property ii) holds if and only if the morphism of presheaves

$$\phi =(h(s_i),i\in I):\coprod _{i}h(X_i)\to h(X)$$

is transformed by the “associated sheaf” functor into an isomorphism, i.e. (4.2) if and only if $a(\phi )$ is an epimorphism and a monomorphism of sheaves. By (4.4), the morphism $a(\phi )$ is an epimorphism if and only if property a) holds. The diagonal morphism

$$\Delta :\coprod _{i}h(X_i)\to \left(\coprod _{i}h(X_i)\right){\times }_{h(X)}\left(\coprod _{i}h(X_i)\right)$$

is the direct sum of a family ${\Delta }_{i,j}$, $(i,j)\in I\times I$, of morphisms defined as follows:

$\beta$) When $i=j$, ${\Delta }_{i,j}$ is the diagonal morphism $h(X_i)\to h(X_i){\times }_{h(X)}h(X_i)$.

$\gamma$) When $i\ne j$, ${\Delta }_{i,j}$ is the morphism ${\varnothing }_{\hat{C}}\to h(X_i){\times }_{h(X)}h(X_j)$, where ${\varnothing }_{\hat{C}}$ denotes the initial object of $\hat{C}$.

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The morphism $a(\phi )$ is a monomorphism if and only if $a(\Delta )$ is an isomorphism, and by (4.1) $a(\Delta )$ is an isomorphism if and only if the $a({\Delta }_{i,j})$ are isomorphisms, i.e. if and only if properties b) and c) hold.

Corollary 4.6.1. Let $C$ be a $U$-category and let $X$ be an object of $C$.

1. Let $\mathcal{G}$ be a $U$-topology on $C$. The object $X$ of $C$ is transformed by the “associated sheaf” functor (for the topology $\mathcal{G}$) into the initial object of ${\tilde{C}}_{\mathcal{G}}$ if and only if the empty sieve covers $X$.

2. When $X$ is a strictly initial object of $C$, the sheaf associated to $X$, for every $U$-topology finer than the canonical topology, is an initial object.

Proof. 1) Take the empty set as the set $I$ in 4.6 2).

2. By 1) and 4.6 1), it suffices to show that the empty sieve covers $X$ for the canonical topology, i.e. (2.6) it suffices to show that for every object $Y\to X$ over $X$, $Y$ is an initial object of $C$, which follows from the definition (4.5).

Corollary 4.6.2. Let $C$ be a $U$-site and let $(s_i:X_i\to X{)}_{i\in I}$ be a family of squarable morphisms of $C$ with the same target and having the following properties:

$\alpha$) The family of the $s_i$, $i\in I$, is covering.

$\beta$) For every $i\in I$, $s_i:X_i\to X$ is a monomorphism.

$\gamma$) For every pair $(i,j)$, $i\ne j$, of elements of $I$, $X_i{\times }_X{X}_j$ is covered by the empty sieve, i.e. for every sheaf $F$ on $C$, $F(X_i{\times }_X{X}_j)$ is a set reduced to one element.

Then ${\epsilon }_C(X)$ is the sum of the ${\epsilon }_C(X_i)$, $i\in I$.

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Proof. This follows immediately from 4.6 2) and from 4.6.1 1).

Corollary 4.6.3. Let $C$ be a $U$-category and let $(s_i:X_i\to X{)}_{i\in I}$ be a family of squarable morphisms with the same target. Suppose that the canonical topology of $C$ is a $U$-topology. The following conditions are equivalent:

i) There exists a $U$-topology $\mathcal{G}$ on $C$, less fine than the canonical topology, such that ${\epsilon }_{\mathcal{G}}(X)$ is the sum of the ${\epsilon }_{\mathcal{G}}(X_i)$, $i\in I$.

ii) The object $X$ is the disjoint and universal sum (4.5) in $C$ of the $X_i$.

Proof. ii) $\Rightarrow$ i). Since $X$ is the universal sum of the $X_i$, $i\in I$, the family of the $s_i$, $i\in I$, is covering for the canonical topology of $C$ (2.6). Condition $\alpha$) of 4.6.2 is therefore satisfied when $C$ is equipped with the canonical topology. Condition $\beta$) is evidently satisfied, and property $\gamma$) follows from 4.6.1 2), whence i).

i) $\Rightarrow$ ii). Let $\mathcal{G}$ be a topology on $C$ less fine than the canonical topology such that ${\epsilon }_{\mathcal{G}}(X)$ is the sum of the ${\epsilon }_{\mathcal{G}}(X_i)$. For every sheaf $F$ for $\mathcal{G}$, the canonical map $\mathrm{Hom}(X,F)\to {\prod }_i\mathrm{Hom}(X_i,F)$ is a bijection. Moreover, every representable presheaf is a sheaf. It follows at once that $X$ is the sum in $C$ of the $X_i$, $i\in I$. Applying this to the case where the set $I$ is empty, one sees that if an object of $C$ is transformed into the initial object of $\tilde{C}$, that object is an initial object of $C$. The functor ${\epsilon }_C:C\to \tilde{C}$ is fully faithful and commutes with finite projective limits. Consequently condition b) of 4.6 2) implies that the $s_i$, $i\in I$, are monomorphisms of $C$, and condition c) implies, by what precedes, that the $X_i{\times }_X{X}_j$, $i\ne j$, are initial objects of $C$. Consequently $X$ is the disjoint sum

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of the $X_i$. Since ${\epsilon }_C$ commutes with fiber products, this latter property is stable under every base change $Y\to X$, and consequently $X$ is the disjoint and universal sum of the $X_i$, $i\in I$.

Corollary 4.6.4. Let $C$ be a $U$-category and let $X$ be an object of $C$. Suppose that the canonical topology on $C$ is a $U$-topology. The following properties are equivalent:

i) There exists on $C$ a topology less fine than the canonical topology such that ${\epsilon }_C(X)$ is an initial object of $\tilde{C}$.

ii) $X$ is a strictly initial object of $C$.

Proof. Take the empty set as the set $I$ in 4.6.3.

Proposition 4.7. Let $C$ be a $U$-category, let $\tilde{C}$ be the category of sheaves on $C$ for the canonical topology, and let ${\epsilon }_C:C\to \tilde{C}$ be the canonical functor (4.4.0). Let $R\rightrightarrows X$ be an equivalence relation in $C$ admitting a cokernel $Y$ in $C$. Let $\pi :X\to Y$ be the canonical morphism and suppose it is squarable. Consider the following properties:

i) ${\epsilon }_C(Y)$ is the quotient of the equivalence relation ${\epsilon }_C(R)\rightrightarrows {\epsilon }_C(X)$.

ii) The equivalence relation $R\rightrightarrows X$ is universally effective (cf. no. 7).

One has ii) $\Rightarrow$ i). When $C$ possesses a $U$-topology less fine than the canonical topology, one has i) $\Rightarrow$ ii).

Proof. ii) $\Rightarrow$ i). The canonical morphism $R\to X{\times }_{Y}X$ is an isomorphism. Let $S\to Y$ be the sieve generated by $\pi :X\to Y$. For every presheaf $F$, one has an exact diagram (I 2.12):

$$\mathrm{Hom}(S,F)\to \mathrm{Hom}(X,F)\rightrightarrows \mathrm{Hom}(R,F).$$

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It therefore suffices to show that, for every sheaf $F$ for the canonical topology, the map

$$\mathrm{Hom}(X,F)\to \mathrm{Hom}(S,F)$$

is a bijection, i.e. it suffices to show that $S$ is covering for the canonical topology, or again that the sieve generated by $\pi :X\to Y$ is covering, which follows from Definition 2.5.

i) $\Rightarrow$ ii). The functor ${\epsilon }_C:C\to \tilde{C}$ commutes with finite projective limits and is fully faithful. Now the canonical morphism

$${\epsilon }_C(R)\to {\epsilon }_C(X){\times }_{{\epsilon }_C(Y)}{\epsilon }_C(X)$$

is an isomorphism (4.3). Consequently the canonical morphism $R\to X{\times }_{Y}X$ is an isomorphism. The morphism ${\epsilon }_C(X)\to {\epsilon }_C(Y)$ is an epimorphism, which implies (4.4) that $\pi :X\to Y$ is a covering morphism of $C$ for the canonical topology, QED.

Proposition 4.8. Let $C$ be a $U$-site. The category of sheaves on $C$ has the following properties:

a) Finite projective limits are representable.

b) Direct sums indexed by an element of $U$ are representable. They are disjoint and universal (4.5).

c) Equivalence relations are universally effective.

This was seen in 4.1 2) and 3), 4.3 3), and 4.5.1.

These properties have been singled out because they will later allow one to characterize the $U$-categories equivalent to categories of sheaves on categories belonging to $U$ (IV 1.2).

Remark. Recall that property a) is equivalent to the property: there exists a final object, and fiber products are representable (I 2.3.1).

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4.9. Let $C$ be a site whose underlying category is a $U$-category. Then the category $\tilde{C}$ of $U$-sheaves on $C$ satisfies the conditions considered in I 7.3 ((i) or (ii), as desired); hence, by loc. cit., the various variants considered in I 7.1 for the notion of generating family in $\tilde{C}$ coincide. This being said:

Proposition 4.10. With the preceding notation, consider the canonical functor $\epsilon :C\to \tilde{C}$ (4.4.0), and let $G\subset \mathrm{ob}C$ be a topologically generating family in $C$ (3.0.1). Then the family $(\epsilon (X){)}_{X\in G}$ of objects of $\tilde{C}$ is a generating family. In particular the family $(\epsilon (X){)}_{X\in \mathrm{ob}C}$ of objects of $\tilde{C}$ is generating.

One must prove that every morphism $u:F\to F^{\prime }$ in $\tilde{C}$ such that

$$(\mathrm{4.10.1})\mathrm{Hom}(\epsilon (X),F)\to \mathrm{Hom}(\epsilon (X),F^{\prime })$$

is a bijection for every $X\in G$, is an isomorphism. But the preceding map identifies with the map

$$(\mathrm{4.10.2})u(X):F(X)\to F^{\prime }(X),$$

and one must show that if this map is bijective for every $X\in G$, then the same holds for every $X\in \mathrm{ob}C$. Now let $(X_i\to X{)}_{i\in I}$ be a covering family of $X$ by objects of $G$, and for every pair of indices $(i,j)$ of $I$, consider the set of all morphisms $X_{ijk}\to X_i{\times }_X{X}_j$ in $C$, with $X_{ijk}\in G$, $k$ varying in an index set $I_{ij}$. One then obtains a homomorphism of exact diagrams of sets

F(X)  -> prod_i F(X_i)  => prod_{ijk} F(X_{ijk})
 |          |                         |
 v          v                         v
F'(X) -> prod_i F'(X_i) => prod_{ijk} F'(X_{ijk}),

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where, by hypothesis, the last two vertical arrows are bijections. The first vertical arrow is therefore also a bijection, QED.

Corollary 4.11. Suppose that $C$ is a $U$-site (3.0.2). Then the category $\tilde{C}$ is a $U$-category and admits a $U$-small generating family.

Indeed, by hypothesis one can take a small generating family for $G$ in 4.10, which proves the existence of a small generating family. On the other hand, if $F,F^{\prime }\in \mathrm{ob}\tilde{C}$, a homomorphism from $F$ to $F^{\prime }$ is known once one knows the homomorphism (4.10.1), i.e. (4.10.2), for every $X\in \mathrm{ob}G$ (I 7.1.1). It follows that the map

$$\mathrm{Hom}(F,F^{\prime })\to \prod _{X\in G}\mathrm{Hom}(F(X),F^{\prime }(X))$$

is injective. Since the second member is $U$-small, the same holds for the first, which proves that $\tilde{C}$ is a $U$-category.

Corollary 4.12. Let $C$ be a $U$-site. Then for every object $X$ of $\tilde{C}$, the set of subobjects of $X$ and the set of quotient objects of $X$ are $U$-small.

This follows from I 7.4 and I 7.5 respectively, which apply thanks to 4.11.

5. Extension of a Topology from $C$ to $\hat{C}$

Proposition 5.1. Let $C$ be a $U$-site and let $f:H\to K$ be a morphism of $\hat{C}$. The following conditions are equivalent:

i) For every $X\to K$, with $X\in \mathrm{ob}C$, the corresponding morphism $H{\times }_{K}X\to X$ has as image a covering sieve of $X$.

ii) The morphism $a(f):aH\to aK$ on associated sheaves is an

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epimorphism of $\tilde{C}$.

ii bis) For every sheaf $F$ on $C$, the map $\mathrm{Hom}(K,F)\to \mathrm{Hom}(H,F)$ deduced from $f$ is injective.