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SGA 4-I – Expose IV: Topoi

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Expose IV – Topoi

By A. Grothendieck and J.-L. Verdier.

0. Introduction

0.1. In II we saw various exactness properties of categories of the form $\tilde{C}$ = category of sheaves of sets on $C$, where $C$ is a small site, properties that can be expressed by saying that in many respects these categories (which we shall call topoi) inherit the familiar properties of the category $(\mathrm{Ens})$ of (small) sets. On the other hand, experience has taught that many situations in mathematics should be considered above all as a technical means for constructing the corresponding categories of sheaves (of sets), i.e. the corresponding “topoi”. It appears that all genuinely important notions attached to a site (for example its cohomological invariants, studied in V, various other “topological” invariants, such as its homotopy invariants recently studied by M. Artin and B. Mazur [1], and the notions studied in J. Giraud’s book on noncommutative cohomology) are in fact expressed directly in terms of the associated topos. From this point of view, two sites should be regarded as essentially equivalent when the associated topoi are equivalent categories, and the datum of a site (at least in the practically most important case where its topology is less fine than its canonical topology) amounts to that of a topos $E$ (namely the associated topos, formed by the sheaves of sets on the site), and of a generating family of elements of $E$

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(cf. II 4.9 and 1.2.1 below). This point of view is analogous to associating a group to any system of generators and any system of relations among these generators, and attaching one’s interest rather to the structure of this group than to the system of generators and relations which served to generate it (regarded as auxiliary data of the situation). Moreover, the “comparison lemma” III 5.1 supplies many examples of pairs of sites $C$, $C^{\prime }$ which are not isomorphic, and not even equivalent as categories, and which give rise to equivalent topoi, so that $C$ and $C^{\prime }$ should be regarded as essentially equivalent.

0.2. In the present expose, we give a characterization of topoi by simple exactness properties (due to J. Giraud), we study the natural notion of morphism of topoi, inspired by the notion of a continuous map from one topological space to another, and we develop, in the framework of topoi, certain familiar constructions from ordinary sheaf theory (internal Hom sheaves, tensor product sheaves, supports). Finally, following M. Artin, we show how one can reconstruct a topos from an “open” of it, from the complementary “closed” part, and from a certain left exact functor relating them, which, up to very little, can moreover be chosen arbitrarily.

0.3. This gives a gluing procedure for topoi which, when applied to topoi coming from ordinary topological spaces, will in general give a topos no longer of the same type. This is a first indication of the remarkable stability of the notion of topos under various natural constructions, a stability lacking in the notion of topological space (from which the notion

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of topos is inspired). For a second remarkable example, let us also mention that of the classifying topos relative to a group of a topos (cf. [1] or 5.9 below), inspired by the classical notion of classifying space of a topological group, and the notion of “modular topos” associated to various “moduli problems” in algebraic geometry or analytic geometry [10] [13].

Other topoi, such as the etale topos of a scheme (studied systematically in the present Seminar starting from Exp. VII) or the crystalline topos of a relative scheme [6], arise naturally when one wants to develop usable cohomology theories for abstract algebraic varieties (and more generally schemes), replacing the classical Betti cohomology of algebraic varieties over the field of complex numbers.

0.4. One may therefore say that the notion of topos, a natural derivative of the sheaf-theoretic point of view in topology, in turn constitutes a substantial enlargement of the notion of topological space (cf. [9], or 4.1 and 4.2 below, for the precise relations between the notion of topos and that of topological space), encompassing a large number of situations which formerly were not considered as belonging to topological intuition. The characteristic feature of such situations is that one has in them a notion of “localization”, a notion formalized precisely by the notion of site and, in the last analysis, by that of topos (via the topos associated to the site). Since the term “topos” itself is intended precisely to suggest this, it seems reasonable and legitimate to the authors of the present Seminar to consider that the object of topology is the study of topoi (and not of topological spaces alone).

0.5. It seemed useful to us to include in this general expose on topoi a fairly large number of examples, many of which have only remote

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relations with the initial aim of this seminar (namely the study of etale cohomology). The hurried reader, interested exclusively in etale cohomology, may of course omit without inconvenience the reading of these examples, as also most of the present expose, to which it will suffice to refer as needed.

1. Definition and Characterization of Topoi

Definition 1.1. A $U$-topos, or simply a topos if no confusion is to be feared, is a category $E$ such that there exists a site $C\in U$ such that $E$ is equivalent to the category $\tilde{C}$ of $U$-sheaves of sets on $C$.

1.1.1. Let $E$ be a $U$-topos. We always regard $E$ as endowed with its canonical topology (II 2.5), which therefore makes it a site, and even, by virtue of 1.1.2 d) below, a $U$-site (II 3.0.2). Unless explicitly stated otherwise, we shall consider no topology on $E$ other than the one just specified.

1.1.2. We saw in II 4.8, 4.11 that a $U$-topos $E$ is a $U$-category (I 1.1) satisfying the following conditions:

1. Finite projective limits are representable in $E$.
2. Direct sums indexed by an element of $U$ are representable in $E$. They are disjoint and universal (II 4.5).
3. Equivalence relations in $E$ are universally effective (I 10.6).
4. $E$ admits a generating family (II 4.9) indexed by an element of $U$.

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In fact, we shall see that these intrinsic properties characterize $U$-topoi:

Theorem 1.2 (J. Giraud). Let $E$ be a $U$-category. The following properties are equivalent:

1. $E$ is a $U$-topos (1.1).
2. $E$ satisfies conditions a), b), c), and d) of 1.1.2.
3. The $U$-sheaves on $E$ for the canonical topology are representable, and $E$ possesses a small generating family (condition 1.1.2 d)).
4. There exist a category $C\in U$ and a fully faithful functor $i:E\to \hat{C}$ (where $\hat{C}$ denotes the category of $U$-presheaves on $C$) admitting a left adjoint $a$ which is left exact.
5. There exists a site $C\in U$, such that projective limits are representable in $C$ and the topology of $C$ is less fine than its canonical topology (II 2.5), such that $E$ is equivalent to the category $\tilde{C}$ of $U$-sheaves of sets on $C$.

Moreover:

Corollary 1.2.1. Let $E$ be a $U$-topos, and let $C$ be a full subcategory of $E$, endowed with the topology induced (III 3.1) by that of $E$ (1.1.1). Consider the functor

E -> C~

which associates to every $X\in \mathrm{ob}E$ the restriction to $C$ of the functor represented by $X$. This functor is an equivalence of categories if and only if $\mathrm{ob}C$ is a generating family of $E$.

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The equivalence i) $⟺$ iv) follows immediately from II 5.5, and we have already recalled above that i) $\Rightarrow$ ii). Since i’) $\Rightarrow$ i) trivially, it remains to prove ii) $\Rightarrow$ iii) and iii) $\Rightarrow$ i’), which will be done in 1.2.4 and 1.2.3 below. The corollary is then obtained by observing that the functor under consideration factors as $E\to \tilde{E}\to \tilde{C}$, so that, since $E\to \tilde{E}$ is an equivalence by virtue of (iii), the question reduces to determining when $\tilde{E}\to \tilde{C}$ is an equivalence. One then concludes by the “comparison lemma” III 4.1, cf. proof 1.2.3 below.

1.2.3. Proof of iii) $\Rightarrow$ i’). Let $\mathfrak{X}=(X_i{)}_{i\in I}$, $I\in U$, be a small generating family of $E$. Since $E$ is a $U$-category, the set of isomorphism classes of finite diagrams in $E$ whose objects are elements $X_i$ is $U$-small. Consequently the smallest set ${\mathfrak{X}}^{\prime }$ of objects of $E$, containing the finite projective limits of objects of $\mathfrak{X}$, is a countable union of small sets and is therefore small (*). Setting ${\mathfrak{X}}^{(n+1)}={\mathfrak{X}}^{(n)}(1)$, and $\mathfrak{X}={\bigcup }_n{\mathfrak{X}}^{(n)}$, one sees that, after enlarging the family of generators if necessary, one may suppose that $\mathfrak{X}$ is stable under finite projective limits. Let $V$ be a universe containing $U$ such that $E$ is $V$-small. For every object $H$ of $E$, denote by $I(H)$ the set ${\coprod }_{i\in I}\mathrm{Hom}(X_i,H)$. Let $H$ be an object of $E$ and let $H^{\prime }\subset H$ be the subsheaf of $H$, for the canonical topology, which is the “union” of the images of the morphisms $u:X_i\to H$, $(u,i)\in I(H)$ (II 4.1). Since $H^{\prime }$ is a subsheaf of a $U$-sheaf, $H^{\prime }$ is a $U$-sheaf. It is therefore representable. Since the family $\mathfrak{X}$ is generating and, for every $i\in I$, the map $\mathrm{Hom}(X_i,H^{\prime })\to \mathrm{Hom}(X_i,H)$ is bijective, the morphism $H^{\prime }\subset H$ is an isomorphism (II 4.9). Consequently (II 6.1) the family $(u:X_i\to H)$, $(u,i)\in I(H)$, is covering for the canonical topology of $E$.

(*) We reason here on classes of objects of $E$ up to isomorphism.

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Thus every object of $E$ can be covered, for the canonical topology of $E$, by objects of $\mathfrak{X}$. Let $C$ be the full subcategory of $E$ defined by the objects of $\mathfrak{X}$, and let $u:C\to E$ be the inclusion functor. The topology $\mathfrak{T}$ induced on $C$ by the canonical topology of $E$ is less fine than the canonical topology of $C$, and when $U$ has an element of infinite cardinality, finite projective limits are representable in $C$. It follows from III 5.1 that the functor $F\mapsto F\circ u$ is an equivalence from $E$ onto the category of $U$-sheaves on $C$ for the topology $\mathfrak{T}$, Q.E.D.

1.2.4. Proof of ii) $\Rightarrow$ iii). The proof has four steps.

Let $V$ be a universe containing $U$, such that $E$ is an element of $V$. Let $\tilde{E}$ be the category of $V$-sheaves on $E$ for the canonical topology, and let $J_E:E\to \tilde{E}$ be the canonical functor.

1.2.4.1. Let $(g_i:G_i\to H)$, $i\in I\in U$, be an epimorphic family in $\tilde{E}$. If the $G_i$ and the fiber products $G_i{\times }_H{G}_j$ are representable, then the sheaf $H$ is representable.

Indeed, the direct sum ${\coprod }_{i\in I}{G}_i$ is representable in $\tilde{E}$ (II 4.1) by a representable sheaf $G$ (property b) and II 4.6). The same is true for the direct sum

K = coproduct_{(i, j) in I x I} G_i x_H G_j.

Moreover, the diagram $K\rightrightarrows G\to H$ is exact and $K$ is the fiber square of $G$ over $H$. This last property is true in the category of presheaves, hence true in the category of sheaves (II 4.1). One deduces from c) and II 4.7 that the sheaf $H$ is representable.

1.2.4.2. Let $X_{\alpha }$, $\alpha \in A\in U$, be a family of generators of $E$. For every sheaf $H$, denote by $I(H)$ the set ${\coprod }_{\alpha \in A}{\mathrm{Hom}}_E(X_{\alpha },H)$. The family

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$(u:X_{\alpha }\to H,(u,\alpha )\in I(H))$ is epimorphic in $\tilde{E}$. When $H$ is a $U$-sheaf, $I(H)$ is an element of $U$.

Indeed, since every sheaf $H$ is the target of an epimorphic family of morphisms whose sources are representable sheaves (I 3.4 and II 4.1), it suffices to prove the first assertion when the sheaf $H$ is representable. Let $G$ then be the image of the family $(u:X_{\alpha }\to H,(u,\alpha )\in I(H))$. The morphism $G\to H$ is a monomorphism. Consequently we are in the situation of the first step, because $X_{\alpha }{\times }_G{X}_{\beta }$ is isomorphic to $X_{\alpha }{\times }_H{X}_{\beta }$, which is representable, and moreover $I(H)$ is an element of $U$. It follows that $G$ is representable. But since the $X_{\alpha }$ are a generating family, $G\to H$ is an isomorphism. The last assertion is trivial by definition of $U$-sheaves.

1.2.4.3. Every subsheaf of a representable sheaf is representable.

Indeed, let $G\to H$ be a subsheaf of a representable sheaf. Then $G$ is a $U$-sheaf. The family $(u:X_{\alpha }\to G,(u,\alpha )\in I(G))$ is therefore epimorphic in $\tilde{E}$ and indexed by an element of $U$. Moreover, the fiber products $X_{\alpha }{\times }_G{X}_{\beta }$ are isomorphic to the fiber products $X_{\alpha }{\times }_H{X}_{\beta }$, which are representable. Consequently we are in the situation of the first step and $G$ is representable.

1.2.4.4. Every $U$-sheaf is representable.

Indeed, by virtue of the first and second steps, it suffices to show that the fiber products $X_{\alpha }{\times }_H{X}_{\beta }$ are representable. But these fiber products are subobjects of the products $X_{\alpha }\times X_{\beta }$. We therefore conclude by the third step. This completes the proof of Theorem 1.2.

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Remark 1.3. Of course, for a given $U$-topos $E$, there is in general no privileged way to represent it, up to equivalence, in the form $\tilde{C}$, with $C$ a small site; or, what amounts essentially to the same thing by virtue of 1.2.1 when one restricts to those $C$ whose topology is less fine than the canonical topology, there is no privileged small generating family in $E$. When one stops imposing on $C$ the condition that $C$ be small, however, by virtue of 1.2 iii) there is a completely canonical choice of a $U$-site $C$ such that $E$ is equivalent to $\tilde{C}$, namely $E$ itself! This is one of the technical reasons why it is not convenient in practice to work only with small sites: in fact, the most important sites of all, namely the $U$-topoi, are not small! Moreover, the topos-generating sites which arise in many questions in algebraic geometry (indeed in topology, cf. 2.5) are not small either; example: the etale site of a scheme (VIII 1).

Proposition 1.4. Let $E$ be a $U$-topos, let $E^{\prime }$ be a category (not necessarily a $U$-category), and let $F$ be a presheaf on $E$ with values in $E^{\prime }$. For $F$ to be a sheaf with values in $E^{\prime }$ (II 6.1), it is necessary and sufficient that $F$ transform $U$-inductive limits in $E$ into projective limits in $E^{\prime }$.

Let $V$ be a universe containing $U$ and such that $E^{\prime }$ is a $V$-category. By composing $F$ with the functors $\mathrm{Hom}(X^{\prime },-):E^{\prime }\to V\text{-}\mathrm{Ens}$ defined by the objects $X^{\prime }$ of $E^{\prime }$, one is reduced to the case where $E^{\prime }=V\text{-}\mathrm{Ens}$, where $V$ is a universe.

Suppose that $F$ transforms $U$-inductive limits into projective limits; then it follows immediately from the definitions that $F$ is a sheaf, since a covering

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family $X_i\to X$ in $E$, by definition of the canonical topology of $E$, makes it possible to regard $X$ as an inductive limit of the diagram $(X_i{\times }_X{X}_j\rightrightarrows X_i)$. Suppose that $F$ is a sheaf, and let us prove that it transforms $U$-inductive limits into projective limits. If $V\subset U$, one may suppose $V=U$ and it suffices to apply criterion 1.2 iii). To treat the general case, a little more work is needed.

Let $C$ be a full subcategory of $E$ generated by a generating family indexed by an element of $U$, and let $u:C\to E$ be the inclusion functor. Endow $C$ with the topology induced by the canonical topology of $E$ (III 4.5). Let ${\tilde{C}}_U$, ${\tilde{C}}_V$ denote the categories of sheaves on $C$, and let ${\tilde{E}}_V$ denote the category of $V$-sheaves on $E$ for the canonical topology; let $J_E:E\to {\tilde{E}}_V$ be the canonical functor, and let $i_{U,V}:{\tilde{C}}_U\to {\tilde{C}}_V$ be the inclusion functor. The diagram

C~_U  --i_{U,V}-->  C~_V
  ^                  ^
  |                  |
  E   ----J_E----->  E~_V

where the vertical arrows are induced by the functor $F\mapsto F\circ u$, is commutative. It follows from the explicit construction of the associated sheaf functor (II 3) and from II 4.1 that the functor $i_{U,V}$ commutes with $U$-inductive limits. Moreover, it follows from III 5.1 and 1.5 that the vertical arrows in (*) are equivalences of categories. Consequently $J_E:E\to {\tilde{E}}_V$ commutes with $U$-inductive limits. For every object $X$ of $E$, one has:

F(X) ~= Hom_{E~_V}(J_E(X), F).

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Therefore $F$ transforms $U$-inductive limits of $E$ into projective limits.

Corollary 1.5. Let $E$ be a $U$-topos, let $E^{\prime }$ be a $U$-category, and let $f:E\to E^{\prime }$ be a functor. For $f$ to admit a right adjoint, it is necessary and sufficient that $f$ commute with $U$-inductive limits.

This is an immediate consequence of 1.4 for the case $E^{\prime }=U\text{-}\mathrm{Ens}$.

Corollary 1.6. Let $E$, $E^{\prime }$ be two $U$-topoi, and let $f:E\to E^{\prime }$ be a functor. The following conditions are equivalent:

1. $f$ commutes with $U$-inductive limits.
2. $f$ admits a right adjoint.
3. $f$ is continuous (III 1.1).

The equivalence i) $⟺$ ii) was seen in 1.5. To prove i) $⟺$ iii), apply the definition of continuous functors, choosing a universe $V$ such that $U\in V$ (so $E$, $E^{\prime }$ are $V$-small sites). One must express that for every $V$-sheaf $F$ on $E^{\prime }$, the composite $F\circ f$ is a sheaf on $E$, i.e. (1.4) that it transforms $U$-inductive limits into projective limits. It is sufficient for this that i) hold, since by virtue of 1.4 $F$ itself transforms $U$-inductive limits into projective limits; it is also necessary, as one sees by taking $F$ to be a representable functor.

Corollary 1.7. With the notation of 1.6, for $f$ to be the inverse image functor $u^{\ast }$ for a morphism of topoi (3.1) $u:E^{\prime }\to E$, it is necessary and sufficient that $f$ be left exact and transform epimorphic families into epimorphic families.

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Necessity is trivial by definition (NB every right exact functor transforms epimorphisms into epimorphisms). Sufficiency follows from III 2.6, which implies that $f$ is continuous under the indicated conditions, hence commutes with $U$-inductive limits by virtue of 1.6.

Remark 1.8. With the notation of 1.5, one can show that $f$ admits a left adjoint if and only if it commutes with $U$-projective limits. In other words, a covariant functor $E\to (U\text{-}\mathrm{Ens})$ is representable if and only if it commutes with $U$-projective limits. A more general statement is found in I 8.12.8, 8.12.9; the sketch of proof in a) to c) corresponds to the proof given in loc. cit. We indicate only the principle of the proof:

1. Standard arguments [5, No. 195, § 3] show that if $F$ commutes with projective limits, it is pro-representable by a strict projective system $(T_i{)}_{i\in I}$, where $I$ is a filtered ordered set, not necessarily small. One may suppose that if $i>j$, then $T_i\to T_j$ is not an isomorphism, and under this hypothesis $F$ is representable if and only if $I$ is small (which in fact implies that the projective system is essentially constant).
2. To prove that $I$ is small, knowing that for every object $X$ of $E$ the set $F(X)={\underleftarrow{\lim }}_i\mathrm{Hom}(T_i,X)$ is small, it suffices to have a small cogenerating family $(X_j{)}_{j\in J}$ (i.e. one which is generating for the opposite category $E^{\circ }$). But one shows that in a $U$-topos $E$ there always exists a small cogenerating family.
3. To prove this last point, one notes by standard arguments [Toh] that every object $X$ of $E$ admits a monomorphism into an “injective” object; then, for any generating family $(L_{\alpha })$ of $E$,

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if one embeds each $L_{\alpha }$ in this way into an injective object $I_{\alpha }$, the family $(I_{\alpha })$ is cogenerating.

2. Examples of Topoi

2.0. We have collected in this number a fairly large number of typical examples of topoi, which the reader will already have had occasion to meet elsewhere, and which are intended to make access to the “yoga” of topoi easier. For other examples (drawn from algebraic geometry) of topologies on sites, giving rise to as many topoi, the reader may consult SGA 3 IV 6 and (for the etale topos) Expose VII of the present seminar. Since in what follows we shall refer to the present number hardly at all except for questions of notation or terminology, we leave to the reader, as an exercise, the verification of the statements with which we have accompanied these examples for general instruction. All the examples of the present number will be made precise in 4, where their functorial dependence on the data will be examined, and in the following numbers as illustrations of the general notions relative to topoi.

2.1. Topos Associated to a Topological Space

Let $X$ be a small topological space, and let $\mathrm{Ouv}(X)$ be the category of open subsets of $X$, endowed with the canonical topology (III 2.9.1). We shall denote by $\mathrm{Top}(X)$ the topos of $U$-sheaves on $\mathrm{Ouv}(X)$. This topos is equivalent to the category of etale topological spaces over $X$, by associating to every such space $X^{\prime }$ over $X$ the sheaf $U\mapsto \Gamma (X^{\prime }/U)={\mathrm{Hom}}_X(U,X^{\prime })$ on $\mathrm{Ouv}(X)$ [TF]. We shall not be able to keep ourselves from sometimes denoting, by abuse

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of language, by the same letter $X$ the topos $\mathrm{Top}(X)$ defined by the topological space $X$.

It is of course the preceding example which served principally as guide and intuitive support for the development of the theory of topoi. One should be careful, however, that the topoi deduced from topological spaces are of a very special nature, due in particular to the fact that they have been described by sites $C=\mathrm{Ouv}(X)$ in which all morphisms are monomorphisms (hence whose underlying category reduces to a preordered set). It follows in particular that the sheaves represented by the objects of $C$ are subsheaves of the final sheaf, and consequently that the subsheaves of the final sheaf form a generating family of the topos under consideration. This property is not shared by most of the topoi which arise naturally in algebraic geometry or algebra, cf. examples below. It is, up to very little, characteristic of topoi of the form $\mathrm{Top}(X)$ (cf. 7.1.9 below).

One verifies easily that the map $\mathrm{Ouv}(X)\to \mathrm{Top}(X)$, which associates to every open subset of $X$ the sheaf it represents, is a bijection from $\mathrm{Ouv}(X)$ onto the set of subobjects of the final object of $\mathrm{Top}(X)$, this bijection even being an isomorphism for the natural order structures, i.e. inducing an isomorphism of the corresponding categories. This suggests that it should be possible to reconstruct, up to homeomorphism, the topological space $X$, when one knows $\mathrm{Top}(X)$ up to equivalence. We shall see below (4.2) that this is indeed so, subject to a slight restriction on $X$.

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2.2. Punctual or Final Topos, and Empty or Initial Topos

When $X$ is a topological space reduced to a single point, the functor

F |-> F(X) : Top(X) -> U-Ens

is an equivalence of categories. This shows in particular that the category $U\text{-}\mathrm{Ens}$ is a $U$-topos. We saw in examples in Exp. II that this $U$-topos is typical from the point of view of exactness properties, in that the verification of many properties (notably exactness properties) of general topoi reduces to this particular topos. The interpretation we give here in terms of the punctual topological space justifies the abuse of language consisting in calling a topos equivalent to the category $U\text{-}\mathrm{Ens}$ a punctual topos (although, as a category, it is not at all equivalent to the punctual category $1$). This is the terminology corresponding to the correct geometric intuition of the role played by these topoi. Sometimes, also by abuse of language, a punctual topos is called a final topos, cf. 4.3; we shall say “the final topos” for the topos $(U\text{-}\mathrm{Ens})$.

When $X$ is reduced to the empty topological space, and hence $\mathrm{Ouv}(X)$ to the punctual category, then a presheaf $F$ on $\mathrm{Ouv}(X)$ is a sheaf if and only if its value at the unique object $\varnothing$ of $\mathrm{Ouv}(X)$ is a one-point set. It follows that $\mathrm{Top}(\varnothing )$ is isomorphic to the category of one-point $U$-sets, a category equivalent to the punctual category. From this one concludes in particular that the punctual category (as well as every $U$-category equivalent to it) is a $U$-topos. It is sometimes called, by abuse

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of language, the empty topos or initial topos (cf. 4.4); one should take care that it is not equivalent to the empty category.

2.3. Topos Associated to a Space with Operators

Let $X$ be a topological space, and let $G$ be a discrete group acting on $X$ by homeomorphisms. In [Toh. 5.1] one then defined the category of $G$-sheaves on $X$, or, as we shall also say, of sheaves on $(X,G)$; these are the sheaves (of sets) on $X$, equipped with operations of $G$ compatible with those of $G$ on $X$. One immediately sees, using Giraud’s criterion 1.2 iii), that this category is a topos (NB $U$ is understood throughout here), which we shall simply denote by $\mathrm{Top}(X,G)$.

When $G$ is reduced to the unit group, one recovers Example 2.1; when $X$ is reduced to the punctual space, one finds the topos of sets on which $G$ acts on the left (or $G$-sets), also called the classifying topos of the discrete group $G$, and denoted $B_G$. One verifies easily that the only subobject of the final object $e$ of the topos $B_G$ is $e$ or the empty sheaf $\varnothing$; in particular, if $G$ is not reduced to the unit group, the subobjects of the final object of the classifying topos $B_G$ do not form a generating family of $B_G$. Thus $B_G$ is then not equivalent to a topos of the type $\mathrm{Top}(X)$ considered in 2.1.

The notion of $G$-sheaf was introduced in loc. cit. to develop the cohomological theory of abelian $G$-sheaves. Interpreting these as the abelian sheaves of the topos $(X,G)$, that theory is included in Exp. V, developed in the framework of general topoi.

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More generally, one may try to attach an appropriate topos to a topological space $X$, equipped with a topological group $G$ (not necessarily discrete) of automorphisms, which would give rise to an adequate cohomological theory. Similarly in the context of differentiable varieties, or real or complex analytic varieties, or schemes. This is indeed possible, cf. 2.5 below.

2.4. Classifying Topos of a Group

Let $E$ be a topos, and let $G$ be a group of $E$. Let $(E,G)$ be the category of objects of $E$ on which $G$ acts. By Giraud’s criterion, one immediately sees that it is a topos. It is called the classifying topos of the group $G$, and is denoted $B_G$. When $E$ is the punctual topos (2.2), i.e. when $G$ is an ordinary group, one recovers the classifying topos of 2.3.

The terminology adopted here is justified by the fact that the topos $B_G$ plays a universal role for the classification of “torsors” (or homogeneous principal bundles) under $G$, or more generally under the $G_{E^{\prime }}=f^{\ast }(G)$, where $E^{\prime }$ is a topos “over $E$”, i.e. equipped with a morphism $f:E^{\prime }\to E$ (cf. 3.1 below). This role, made explicit in [3, Chap. V] or in 5.9 below, shows that $B_G$ plays, in the context of topoi, the same role as the classical classifying spaces of topological groups in the homotopy theory of topological spaces. The latter may be regarded (cf. 2.5) as a weakened version of the former, obtained by retaining from the classifying topos only the “homotopy type” of that topos, in a suitable sense that need not be specified here.

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2.5. “Big Site” and “Big Topos” of a Topological Space. Classifying Topos of a Topological Group (*)

Let $U\text{-}\mathrm{Esp}$, or simply $(\mathrm{Esp})$, be the category of topological spaces $\in U$. We know that in $(\mathrm{Esp})$ finite projective limits are representable. Consider on $(\mathrm{Esp})$ the pretopology (I 1.3) for which $\mathrm{Cov}(X)$ is the set of surjective families of open immersions $u_i:X_i\to X$. We shall regard $(\mathrm{Esp})$ as a site by means of the topology generated by the preceding pretopology. For every object $X$ of $(\mathrm{Esp})$, consider the category

(Esp)/X

of objects of $(\mathrm{Esp})$ over $X$, i.e. of topological spaces over $X$, as a site, by means of the topology induced by that of $(\mathrm{Esp})$ via the forgetful functor $(\mathrm{Esp})/X\to (\mathrm{Esp})$ (III 5.2 4)). This site is called the big site associated to $X$. One should be careful that it is not $\in U$; it is not a $U$-site in the sense of II 3.0.2 either, so precautions are necessary in order to apply the usual results to it. To remedy this inconvenience, one may choose a universe $V$ such that $U\in V$, so that $(\mathrm{Esp})/X$ becomes a $V$-site, and one may work with the associated $V$-topos $((\mathrm{Esp})/X{)}^{\sim }$, which may be denoted $\mathrm{TOP}(X)$ and will be called the big topos of $X$. If one is reluctant to enlarge $U$, one may choose a cardinal $c$ bounding the cardinals of $X$ and of all the topological spaces one expects to involve in the arguments (most often, $\mathrm{Sup}(\mathrm{card}X,\mathrm{card}R)$ will suffice!), and replace $(\mathrm{Esp})/X$ by the subcategory $(\mathrm{Esp}{)}^{\prime }/X$ formed by the $X^{\prime }$ over $X$ such that $\mathrm{card}{X}^{\prime }\le c$, endowed with the induced topology, and denote by $\mathrm{TOP}(X)$ the topos of sheaves on

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this site. To fix ideas, suppose that the first definition has been adopted.

(*) The introduction of these sites and topoi is due to M. Giraud, who also brought to light their advantages over the traditional “small” site.

The advantage of the big topos of $X$ over the small one is that the site defining it contains $(\mathrm{Esp})/X$ as a full subcategory; since the topology of this site is manifestly less fine than the canonical topology, one sees that the canonical functor from $(\mathrm{Esp})/X$ to $\mathrm{TOP}(X)$, associating to every space $X^{\prime }$ over $X$ the sheaf it represents, is fully faithful. Consequently, a space $X^{\prime }$ over $X$ is known up to $X$-isomorphism when one knows the sheaf ($\in \mathrm{TOP}(X)$) it defines; thus the notion of sheaf on (the big site of) $X$ may be regarded as a generalization of the notion of topological space over $X$, by means of which all constructions of sheaf theory take on a meaning for topological spaces over $X$.

Thus, when $G$ is a group object of the category $(\mathrm{Esp})/X$ of topological spaces over $X$, one can associate to it the classifying topos $B_G$ (2.4), hence classifying cohomology groups, classifying homotopy groups, etc. (defined as the corresponding invariants of the $V$-topos $B_G$). In particular, when $X$ is a punctual space, $G$ is identified with an ordinary topological group. One can verify, subject to the usual local conditions ensuring that the singular cohomology of the cartesian products $G^n$ coincides with sheaf cohomology (for constant coefficients, say), for example if $G$ is locally contractible, that the cohomology of the classifying topos of $G$ is canonically isomorphic to that of the classifying space of $G$ in the sense of topologists.

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The introduction of classifying topoi (via the “big sites”) has the advantage over classifying spaces of providing a richer theory, since in particular they provide useful cohomological invariants for coefficients more general than constant or locally constant coefficients. Moreover, the definition considered here adapts in an evident way to the other usual contexts: differentiable varieties, real or complex analytic varieties or spaces, schemes. This point of view makes it possible in particular to connect the study of characteristic classes from the traditional point of view and from the “arithmetic” point of view, by considering the “classical groups” as coming from schemes defined over the ring of integers; cf. [7] for indications in this direction. Similarly, J. Giraud’s general results [3] on the classification of extensions of groups, developed in the very general and very flexible framework of topoi, can, thanks to the “big topoi”, be specialized to results on the classification of extensions of topological groups, or of real or complex Lie groups, which seem to have been hardly known to topologists except in the case of extensions with abelian kernel [11].

2.6. Topoi of the Form $\hat{C}$

Let $C$ be a small category. Then the category $\hat{C}$ of $U$-presheaves on $C$ is evidently a $U$-topos, since it is of the form $C^{\wedge }$, where $C$ is endowed with the chaotic topology. We shall give below a few details on the relations between $C$ and $\hat{C}$. Let us only note here that a topos $E$ equivalent to a topos of the form $\hat{C}$ is of a rather special nature, because it admits a small generating family $(X_i)$ formed of connected projective objects, i.e. objects $X$ such that the functor

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Y |-> Hom(X, Y)

transforms epimorphisms into epimorphisms and sums into sums: indeed it suffices to take in $\hat{C}$ the generating family formed by the functors represented by the $X\in \mathrm{ob}C$. Note moreover that if in a topos $E$ one has a covering family $X_i\to X$, with the $X_i$ projective and connected, then every other covering family of $X$ is majorized (I 4.3.2, 4.3.3) by the preceding one. Consequently, in a topos $E$ of the form $\hat{C}$ every object $X$ admits a covering family majorizing all the others. A topos of the form $\mathrm{Top}(X)$ (2.1), with $X$ a topological space whose points are closed, has the preceding property only if $X$ is discrete.

When the category $C$ has a single object, $C$ is identified with a monoid $G$. A presheaf on $C$ is then identified with a set on which $G$ acts on the right (since it is a functor $G^{\circ }\to (\mathrm{Ens})$), and $\hat{C}$ is the topos of sets with a monoid of right operators, which may also be denoted $B_{G^{\circ }}$. Taking 2.3 into account: this is the topos of sets with monoid of operators $G^{\circ }$ (the monoid opposite to $G$). When $G$ is a group, using the isomorphism $g\mapsto g^{-1}$ from $G$ onto $G^{\circ }$, one recovers the classifying topos $B_G$ of 2.3.

2.7. Classifying Topos of a Pro-Group

2.7.1. Let $G=(G_i{)}_{i\in I}$ be a projective system of groups, with $G_i\in U$. Suppose the projective system is strict, i.e. the transition morphisms $G_j\to G_i$ are surjective. If $E$ is a set, an operation of $G$ on $E$ (on the left, say) means the following structure:

1. a family $(E_i{)}_{i\in I}$ of subsets of $E$, with union $E$;
2. for every $i\in I$, an action of the group $G_i$ on the set $E_i$.

These data are moreover supposed subject

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to the following condition: for $j\ge i$, $E_i$ is the subset of $E_j$ formed by the elements fixed under the kernel group of $G_j\to G_i$. We also say that $G$ acts on $E$ (on the left) if an action of $G$ on $E$ (on the left) has been given. The sets $\in U$ equipped with an action of $G$ form a category in the evident way. By Giraud’s criterion, one immediately sees that this category is a $U$-topos. It is denoted $B_G$, and is called the classifying topos of $G$. When $I$ admits an initial object $i_0$, setting $G=G_{i_0}$, one recovers the classifying topos of 2.3.

2.7.2. Another important example is the one where the groups $G_i$ are finite, so that

G = lim<-_i G_i

is a compact totally disconnected topological group, or profinite group. An action of $G$ on $E$ then amounts to an action of $G$ on $E$ which is continuous, or equivalently, such that the stabilizer of every point of $E$ is an open subgroup of $G$. The classifying topos of the topological group $G$ will also be denoted $B_G$, where, of course, $G$ must be regarded as endowed with its profinite topology.

2.7.3. It is easy to verify, using the remarks of 2.6, that the topos $B_G$ defined by a strict projective system $G=(G_i{)}_{i\in I}$ of groups is equivalent to a topos of the form $\hat{C}$ only if this projective system is essentially constant; in the case of a profinite group, this means that this group is in fact finite.

2.7.4. The following geometric interpretation of the classifying topos $B_G$ of a discrete group $G$ is useful for giving correct geometric intuition about these topoi. (Cf. also, in the same direction, 4.5, 5.8, 5.9, and 7.2 below.) Let $X$ be a connected, locally connected

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and locally simply connected topological space, let $x$ be a point of $X$, and let $G$ be its fundamental group at $x$. (NB it is known that, up to isomorphism, every discrete group $G$ can be obtained in this way.) Then Galois theory of covering spaces of $X$ gives an equivalence between the category $B_G$ of $G$-sets and the category of etale coverings of $X$, i.e. of spaces $X^{\prime }$ over $X$ which are locally $X$-isomorphic to $X$-spaces of the form $X\times I$, where $I$ is a discrete space. (Compare SGA 1 V 4.5.) This latter category may moreover be interpreted as the category of locally constant sheaves on $X$, i.e. the category of locally constant objects (IX 2.0) of the topos $\mathrm{Top}(X)$.

When $G$ is a profinite group, there is an analogous geometric interpretation of $B_G$ as the category of $X$-schemes which are sums of finite etale coverings of a connected scheme $X$, equipped with a geometric point $x$ and with an isomorphism $G\simeq {\pi }_1(X,x)$. It is again known that every profinite group can be obtained as the fundamental group of a suitable connected scheme (the spectrum of a field, if desired). Finally, one also encounters pro-groups (not necessarily profinite nor essentially constant) for the classification of coverings of connected and locally connected spaces which are not locally simply connected, or the classification of etale coverings, not necessarily finite or ind-finite, of nonnormal connected schemes. For this last case, cf. SGA 3 X 6.

Exercise 2.7.5. Define for a topos $E$ the notions of connectedness, local connectedness (*), simple connectedness, and local simple connectedness. Define the notion of a constant object and of a locally constant object of $E$ (cf. IX 2.0). Let

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$f:E^{\prime }\to E$ be a morphism of topoi (3.1), with $E^{\prime }$ simply connected (for example $E^{\prime }$ the punctual topos (2.2)), and $E$ connected and locally connected. Define a strict pro-group ${\pi }_1(E,f)=(G_i{)}_{i\in I}=G$ (called the fundamental pro-group of $E$ at $f$) and an equivalence of categories from $B_G$ to the category of locally constant objects of $E$. (*) Show that when $E$ is locally simply connected, ${\pi }_1(E,f)$ is essentially constant and is therefore identified with an ordinary discrete group ${\pi }_1(E,f)$, called the fundamental group of $E$ at $f$. Show that every strict pro-group $G$ is isomorphic (as a pro-group) to the fundamental pro-group of a suitable connected and locally connected topos at a suitable $f$, with $E^{\prime }$ the punctual topos (take $E=B_G$, and $f:E^{\prime }\to E$ defined by the forgetful functor $f^{\ast }:E\to E^{\prime }=(\mathrm{Ens})$). When $G$ is essentially constant, i.e. isomorphic (as a pro-group) to an ordinary discrete group, prove that one can take $E$ above to be locally simply connected (again take $E=B_G$).

(*) One may draw inspiration from SGA 3 X 6.

2.8. Example of a False Topos

Let $G=(G_i{)}_{i\in I}$ be a strict pro-group, where $I$ is filtered ordered and where $i>j$ implies that $G_i\to G_j$ is not an isomorphism. Suppose that $\mathrm{card}(I)\notin U$. Consider the category of sets $E\in U$ on which $G$ acts on the left (2.7.1). It is a $U$-category, and one sees as in 2.7.1 that this category satisfies conditions a), b), c) of 1.1.2. However it is not a $U$-topos, because one sees that it admits no generating family which is $U$-small. Similarly one sees that it is not a $V$-topos for any universe $V$.

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3. Morphisms of Topoi

Definition 3.1. Let $E$ and $E^{\prime }$ be two $U$-topoi. A morphism from $E$ to $E^{\prime }$, or sometimes (by abuse of language) a continuous map from $E$ to $E^{\prime }$, is a triple $u=(u_{\ast },u^{\ast },\phi )$, formed by functors

u_* : E -> E',        u^* : E' -> E

and by an “adjunction” isomorphism $\phi$ of bifunctors in $X^{\prime }\in \mathrm{Ob}{E}^{\prime }$, $Y\in \mathrm{Ob}E$:

phi : Hom_E(u^*(X'), Y) ~= Hom_{E'}(X', u_*(Y)),

the functor $u^{\ast }$ being moreover subject to the condition of being left exact, i.e. of commuting with finite projective limits. The functor $u_{\ast }$ is called the direct image functor for the morphism of topoi $u$, the functor $u^{\ast }$ is called the inverse image functor for the morphism of topoi $u$, and the isomorphism $\phi$ is called the adjunction isomorphism for $u$.

3.1.1. Unless explicitly stated otherwise, for a morphism of topoi $u:E\to E^{\prime }$ we shall subsequently denote by $u_{\ast }$ and $u^{\ast }$ the corresponding direct image and inverse image functors (*). Note that, since $u_{\ast }$ is right adjoint to $u^{\ast }$ and $u^{\ast }$ is left adjoint to $u_{\ast }$ by the adjunction isomorphism $\phi$, each of the two functors $u_{\ast }$, $u^{\ast }$ determines the other up to unique isomorphism, by the well-known sorites of adjoint functors [14]. In practice, depending on the case, it may be more convenient to define a morphism of topoi $u:E\to E^{\prime }$ either by giving $u^{\ast }:E^{\prime }\to E$, or by giving $u_{\ast }:E\to E^{\prime }$; in the first case, one must simply verify that the given functor $u^{\ast }$

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admits a right adjoint and is left exact. In the second, that the given functor $u_{\ast }$ admits a left adjoint which is left exact. In either case, from the partial datum, by choosing an adjoint functor and an adjunction isomorphism, one deduces a morphism of topoi $u:E\to E^{\prime }$, and this latter will be “unique up to unique isomorphism” in terms of the datum $u^{\ast }$ resp. $u_{\ast }$, in a fairly clear sense, which will moreover be made entirely explicit below (3.2.1).

(*) Sometimes one should also write $u^{-1}$ instead of $u^{\ast }$.

3.1.2. If $u:E\to E^{\prime }$ is a morphism of topoi, it follows from the properties of adjoint functors (I 2.11) that the direct image functor $u_{\ast }:E\to E^{\prime }$ commutes with projective limits, and the functor $u^{\ast }:E^{\prime }\to E$ commutes with inductive limits (+). Since this latter is moreover supposed to be left exact, i.e. to commute with finite projective limits, one sees in particular that $u^{\ast }$ is exact. Thus one may say that it is the inverse image functor $u^{\ast }$, in the pair $(u_{\ast },u^{\ast })$, which has the most remarkable exactness properties. These properties ensure that for every species of algebraic structure $\Sigma$ whose data can be described in terms of data of arrows between the base sets and sets deduced from these by repeated application of operations of finite projective limits and arbitrary inductive limits, and for every “object of $E^{\prime }$ equipped with a structure of species $\Sigma$” (a notion which makes sense thanks to the internal exactness properties of the topos $E^{\prime }$ (II 4.1)), its image by $u^{\ast }$ is equipped with the same structures. Rather than undertaking the unappealing task of giving a precise meaning to this statement and

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justifying it formally, we advise the reader to make it explicit and convince himself of its validity for species of structure such as that of group, ring, module over a ring, comodule over a ring, bialgebra over a ring, torsor under a group. (In these examples, the first three species of structure are defined exclusively in terms of finite projective limits, while the other notions implicitly involve constructions also appealing to inductive limits.) Moreover, the functor $u^{\ast }$ “commutes” with all the usual functorial operations made in terms of such structures, more precisely with all operations that can be expressed in terms of $\underrightarrow{\lim }$ and finite $\underleftarrow{\lim }$: constructions of free objects (free groups or modules, for example) generated by an object, tensor products (cf. § 12 below), etc.

(+) Moreover (1.5 and 1.6), for a given functor $u_{\ast }:E\to E^{\prime }$ resp. $u^{\ast }:E^{\prime }\to E$, this functor admits a left adjoint (resp. right adjoint) if and only if it commutes with $U$-projective limits (resp. $U$-inductive limits).

As for the direct image functor $u_{\ast }:E\to E^{\prime }$, which commutes with projective limits, it consequently “respects” every algebraic structure on an object (or a family of objects) of $E$ definable exclusively in terms of projective limits (such as structures of group, ring, or module over a ring, among the preceding examples). On the other hand, the functor $u_{\ast }$ is in general not right exact, i.e. it does not in general commute with finite inductive limits, and it does not even in general transform epimorphisms into epimorphisms (indeed this lack of exactness of the functor $u_{\ast }$ is the source of its cohomological properties, which will be studied (from the point of view of commutative homological algebra) in the following expose). Consequently, it does not extend, in general, to a functor on objects of the type comodule, or bialgebra, or torsor under a group, and it does not in general commute with operations such as “free generated module”, tensor product of modules, etc.

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3.1.3. In practice, when one has a functor $f:E\to F$ from one $U$-topos to another, one should make explicit all its exactness properties, including the possible existence of left or right adjoint functors (cf. footnote 26), in order to arrive at an understanding of the “geometric nature” of $f$, an understanding that will generally be an indispensable guide for correct geometric intuition about the situation. Thus, if it turns out that $f$ commutes with arbitrary inductive limits and finite projective limits, one should write $f$ in the form

f = u^*,

where

u : F -> E

is a morphism of topoi, i.e. interpret $f$ as an “inverse image” functor by a “continuous map” of topoi. When $f$ commutes with arbitrary projective limits, hence admits a left adjoint, and if this latter (which a priori commutes with arbitrary inductive limits) moreover commutes with finite projective limits, one should write $f$ in the form

f = v_*,

where

v : E -> F

is a morphism of topoi. In some cases, it may happen that $f$

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satisfies both of the two properties considered (cf. 4.10 for an example). In this case one should introduce simultaneously the morphisms of topoi

u : F -> E      and      v : E -> F,

which give rise to a sequence of three adjoint functors (I 5.3):

v^*,  v_* = u^* = f,  u_*.

One should then carefully distinguish between the morphisms of topoi $u$ and $v$, under penalty of losing the geometric intuition of the situation.

In this connection, note that when between two topoi $E$, $F$ one has a sequence of three adjoint functors

e, f, g      (e, g : F -> E,  f : E -> F),

then $f$ commutes with arbitrary inductive and projective limits, so it can always be put in the form $u^{\ast }$, where $u:F\to E$ is a morphism of topoi. Then $g$ is therefore written $g=u_{\ast }$. On the other hand, of course, $e$ can be written in the form $v^{\ast }$ (and then $f$ in the form $v_{\ast }$) only if it moreover commutes with finite projective limits. This will obviously be the case if it is itself the right adjoint of a fourth functor $d$. Without a condition of this nature, one often writes:

e = u_!

for the left adjoint of an inverse image functor $f=u^{\ast }$, when such a left adjoint exists, this notation being suggested by the example of an open immersion $u:X\to Y$ of topological spaces. Similarly, if $e$ is of the form $v^{\ast }$, i.e. $f$ of the form $v_{\ast }$, one sometimes writes $g=v^{!}$ for the right adjoint of a direct image functor $v_{\ast }$, when this adjoint exists (*).

(*) Cf. below, in the case of abelian sheaves.

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3.2. Let $E$, $E^{\prime }$ be two $U$-topoi, and let

u = (u_*, u^*, phi),      v = (v_*, v^*, psi) : E => E'

be two morphisms of topoi from $E$ to $E^{\prime }$. A morphism from $u$ to $v$ means any morphism from $u_{\ast }$ to $v_{\ast }$ (in the sense of the category $\mathrm{Hom}(E,E^{\prime })$ of functors from $E$ to $E^{\prime }$). Morphisms of morphisms of topoi compose in the evident way, and one thus defines a category, which is in fact a $U$-category (I 7.8), denoted

Homtop(E, E'),

called the category of morphisms (or continuous maps) from $E$ to $E^{\prime }$. One then defines in the evident way a functor

u |-> u_* : Homtop(E, E') -> Hom(E, E'),

a functor which is fully faithful by definition of the arrows in the source (but is not injective on objects).

3.2.1. One should be careful that, if $u$ and $v$ are given as above, the theory of adjoint functors supplies a canonical bijection

Hom(u_*, v_*) ~= Hom(v^*, u^*) ;

in particular, one obtains a contrafunctor on $\mathrm{Homtop}(E,E^{\prime })$:

u |-> u^* : Homtop(E, E')^circ -> Hom(E', E).

In accordance with the geometric intuition according to which the direct image functor $u_{\ast }$ “goes in the same direction” as the continuous map giving rise to it, one should therefore define the direction of arrows for morphisms between morphisms of topoi in terms of direct image functors, and not in terms of inverse image functors (although it is the latter which, as we have seen, possess the exactness properties characteristic of the notion of morphism of topoi).

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3.2.2. Having defined the category $\mathrm{Homtop}(E,E^{\prime })$ of morphisms from the topos $E$ to the topos $E^{\prime }$, the notion of isomorphism between two morphisms $u$, $v$ from $E$ to $E^{\prime }$ is also defined. In practice, there is no need to distinguish essentially between two isomorphic morphisms of topoi, at least when one has a canonical isomorphism between the two (just as there is often no need to distinguish between two elements of a category when one is given a canonical isomorphism between them).

Let us point out in this connection that most often, when one treats diagrams of morphisms of topoi and questions of commutativity of such diagrams (a notion which makes sense thanks to (3.3.)), what is meant is only commutativity up to (“canonical”) isomorphism; by abuse of language, one then treats these diagrams as actually commutative diagrams.

One might think of justifying this abuse of language by introducing the set $\mathrm{Homtop}(E,E^{\prime })/\mathrm{Isom}$ of morphisms from $E$ to $E^{\prime }$ up to isomorphism, and calling such an isomorphism class a morphism, instead of following Definition 3.2. But this runs into the very serious drawbacks that arise every time one tries to identify two isomorphic objects of a category without having a canonical isomorphism between them. Experience proves that such a point of view is impracticable, and that one must keep the “fine” notion 3.1 and the notion of morphism between morphisms of topoi, even if one is sometimes forced to fight with compatibilities among canonical isomorphisms (*).

(*) For examples of such battles (victorious, it seems), we refer the reader to Mme M. Hakim’s book on relative schemes [9].

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3.3.1. Let $E$, $E^{\prime }$, $E^{\prime \prime }$ be three $U$-topoi, and consider morphisms of topoi

u : E -> E',      v : E' -> E''.

The theory of adjoint functors then gives us an adjunction isomorphism between the composite functors $v_{\ast }{u}_{\ast }$ and $u^{\ast }{v}^{\ast }$, in terms of the adjunction isomorphisms for the pairs $(u_{\ast },u^{\ast })$ and $(v_{\ast },v^{\ast })$. On the other hand, the functor $u^{\ast }{v}^{\ast }$ is left exact, as a composite of two left exact functors. Consequently, one obtains a morphism from $E$ to $E^{\prime \prime }$, called the composite of the morphisms $u$ and $v$, and denoted $vu$:

vu : E -> E''.

One then verifies trivially that composition of morphisms is associative, and that for every $U$-topos $E$ there is a morphism from $E$ to itself which is a two-sided unit for composition: it is the morphism $({\mathrm{id}}_E,{\mathrm{id}}_E,\phi )$, where $\phi$ is the evident adjunction isomorphism of ${\mathrm{id}}_E$ with itself. Let $V$ be a universe such that $U\in V$. One defines a category

(V-U-Top),

whose objects are the $U$-topoi which are $\in V$, whose arrows are the morphisms between such $U$-topoi, and whose composition of arrows is the one just made explicit.

3.3.2. In fact, the composition map

Homtop(E, E') x Homtop(E', E'') -> Homtop(E, E'')

is the map induced on objects by a “composition of morphisms” functor:

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Homtop(E, E') x Homtop(E', E'') -> Homtop(E, E''),

whose effect on arrows is the usual “convolution product” operation for morphisms between functors (here direct image functors). These composition functors satisfy a (strict) associativity property, for four topoi $E$, $E^{\prime }$, $E^{\prime \prime }$, $E^{\prime \prime \prime }$, making precise the associativity of composition of morphisms of topoi. One may also say, in language that is beginning to become familiar [2] [9], that the $U$-topoi are the objects (or $0$-arrows) of a 2-category, whose $1$-arrows are morphisms of topoi and whose $2$-arrows are morphisms of morphisms of topoi.

The fact that the $U$-topoi (elements of a universe $V$) form a 2-category, and no longer merely an ordinary category like ordinary topological spaces, is from the technical point of view the most important difference between the theory of topoi and that of topological spaces. This fact is the source of certain technical complications to which we have already alluded, but also of facts essentially new in relation to traditional topology.

3.4. The fact that the $U$-topoi (elements of a universe $V$) form a 2-category (3.3.2) makes it possible in particular to define the notion of equivalence of two $U$-topoi $E$, $E^{\prime }$: we shall say that $E$ and $E^{\prime }$ are equivalent if there exist morphisms of topoi $u:E\to E^{\prime }$ and $v:E^{\prime }\to E$ such that the composites $uv$ and $vu$ are isomorphic, respectively, to the identity morphism of $E$ and of $E^{\prime }$; one then says that the morphisms $u$ and $v$ are

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quasi-inverse equivalences of one another.

One immediately sees that for the morphism of topoi $u:E\to E^{\prime }$ to be an equivalence, it is necessary and sufficient that $u_{\ast }$ be an equivalence, or equivalently, that $u^{\ast }$ be an equivalence. (Use the fact that a functor $f:E\to E^{\prime }$ between two topoi which is an equivalence is both of the form $u_{\ast }$ and of the form $v^{\ast }$, with $u$ and $v$ morphisms of topoi); and for $E$ and $E^{\prime }$ to be equivalent in the sense of the preceding paragraph, it is necessary and sufficient that they be equivalent as categories (i.e. as objects of the 2-category $(V\text{-}\mathrm{Cat})$). As expected, this shows that the notions of equivalence introduced do not depend on the choice of the universe $V$ in 3.3.1.

3.4.1. In practice, there is usually no need to distinguish essentially between equivalent $U$-topoi, just as there is often no need to distinguish essentially between two equivalent categories, provided however that one has an explicit equivalence from one to the other, or at least an equivalence defined up to unique isomorphism. It is the notion of equivalence of topoi which replaces here the traditional notion of homeomorphism between two topological spaces. See Example 4.2 below for the precise relations between these two notions.

4. Examples of Morphisms of Topoi

We return here to the examples of 2, using the notion of morphism of topoi. The comments of 2.0 apply equally to the present number. The universe $U$ will generally be understood.

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4.1. The Topos $\mathrm{Top}(X)$ for a Variable Topological Space $X$

4.1.1. Let

f : X -> Y

be a continuous map of topological spaces. We shall associate to it canonically a morphism of topoi

Top(f) or f : Top(X) -> Top(Y),

with the notation of 2.1. When one defines $\mathrm{Top}(X)$ as $\mathrm{Ouv}(X{)}^{\sim }$, the most convenient description of $\mathrm{Top}(f)$ is by the direct image functor of sheaves

f_* : Top(X) -> Top(Y),

defined by the formula

f_*(F) = F circ f^{-1},

where

f^{-1} : Ouv(Y) -> Ouv(X)

is the evident functor $U\mapsto f^{-1}(U)$. We already noted that this functor is continuous and left exact, hence indeed defines a functor $f_{\ast }$ above, admitting a right adjoint $f^{\ast }$ which is left exact (III 1.9.1). Of course, in all strictness, the morphism of topoi $\mathrm{Top}(f)$ depends on the choice of the right adjoint $f^{\ast }$ of $f_{\ast }$, and is therefore defined only up to canonical isomorphism. We shall henceforth dispense with explicitly mentioning phenomena of this kind.

When one adopts the point of view of “etale spaces” to define $\mathrm{Top}(X)$, it is the inverse image functor

f^* : Top(Y) -> Top(X)

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which is the most convenient one for defining the morphism of topoi $\mathrm{Top}(f)$, by simply setting

f^*(Y') = X x_Y Y'

for every etale space $Y^{\prime }$ over $Y$; it is evident that the fiber product is indeed an etale space over $X$, and that the functor $f^{\ast }$ thus obtained is left exact and commutes with arbitrary $\underrightarrow{\lim }$, and consequently defines a morphism of topoi $\mathrm{Top}(f)$. For the compatibility of the two definitions obtained, we refer to [TF].

When one has two composable continuous maps

X --f--> Y --g--> Z,

one finds a canonical isomorphism

Top(gf) ~= Top(g) Top(f)

of morphisms of topoi. These transitivity isomorphisms, for three composable continuous maps $f$, $g$, $h$, satisfy a compatibility relation which we shall not write here, and which is precisely the one considered in SGA 1 VI 7.4 B) (for $C=(\mathrm{Esp}{)}^{\circ }$). This may be expressed by saying that, for variable $X$ in the category $(\mathrm{Esp})$,

X |-> Top(X)

is a “pseudo-functor”

(4.1.1.1)    (U-esp) -> (V-U-top) ;

or also, in the terminology of 2-categories, that one has a non-strict functor of 2-categories [9]. In practice, we shall most often allow ourselves the abuse of language consisting in identifying $\mathrm{Top}(gf)$ and $\mathrm{Top}(g)\mathrm{Top}(f)$, i.e. in reasoning as if (4.1.1.1) were a true functor of

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ordinary categories. We shall allow ourselves analogous abuses of language in the other examples treated below.

4.1.2. The preceding considerations extend immediately to the case of topoi associated to topological spaces with groups of operators (2.3). If $f=(f^{\mathrm{es}},f^{\mathrm{gr}})$,

f : (X, G) -> (Y, H)

is a morphism of spaces with operators (where

f^{es} : X -> Y      and      f^{gr} : G -> H

are respectively continuous maps and group morphisms, compatible in the evident sense), one associates to it a morphism of topoi

Top(f) or f : Top(X, G) -> Top(Y, H),

whose definition is left to the reader. When the groups $G$ and $H$ are the unit groups, one recovers the definition of 4.1.1; when, on the other hand, it is the spaces $X$ and $Y$ which are reduced to a point, one finds as inverse image functor $f^{\ast }$ the functor “restriction of the group of operators”

f^* : B_H -> B_G,

associating to every $H$-set the $G$-set it defines by means of $f:G\to H$. We shall encounter this example again in other forms in 4.5 and 4.6.1.

4.1.3. Given a continuous map $f:X\to Y$ of topological spaces, one also associates to it a morphism on the corresponding “big topoi” (2.5)

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TOP(f) or f : TOP(X) -> TOP(Y),

defined most conveniently by the inverse image functor

f^* : TOP(Y) -> TOP(X),

which is none other than the restriction functor. This morphism $\mathrm{TOP}(f)$ is a particular case of the so-called “inclusion” morphism for an induced topos, which will be studied in 5. Thus one sees that, subject to the same reservations as in 4.1.1, the topos $\mathrm{TOP}(X)$ may be regarded as a functor in $X$, for $X$ variable in $(\mathrm{Esp})$.

4.2. Faithfulness Properties of $X\mapsto \mathrm{Top}(X)$

We propose to make precise to what extent a topological space $X$ can be reconstructed in terms of the topos $\mathrm{Top}(X)$, and for this purpose it is appropriate to describe, for two spaces $X$ and $Y$, the category of morphisms from $\mathrm{Top}(X)$ to $\mathrm{Top}(Y)$ (3.2), in order to make precise the faithfulness properties of the “functor” $X\mapsto \mathrm{Top}(X)$. We restrict ourselves to stating the results one obtains, referring the reader to [9] for details. The reader who wants to carry out the verification exercise himself may consult Exerc. 7.8.

4.2.1. Recall that a topological space $X$ is called sober if every irreducible closed subset of $X$ has exactly one generic point. Let us point out that almost all spaces used in practice are sober; this holds in particular for a separated space, more generally for a space all of whose points are closed, or for the underlying space of a scheme. If $X$ is a topological space, one associates to it (loc. cit. or EGA $0_{\mathrm{I}}$, re-edition) a sober topological space $X_{\mathrm{sob}}$ and a continuous map

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(4.2.1.1)    phi : X -> X_sob

which is universal for continuous maps from $X$ to sober spaces; in other words, one constructs a left adjoint functor $X\mapsto X_{\mathrm{sob}}$ to the inclusion functor $({\mathrm{Esp}}_{\mathrm{sob}})\to (\mathrm{Esp})$ from the category of sober spaces into that of “arbitrary” topological spaces (the quotation marks recalling that there is a universe!). The explicit construction is made by taking as points of $X_{\mathrm{sob}}$ the irreducible closed subsets of $X$, and as open sets the sets of the form $U^{\prime }$, where $U$ is an open subset of $X$ and $U^{\prime }\subset X_{\mathrm{sob}}$ denotes the set of irreducible closed subsets of $X$ which meet $U$. The map (4.2.1.1) is obtained by associating to every $x\in X$ the closure of $\{x\}$. The space $X$ is sober if and only if the preceding map is bijective, hence a homeomorphism.

One observes that the functor

phi^{-1} : Ouv(X_sob) -> Ouv(X)

induced by $\phi$ is an isomorphism, which implies that the morphism of topoi

Top(phi) : Top(X) -> Top(X_sob)

defined by $\phi$ is also an isomorphism. This explains a priori why $X_{\mathrm{sob}}$ must necessarily enter the question of reconstructing $X$ from $\mathrm{Top}(X)$: since the latter depends only on $X_{\mathrm{sob}}$ up to isomorphism, the question can have an affirmative answer only if $X$ is sober. We shall make precise below (7.1) how $X_{\mathrm{sob}}$ can effectively

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be reconstructed in terms of $\mathrm{Top}(X)$, by interpreting its points as “points” of the topos $\mathrm{Top}(X)$ (or again as fiber functors).

4.2.2. On every topological space one should introduce the order relation $\le$ for which

x <= y  <=>  closure({x}) subset closure({y}),    i.e. x in closure({y})

(which is also expressed by saying that $x$ is a specialization of $y$, or that $y$ is a generalization of $x$). For a space of the form $X_{\mathrm{sob}}$, this is simply the inclusion relation between irreducible closed subsets of $X$.

This being said, one should introduce on the set of maps from a space $X$ to another space $Y$ the order relation called “specialization”, deduced from that of $Y$, namely

f <= g  <=>  f(x) <= g(x) for every x in X.

With these conventions, one has the following result:

4.2.3. Let $X$, $Y$ be two topological spaces, with $Y$ sober, and let $f$ and $g$ be two continuous maps from $X$ to $Y$. Then there is at most one morphism from $\mathrm{Top}(f)$ to $\mathrm{Top}(g)$, and for there to be one, it is necessary and sufficient that $g$ be a specialization of $f$. Finally, every morphism of topoi $\mathrm{Top}(X)\to \mathrm{Top}(Y)$ is isomorphic to a morphism of the form $\mathrm{Top}(f)$, where $f:X\to Y$ is a continuous map (uniquely determined by the first assertion).

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If one defines the category $\mathrm{cat}(I)$ associated to an ordered set $I$ by declaring that for $i\ge j$ there is exactly one arrow from $i$ to $j$, the preceding result may be summarized by saying that there is a canonical equivalence of categories

cat(Hom_{(Esp)}(X, Y)) ~= Homtop(Top(X), Top(Y))

(where the second member is defined in 3.2).

These results formally imply:

Corollary 4.2.4.

1. Let $f:X\to Y$ be a continuous map. Then $\mathrm{Top}(f):\mathrm{Top}(X)\to \mathrm{Top}(Y)$ is an equivalence of topoi if and only if $f_{\mathrm{sob}}:X_{\mathrm{sob}}\to Y_{\mathrm{sob}}$ is a homeomorphism (hence, when $X$ and $Y$ are sober, if and only if $f$ is a homeomorphism).
2. Let $X$ and $Y$ be topological spaces. For $\mathrm{Top}(X)$ and $\mathrm{Top}(Y)$ to be equivalent, it is necessary and sufficient that $X_{\mathrm{sob}}$ and $Y_{\mathrm{sob}}$ be homeomorphic (hence, if $X$ and $Y$ are sober, it is necessary and sufficient that $X$ and $Y$ be homeomorphic).

4.3. Morphisms into the Final Topos: Constant Objects of a Topos, Section Functors

Denote by $P$ (initial of “point”) the standard final topos, i.e. $P=(\mathrm{Ens})$ (2.2). Let $E$ be any topos; we shall see that, up to unique isomorphism, there exists a unique morphism of topoi

f : E -> P.

More precisely, the category $\mathrm{Homtop}(E,P)$ is equivalent to the punctual category: for two objects of this category, there therefore exists

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a unique arrow from one to the other, and it is an isomorphism. This justifies to some extent the name “final topos”.

For this, recall (3.2.1) that $\mathrm{Homtop}(E,P)$ is equivalent to the full subcategory of $\mathrm{Hom}(P,E{)}^{\circ }$ formed by the functors

f* : P = (Ens) -> E

which commute with inductive limits and are left exact. Let $e$ be a one-point set; then every set $X$ is written canonically as the “sum of $X$ copies of $e$”, whence it follows that the category of functors $g:(\mathrm{Ens})\to E$ which commute with inductive limits is equivalent to the category $E$, by associating to every $g$ the object $g(e)$ of $E$. One reconstructs $g$ in terms of $T=g(e)$, up to unique isomorphism, by $g(I)=T\times I$, where one sets

T x I = coproduct_{i in I} T_i,    with T_i = T for every i in I.

That the functor in $I$ thus defined by $T$ indeed commutes with inductive limits follows from the fact that it is manifestly right adjoint to the functor $X\mapsto \mathrm{Hom}(T,X)$ from $E$ to $(\mathrm{Ens})$. This being said, for $g$ to be left exact, it is evidently necessary that $g(e)=T$ be a final object of $E$ (since $e$ is a final object of $(\mathrm{Ens})$), and it follows easily from the fact that in $E$ “sums are universal” (1.1.2 b)) that this condition is also sufficient. Thus one finds that the category of inverse image functors $f^{\ast }$ is equivalent to the category of final objects of $E$, which is obviously itself equivalent to the final category.

From what precedes, one sees that the choice of a morphism (4.3.1) is essentially equivalent to that of a final object of $E$, say $e_E$. In terms of this, one then has canonical isomorphisms of functors

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(4.3.2)    f*(I) ~= e_E x I = sum of I copies of e_E,    for I in ob(Ens),

and

(4.3.3)    f_*(X) = Hom(e_E, X)                           for X in Ob E.

4.3.4. The two preceding functors will play an important role later. For a set $I$, the object $f^{\ast }(I)=e_E\times I$ of (4.3.2) is called the constant object of value $I$ in $E$ (or, when $E$ is realized as a category $\tilde{C}$ in terms of a site $C$, the constant sheaf of value $I$ on $C$). It will also often be denoted $I_E$, or $I_C$ when $E$ is defined by the site $C$.

The fact that $I\mapsto I_E$ is the inverse image functor of a morphism of topoi makes precise its exactness properties, which imply in particular that this functor respects all the usual species of algebraic structures, transforming a group into a group object of $E$, etc. (3.1.2). When, for example, one has a group $G$ of $E$, one says that it is a constant group (or, if appropriate, a constant sheaf of groups) if it is isomorphic to a group of the form $G_{0E}$, where $G_0$ is an ordinary group. The same terminology applies for every other “algebraic” species of structure, in the sense made precise (more or less) in 3.1.2.

4.3.5. One should be careful that the functor $I\mapsto I_E$ is not necessarily fully faithful (nor even faithful: take for $E$ the “empty topos” (2.2)), so a constant object of $E$ does not in general determine, up to unique isomorphism, the set $I$ which gives rise to it. One says that $E$ is $0$-acyclic, or connected-nonempty, if the functor $I\mapsto I_E$ is fully faithful. By the general properties of adjoint functors, this is equivalent to saying that the adjunction morphism

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I -> f_*(f*(I)) = f_*(I_E) = Hom(e_E, I_E)

is an isomorphism (so that the functor (4.3.3) makes it possible to recover the “value” of a constant object of $E$). One verifies easily that for this it is necessary and sufficient that $e_E$ not be the initial object ${\varnothing }_E$ of $E$, i.e. that $E$ not be an “empty topos” (which expresses the faithfulness of the functor $I\mapsto I_E$ (*)), and that $e_E$ be a connected object of $E$, i.e. not be the sum of two objects of $E$ which are not “empty” (i.e. which are not initial objects of $E$).

4.3.6. The functor (4.3.3) is also often called the sections functor and denoted ${\Gamma }_E$, or $\Gamma (E,-)$, or simply $\Gamma$:

(4.3.6.1)    Hom(e_E, X) = Gamma_E(X) = Gamma(E, X) = Gamma(X).

It is a functor commuting with arbitrary projective limits, but not right exact in general; its derived functors (on abelian group objects) will be studied in the next expose.

(*) Or also the fact that this functor is conservative (I 6.3).

4.4. Morphisms from the “Empty Topos”

Let ${\varnothing }_{\mathrm{top}}$ be an empty topos, which therefore corresponds to a category of sheaves $\Phi$ equivalent to the final category (2.2). Let $E$ be a topos. The category of functors from $E$ to $\Phi$ is evidently equivalent to the punctual category, and every such functor commutes with inductive and projective limits (with no merit, moreover), so can be regarded as an inverse image functor $f^{\ast }$ for a morphism of topoi ${\varnothing }_{\mathrm{top}}\to E$. It follows that the category $\mathrm{Homtop}({\varnothing }_{\mathrm{top}},E)$ is equivalent to the punctual category, and in particular that, up to

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unique isomorphism, there exists one and only one morphism of topoi

(4.4.1)    empty_top -> E.

This justifies to some extent the terminology “initial topos” introduced in 2.2.

One can also determine the morphisms of topoi

(4.4.2)    E -> empty_top ;

one immediately verifies that such a morphism exists if and only if the initial object of $E$ is also a final object, i.e. if and only if $E$ itself is an “empty topos”, and that in this case the category $\mathrm{Homtop}(E,{\varnothing }_{\mathrm{top}})$ is again equivalent to the punctual category. The unique morphism (4.4.2) (modulo isomorphism) is then an equivalence of topoi.

4.5. The Classifying Topos $B_G$ for Variable Group $G$

4.5.1. Let $E$ be a topos, and let

f : G -> H

be a morphism of groups in $E$. One deduces from it the functor “restriction of the group of operators”

f* : B_H -> B_G,

where the notation is that of 2.4. It is trivial that this functor commutes with inductive limits and projective limits; a fortiori it can be interpreted as an inverse image functor associated to a morphism of topoi

B_f, or f : B_G -> B_H.

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One easily makes explicit the corresponding direct image functor

f_* : B_G -> B_H

by the formula

f_*(X) = Hom_G(H_s, X),

where $X$ is an object of $E$ with $G$ acting on the left, where $H_s$ is $H$ regarded as equipped with the left actions by $G$ deduced from $f$, and where ${\mathrm{Hom}}_G$ denotes the expected subobject of the object $\mathrm{Hom}$ defined below (10.); one makes $H$ act on the left on ${\mathrm{Hom}}_G(H_s,X)$ by means of the right actions of $H$ on $H_s$, by right translations.

Since the inverse image functor $f^{\ast }$ commutes with arbitrary limits (and not only with finite limits), it is itself the left adjoint of a functor

f_! : B_G -> B_H,

so that one has a sequence of three adjoint functors as in 3.1.3:

f_!,  f*,  f_*.

One easily makes $f_{!}$ explicit by the formula

f_!(X) = H x^G X,

where the second member denotes the “contracted product”, deduced from the actions of $G$ on $X$ (on the left) and on $H$ (on the right via right translations and $f$), defined as the quotient of $H\times X$ by $G$ acting by the formula

g . (h, x) = (h g^{-1}, gx).

The functor $f_{!}$, being a left adjoint, evidently commutes with inductive limits, but it is not in general left exact (i.e.

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it cannot in turn be regarded as an inverse image functor by a morphism of topoi $B_H\to B_G$). In fact, one verifies easily that it can be left exact only if $f:G\to H$ is an isomorphism. Similarly, the functor $f_{\ast }$, which commutes with projective limits, is not in general right exact, and a fortiori does not in general admit a right adjoint. At least when $E$ is the punctual topos, i.e. when $G$ and $H$ are ordinary groups, $f_{\ast }$ is right exact only if $f$ is an isomorphism.

When one has a second morphism of groups $g:H\to K$, one finds, as in 4.1.1, a transitivity up to canonical isomorphism, so that, subject to the usual reservation, one may regard the classifying topos $B_G$ as depending functorially on the group $G$. We leave to the reader the task of generalizing this functorial behavior to the case where $G$ and the topos $E$ are varied simultaneously.

4.5.2. The topos $B_G$ for a variable pro-group $G$.

We leave to the reader the task of making precise the covariant character of the topos $B_G$ (2.7) with respect to $G$, by modeling the exposition we gave in 4.5.1. One should be careful, however, that in the case of a morphism $f:G\to H$ of pro-groups which are not essentially constant, the corresponding morphism of topoi $B_G\to B_H$ does not in general allow the definition of a functor $f_{!}$ (whose right adjoint would be the functor $f^{\ast }$ of restriction of the pro-group of operators). Supposing, to simplify the statement, that $G$ and $H$ are defined by profinite groups $G$ and $H$, one sees easily that $f_{!}$ exists if and only if the image of the morphism under consideration $f:G\to H$ has finite index in $H$, and

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in this case $f_{!}$ is given by the same formula as in 4.5.

4.6. The Topos $\hat{C}$ for Variable Category $C$

4.6.1. Let

f : C -> C'

be a functor from a category $C\in U$ to another $C^{\prime }$, giving a functor

f* : C'^ -> C^,    f*(F) = F circ f.

It is trivial that this functor commutes with projective limits and inductive limits; a fortiori it can be regarded as the inverse image functor for a morphism of topoi

f^ or f : C^ -> C'^.

The corresponding direct image functor

f_* : C^ -> C'^

is none other than the functor also denoted $f_{\ast }$ in I 5.1. Moreover (as was to be expected from the fact that $f^{\ast }$ also commutes with arbitrary limits), $f^{\ast }$ also admits a left adjoint

f_! : C^ -> C'^,

(which was also denoted $f_{!}$ in I 5.1). One thus obtains a sequence of three adjoint functors

f_!,  f*,  f_*,

the first moreover being an extension of $f:C\to C^{\prime }$ to the categories of presheaves (for the usual embedding of $C$, $C^{\prime }$ in the categories $\hat{C}$, ${\hat{C}}^{\prime }$). One should be careful that the functor $f_{!}$ is not in general left exact, nor $f_{\ast }$ right exact, which therefore removes any ambiguity about the

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direction of variance of the topos $\hat{C}$ associated to the variable category $C$: this topos is a covariant functor in $C$, subject to the usual reservation coming from transitivity isomorphisms (cf. 4.1.1). When the sets of objects of $C$ and $C^{\prime }$ are reduced to one element, so that $C$ and $C^{\prime }$ are identified with monoids $G$, $G^{\prime }$, the corresponding topoi are the classifying topoi $B_{G^{\circ }}$ and $B_{G^{\prime \circ }}$, and one recovers the variance of these for variable $G$ already encountered from other points of view in 4.1.2 and 4.5 (where the restriction to groups rather than monoids was not essential).

4.6.2. One can make precise the 2-functorial dependence of the topos $\hat{C}$ with respect to $C$, by introducing for two categories $C$, $C^{\prime }\in U$ a canonical functor

(4.6.2.1)    Hom(C, C') -> Homtop(C^, C'^).

It remains to define this functor on arrows, and for this one notes that $f\mapsto f^{\ast }$ makes it possible to identify (up to equivalence of categories) the second member with a full subcategory of $\mathrm{Hom}({\hat{C}}^{\prime },\hat{C}{)}^{\circ }$ (3.2.1). Now, if $f,g:C\to C^{\prime }$ are two functors, every morphism $u:f\to g$ of functors defines a morphism $F\circ g\to F\circ f$ of functors in $F\in \mathrm{ob}{\hat{C}}^{\prime }$ ($F$ being a contrafunctor), hence a morphism $g^{\ast }\to f^{\ast }$, and consequently defines a morphism $\hat{f}\to \hat{g}$ as

announced.

When $C$ is the punctual category, $\hat{C}$ is the punctual topos (2.2), denoted $P$, and (4.6.2.1) is interpreted as a natural functor

(4.6.2.2)    C' -> Homtop(P, C'^) = Points(C'^)

from $C^{\prime }$ to the “category of points” of ${\hat{C}}^{\prime }$, which will be studied in 6.

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This functor is not necessarily an equivalence of categories (7.3.3); a fortiori (4.6.2.1) is not necessarily an equivalence of categories. On the other hand, the functor (4.6.2.1) is always fully faithful.

To see this, note that if $f,g:E\to E^{\prime }$ are two morphisms of topoi such that $f_{!}$ and $g_{!}$ are defined (3.1.3), it follows from the theory of adjoint functors that one has canonical isomorphisms

Hom(f_!, g_!) ~= Hom(g*, f*) ~= Hom(f_*, g_*)  def= Hom(f, g).

Applying this to functors of the form $\hat{f}$, $\hat{g}$ associated to $f,g:C\to C^{\prime }$, one finds the announced result, taking into account that the natural extension map

Hom(f, g) -> Hom(f_!, g_!)

is bijective (I 7.8).

4.6.3. One may ask when the functor (4.6.2.1) is an equivalence of categories, i.e. when it is essentially surjective, which is a particular case of the question of determining all morphisms of topoi $\hat{C}\to {\hat{C}}^{\prime }$. More generally, if $C$ is a category $\in U$ and $E$ a topos, one may seek to determine the morphisms of topoi

f : C^ -> E.

The category of these morphisms is equivalent to the opposite category of that of functors

f* : E -> C^

commuting with arbitrary inductive limits and finite projective limits. Interpreting functors $E\to \hat{C}$ as functors $E\times C^{\circ }\to (U\text{-}\mathrm{Ens})=(\mathrm{Ens})$, or again as functors $F:C^{\circ }\to \mathrm{Hom}(E,(\mathrm{Ens}))$, the exactness property under consideration is expressed by the condition that for every object $X$ of $C$, the functor $F(X):E\to (\mathrm{Ens})$ commute with arbitrary inductive limits and finite projective limits (or, as we shall say in 6, that $F(X)$ is a

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“fiber functor” on $E$). This is equivalent to saying that $F(X)$ is the inverse image functor for a morphism of topoi $P\to E$, so that one finally obtains a canonical equivalence of categories

(4.6.3.1)    Homtop(C^, E) ~= Hom(C, Points(E)),

where, for short, we denote by

Points(E) = Homtop(P, E)

the category of points of the topos $E$.

When $E$ is of the form ${\hat{C}}^{\prime }$, one sees immediately from the definitions that the composite of (4.6.2.1) and the preceding equivalence (4.6.3.1) is the functor

(4.6.3.2)    Hom(C, C') -> Hom(C, Points(C'^))

defined by $F\mapsto i\circ F$, where $i:C^{\prime }\to \mathrm{Points}({\hat{C}}^{\prime })$ is the canonical embedding (4.6.2.2). It follows immediately that for (4.6.2.1) to be an equivalence, it is necessary and sufficient that $C$ be empty or that (4.6.2.2) be essentially surjective (hence an equivalence of categories). This last condition on $C^{\prime }$ is satisfied in certain interesting cases, notably when $C^{\prime }$ is the category with one object defined by a group $G$ (7.2.5).

Remark 4.6.4. The fact of having associated a topos $\hat{C}$ to an arbitrary category $C$ suggests that a category $C$ admits invariants of topological nature (cohomology groups, homotopy groups, etc.), just like a topos. The cohomology groups of $\hat{C}$ with coefficients in an abelian group object $F$ (in the general sense studied in V) are none other than the values of the derived functors ${\underleftarrow{\lim }}^n$ of the functor $\underleftarrow{\lim }$, already familiar to topologists.

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J. L. Verdier and (independently) D. G. Quillen have verified that when one restricts to constant coefficients, or more generally locally constant coefficients, these cohomology groups are identified with the cohomology groups of the semisimplicial set $\mathrm{Nerf}(C)$ canonically associated to $C$ (ref. [5], No. 212, Prop. 4.1), and that moreover, up to isomorphism in the “homotopy category” of [2], every semisimplicial set can be obtained by means of a category $C$.

By means of a suitable notion of homotopy type for topoi, which we do not make precise here, one can say that the semisimplicial homotopy types of topologists are none other than the homotopy types of topoi of the special form $\hat{C}$, more generally of topoi $E$ in which every object admits a covering refining all others (2.6) (*). In the absence of this condition on $E$, one can at best express its homotopy type by a suitable projective system of semisimplicial sets [1].

(*) And, more generally still, of topoi which are “locally $\infty$-connected”, in an evident sense which we leave to the reader to make precise.

4.7. The Topos $\tilde{C}$ for a Variable Site $C$ (Cocontinuous Functors)

Let

f : C -> C'

be a cocontinuous functor (III 2.1) between sites $\in U$, i.e. a functor such that the functor $f_{\ast }$, or ${\hat{f}}_{\ast }:F\mapsto F\circ f$, from $\hat{C}$ to ${\hat{C}}^{\prime }$ (4.6.1), carries $\tilde{C}$ into ${\tilde{C}}^{\prime }$, that is, induces a functor

(4.7.1)    f~_* : C~ -> C'~

making the diagram of functors

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(4.7.2)
        f~_*
  C~ --------> C'~
  |            |
i |            | i'
  v            v
  C^ ---------> C'^
        f^_*

commutative, where $i$, $i^{\prime }$ are the inclusion functors. We then saw (III 2.3) that the functor ${\tilde{f}}_{\ast }$ admits a left adjoint ${\tilde{f}}^{\ast }$, and that the latter is left exact. In other words, ${\tilde{f}}_{\ast }$ is the direct image functor associated to a morphism of topoi

f~ or f : C~ -> C'~,

the corresponding inverse image functor being of course ${\tilde{f}}^{\ast }$. Taking left adjoints of the functors in question, the commutative diagram (4.7.2) moreover gives a diagram commutative up to isomorphism

(4.7.3)
        f~*
  C~ <-------- C'~
  ^            ^
a |            | a'
  |            |
  C^ <--------- C'^
        f^*

where $a$ and $a^{\prime }$ are the “associated sheaf” functors, a diagram which immediately recovers the formula (III 2.3)

f~* = a f^* i'.

The transitivity property for the morphisms of topoi $\hat{f}:\hat{C}\to {\hat{C}}^{\prime }$ implies the same property for the morphisms of topoi $\tilde{f}:\tilde{C}\to {\tilde{C}}^{\prime }$ associated to cocontinuous functors, so that one can say that the topos $\tilde{C}$ varies functorially in $C$, covariantly, when one takes cocontinuous functors as “morphisms” of sites.

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In all cases encountered up to now, the cocontinuous functor $f$ used is also continuous, i.e. (III 1.1) “extends” to a functor

f~_! : C~ -> C'~

commuting with inductive limits, and which is left adjoint to ${\tilde{f}}^{\ast }$, so that one has a sequence of three adjoint functors

f~_!,  f~*,  f~_*.

One should be careful that the functor ${\tilde{f}}_{!}$ is not in general left exact, nor ${\tilde{f}}_{\ast }$ right exact, which removes any ambiguity about the direction of variance of the topos $\tilde{C}$, for $C$ variable by functors which are supposed only to be cocontinuous, or even continuous and cocontinuous.

Remark 4.7.4. Given a morphism of topoi

F : E -> E',

for there to exist a functor $F_{!}$, left adjoint to $F^{\ast }$, it is necessary and sufficient that one be able to “realize” (up to equivalence) $E$ and $E^{\prime }$ in the form $\tilde{C}$ and ${\tilde{C}}^{\prime }$, for two sites $C$, $C^{\prime }\in U$, and be able to find a continuous and cocontinuous functor $f:C\to C^{\prime }$ such that $F$ is identified with $\tilde{f}$. This is evidently sufficient, and for necessity it suffices to take for $C$ and $C^{\prime }$ small full generating subcategories of $E$ and $E^{\prime }$ respectively, such that

F_!(ob C) subset ob C',

endowed with the topologies induced by those of $E$ and $E^{\prime }$, and to take for $f$ the functor induced by $F_{!}$ (cf. 1.2.1). Recall (1.8) that the existence of $F_{!}$ also means that $F^{\ast }$ commutes with projective limits,

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or, what is the same here since this functor is left exact, that it commutes with products.

4.7.5. If one asks which morphisms of topoi $F:E\to E^{\prime }$ can be realized by a cocontinuous (not necessarily continuous) functor of sites, one sees similarly that the answer is the following: the set of objects $X$ of $E$ such that the functor

X' |-> Hom(X, F*(X'))

from $E^{\prime }$ to $(\mathrm{Ens})$ commutes with projective limits (or, equivalently, with products) must be a generating family of $E$.

4.8. The Morphism of Topoi $\tilde{C}\to \hat{C}$ for a Site $C$

Let $C$ be a small site, to which are therefore associated the two topoi $\tilde{C}$ and $\hat{C}$ (the second not depending on the topology put on $C$). In II 3.4 we defined the “associated sheaf” functor

a : C^ -> C~

and established that it is left exact and commutes with inductive limits. It is therefore the inverse image functor associated to a morphism of topoi

p : C~ -> C^,    p* = a,

the corresponding direct image functor being the canonical inclusion

i = p_* : C~ -> C^.

It is well known that this latter functor is not in general right exact (its derived functors on abelian objects give rise to the cohomology presheaves $R^{r}i(F)$ of V 2), and that $a$ does not in general commute with arbitrary limits, which removes any ambiguity about the direction of the “natural” morphism of topoi between $\hat{C}$ and $\tilde{C}$.

When one has a cocontinuous functor

f : C -> C'

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of sites, one deduces a diagram of morphisms of topoi

        f~
  C~ -------> C'~
  |           |
p |           | p'
  v           v
  C^ --------> C'^
        f^

which is commutative up to canonical isomorphism: this is indeed what is expressed by the commutativity of the diagram of functors (4.7.2). Thus one can say that the morphism of topoi $p:\tilde{C}\to \hat{C}$ is functorial in $C$, when $C$ is varied by cocontinuous functors between sites.

4.9. Effect of a Continuous Functor of Sites. Morphisms of Sites

4.9.1. If

f : C -> C'

is a continuous functor from $C$ to $C^{\prime }$, i.e. such that the functor

f^* : C'^ -> C^

carries ${\tilde{C}}^{\prime }$ into $\tilde{C}$, hence induces a functor

f_s : C'~ -> C~,

we saw (III 1.2) that this latter admits a left adjoint

f^s : C~ -> C'~,

which moreover “extends” $f$ in an evident sense. One should be careful that in general $f^s$ is not right exact (even if $f$ is also cocontinuous), nor does $f_s$ commute with inductive limits, so that the datum of $f$ does not, without further hypothesis, describe a morphism of topoi in either direction between $\tilde{C}$ and ${\tilde{C}}^{\prime }$. The case where $f$ is cocontinuous, i.e. where $f_s$ commutes with inductive limits and

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can therefore be regarded as an inverse image functor for a morphism of topoi $\tilde{f}:\tilde{C}\to {\tilde{C}}^{\prime }$, was examined in 4.7. We shall examine the case where the functor $f^s$ is left exact, and can therefore be regarded as an inverse image functor for a morphism of topoi in the opposite direction:

(4.9.1.1)    Top(f) = g : C'~ -> C~.

One should be careful that we have taken care not to denote this morphism by the letter $\tilde{f}$ or $f$, to avoid confusions with the situation of 4.7, in accordance with the general recommendations of 3.1.3. One sometimes says that the functor $f:C\to C^{\prime }$ is a morphism of sites from $C^{\prime }$ to $C$ (attention, not from $C$ to $C^{\prime }$) if it is continuous and if the functor $f^s$ is left exact, in other words if there exists a morphism of topoi (4.9.1.1) such that the corresponding inverse image functor

g* : C~ -> C'~

extends the functor $f$, i.e. makes the diagram of functors

        f
  C --------> C'
  |           |
eps |         | eps'
  v           v
  C~ -------> C'~
        g*

commute up to isomorphism, where $\epsilon$, ${\epsilon }^{\prime }$ are the canonical functors of II 4.4.0.

4.9.2. In practice, one recognizes that a continuous functor $f:C\to C^{\prime }$ is a morphism of sites from $C^{\prime }$ to $C$, by the fact that finite projective limits are representable in $C$, and that $f$ commutes with them (III 1.3.5).

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Subject to the indicated condition on $C$ (almost always verified in practice), and supposing moreover that the topology of $C^{\prime }$ is less fine than the canonical topology (also almost always verified), the preceding sufficient condition ($f$ left exact) for $f$ to be a morphism of sites from $C^{\prime }$ to $C$ is moreover also necessary.

4.9.3. In the spirit of what precedes, if $C$ and $C^{\prime }$ are two $U$-sites, one should define the category of morphisms of sites $\mathrm{Morsite}(C^{\prime },C)$ from $C$ to $C^{\prime }$ as the full subcategory of the opposite category $\mathrm{Hom}(C,C^{\prime }{)}^{\circ }$ of the category of functors from $C$ to $C^{\prime }$, formed by those functors which are willing to be morphisms of sites (from $C^{\prime }$ to $C$). In this way one obtains a canonical functor (defined up to unique isomorphism)

Morsite(C', C) -> Homtop(C'~, C~).

Proposition 4.9.4. Let $E$ be a $U$-topos and let $C$ be a $U$-site. Then the functor

f |-> f*|_C = f* circ eps_C

associating to every morphism of topoi $f:E\to \tilde{C}$ the “restriction” to $C$ of the associated inverse image functor $f^{\ast }:\tilde{C}\to E$, induces an equivalence of categories

Homtop(E, C~) ~= Morsite(E, C) (subset Hom(C, E)^circ).

When finite limits are representable in $C$, the corresponding fully faithful functor

Homtop(E, C~) -> Hom(C, E)^circ

has as essential image the set of functors $g:C\to E$ which are left exact and continuous, or equivalently left exact and transform covering families into covering families.

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The last assertion follows from the first by means of 4.9.2. On the other hand, one deduces from III 1.2 iv) and IV 1.2 iii) that the functor

G |-> G circ eps

induces an equivalence between the category of continuous functors from $\tilde{C}$ to $E$ and the category of continuous functors from $C$ to $E$, a quasi-inverse functor being obtained by $g\mapsto g^s$ in the notation of loc. cit. On the other hand, by the very definition, $g$ is a morphism of sites if and only if $g^s$ is the inverse image functor associated to a morphism of topoi $E\to \tilde{C}$, whence the conclusion by 3.2.1; the “or equivalently” comes from III 1.6.

One should retain above all from 4.9.4 that (when finite limits are representable in $C$) “it amounts to the same thing” to give a morphism of topoi $f:E\to \tilde{C}$ or a functor $g:C\to E$ which is left exact and transforms covering families into covering families.

4.9.5. Using the developments of 4.9.1 and 4.9.3, one sees as usual that for a variable $U$-site $C$, via the notion of morphism of sites and of morphism of morphisms of sites just made explicit, the topos $\tilde{C}$ depends functorially (or more exactly, 2-functorially) on the site $C$. Note that thanks to the terminology introduced, $\tilde{C}$ depends covariantly on the site $C$.

4.9.6. It is immediate that (unlike what happens for the notion of cocontinuous functor, cf. 4.7.4) every morphism of topoi $F:E\to E^{\prime }$ can be realized by means of a morphism of sites $f:C\to C^{\prime }$ (i.e. a continuous functor $f:C^{\prime }\to C$ such that…): it suffices to choose in $E$ and $E^{\prime }$ small full generating subcategories

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$C$ and $C^{\prime }$ respectively, endowed with the topologies induced by those of $E$ and $E^{\prime }$, such that one has

F*(ob C') subset ob C,

and to take for $f$ the functor induced by $F^{\ast }$. This explains why most morphisms of topoi encountered in practice are indeed described by means of morphisms of sites (rather than by means of cocontinuous functors as in 4.7).

4.10. Relations Between the Small and Big Topoi Associated to a Topological Space $X$

We return to the notation of 2.5. In particular, $V$ is a universe such that $U\in V$. We depart from the convention of 2.1, denoting by $\mathrm{Top}(X)$ the topos of $V$-sheaves (and not $U$-sheaves) on $X$. Thus we shall reason with $V$-topoi and not $U$-topoi. (NB: it would be possible to keep the $U$-topoi, by adopting the appropriate convention for the definition of $\mathrm{TOP}(X)$, so that the latter is a $U$-topos; cf. 2.5.)

This being said, we shall define two morphisms of topoi

(4.10.1)    f : Top(X) -> TOP(X)
            g : TOP(X) -> Top(X),    g f ~= id_{Top(X)},

giving rise to a sequence of three adjoint functors

(4.10.1.1)    g* = f_!,    g_* = f*,    g^! = f_*,

the notation being that of 3.1.3. We shall successively define

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the first two functors of this sequence, the third then being defined as the right adjoint of the second.

4.10.2. The functor

g_* = f* : TOP(X) -> Top(X)

is defined simply as the restriction functor from $(\mathrm{Esp})/X$ to the site $\mathrm{Ouv}(X)$ of open subsets of $X$, which indeed transforms sheaves into sheaves, as follows immediately from the definition. Denoting sheaves on the big site of $X$ by underlined letters, for such a sheaf $F$ we denote by $F_X$ its restriction to the site $\mathrm{Ouv}(X)$, whence a functor

(4.10.2.1)    Restr : F |-> F_X : TOP(X) -> Top(X).

It is evident that this functor commutes with $V$-projective limits, since these are computed argument by argument (one could also invoke the existence of the left adjoint, constructed in 4.10.4 below). I claim that it also commutes with $V$-inductive limits. To be convinced of this, we shall give a very convenient interpretation of “big” sheaves on $X$, i.e. of objects of $\mathrm{TOP}(X)$, in terms of ordinary sheaves (NB: $V$-sheaves are meant) on topological spaces $X^{\prime }$ over $X$.

4.10.3. For a big sheaf $F$ on $X$, and for every space $X^{\prime }$ over $X$ (understood: $X^{\prime }\in U$), define in the evident way, as in 4.10.2, the “small” sheaf $F_{X^{\prime }}$, restriction of $F$ to $X^{\prime }$. If

u : X'' -> X'

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is a morphism of $(\mathrm{Esp})/X$, one then defines in the evident way a morphism $u^{\ast }(F_{X^{\prime }})\to F_{X^{\prime \prime }}$, or equivalently, a “transition morphism”

(4.10.3.1)    phi_u : u*(F_{X'}) -> F_{X''}.

These morphisms, for variable $u$, satisfy an evident transitivity condition for a composite

X''' --v--> X'' --u--> X'

of morphisms in $(\mathrm{Esp})/X$, which we leave to the reader to make explicit. In this way one obtains a natural functor from the category $\mathrm{TOP}(X)$ of big sheaves on $X$ to the category of systems

(F_{X'})_{X' in ob (Esp)/X},    (phi_u)_{u in Fl (Esp)/X},

satisfying the transitivity condition under consideration, and such moreover that for every morphism $u:X^{\prime \prime }\to X^{\prime }$ which is an open immersion (or, more generally, an etale space), the transition morphism ${\phi }_u$ is an isomorphism.

Since the functors $u^{\ast }:\mathrm{Top}(X^{\prime })\to \mathrm{Top}(X^{\prime \prime })$ used for the description of the ${\phi }_u$ commute with $V$-inductive limits, it follows immediately that, in the preceding description of objects of $\mathrm{TOP}(X)$ in terms of “small” sheaves $F_{X^{\prime }}$, $V$-inductive limits are computed argument by argument, i.e. the functors of the form $F\mapsto F_{X^{\prime }}$ commute with $V$-inductive limits.

In particular, this holds for the functor $F\mapsto F_X$ considered in 4.10.2. It therefore indeed admits a right adjoint (1.5). It remains to construct a left adjoint for it (whose existence follows a priori from 1.8), and to verify that the latter is left exact.

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This will complete the definition of the announced morphisms of topoi (4.10.1) in terms of the functors (4.10.1.1).

4.10.4. The functor

g* = f_! : Top(X) -> TOP(X)

is obtained by associating to every small sheaf $F$ on $X$ the system of its inverse images $(F_{X^{\prime }})$, $X^{\prime }\in \mathrm{ob}(\mathrm{Esp})/X$, by the structural morphisms $X^{\prime }\to X$, the transition morphism ${\phi }_u$, for $u:X^{\prime \prime }\to X^{\prime }$, being the transitivity isomorphism for inverse images of sheaves (4.1.1). One immediately sees that one thus obtains a functor

Prol : Top(X) -> TOP(X)

which is fully faithful, and whose essential image is formed by the big sheaves $F$ on $X$ for which all the transition morphisms ${\phi }_u$ of 4.10.3 are isomorphisms. We leave to the reader the task of defining an adjunction isomorphism between this canonical prolongation functor and the restriction functor of 4.10.2, proving that the latter is right adjoint to the former. This is immediate in terms of the description 4.10.3 of the category $\mathrm{TOP}(X)$.

Remarks 4.10.5.

a) The construction 4.10.4 shows at the same time that $g^{\ast }=f_{!}$ is fully faithful (or, equivalently, that $g_{\ast }=f^{\ast }$ is a functor of passage to a category of fractions, or finally that $g^{!}=f_{\ast }$ is fully faithful). The big sheaves on $X$ which belong to the essential image of the functor $g^{\ast }=f_{!}=\mathrm{Prol}$ deserve the name of etale big sheaves on $X$, since they form a category equivalent to that of ordinary sheaves on $X$, or again to that of etale spaces

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over $X$. One also sees easily that if $X^{\prime }$ is a topological space over $X$, then the big sheaf on $X$ represented by it is etale in the preceding sense if and only if $X^{\prime }$ is an etale space over $X$.

b) One may also express the full faithfulness of $f_{!}$ by saying that the adjunction morphism $f^{\ast }{f}_{!}\to {\mathrm{id}}_{\mathrm{Top}(X)}$ is an isomorphism, i.e. that $g_{\ast }{g}^{\ast }\simeq \mathrm{id}$, i.e. that $gf\simeq \mathrm{id}$. Thus $g$ makes $\mathrm{TOP}(X)$ a topos over $\mathrm{Top}(X)$, admitting a “section” $f$ over $\mathrm{Top}(X)$.

c) The fact that the functor $g^{\ast }=f_{!}$ is fully faithful, exact, and commutes with $V$-inductive limits partially justifies the very convenient point of view (due to J. Giraud) according to which, in practically all questions of sheaf theory on $X$, it is harmless to replace usual sheaves, or “small” sheaves, by the associated “big” sheaves. This is so in particular for cohomological questions, since $g_{\ast }$ is exact, the functors $R^i{g}_{\ast }$ vanish for $i>0$, and therefore for every big sheaf $F$ on $X$ one has canonical isomorphisms

H^i(TOP(X), F) ~= H^i(Top(X), F_X) = H^i(X, F_X).

Applying this to a sheaf of the form $\mathrm{Prol}(F)$, where $F$ is a small sheaf on $X$, one concludes (since $\mathrm{Prol}(F{)}_X\simeq F$ canonically) a canonical isomorphism:

H^i(X, F) = H^i(TOP(X), Prol(F)).

Thus the cohomological invariants of $X$, computed via the small or the big topos of $X$, are essentially identical. The same result is moreover valid in noncommutative cohomology.

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Exercise 4.10.6. Let $S$ be a $U$-site whose topology is less fine than the canonical topology, and let $M$ be a subset of $\mathrm{Fl}S$ satisfying the following conditions:

a) The morphisms of $M$ are quarrable (I 10.3), and $M$ is stable under base change.

b) $M$ contains the identity arrows and is stable under composition.

c) An arrow $u:X\to Y$ such that there exists a covering family $Y_i\to Y$, with the morphisms $X{\times }_Y{Y}_i\to Y_i$ in $M$, is itself an element of $M$.

d) For every $X\in \mathrm{ob}S$, every covering family of $X$ is refined by a covering family $(f_i:X_i\to X)$, with the $f_i\in M$.

For every $X\in \mathrm{ob}S$, consider the site $S(X)$ (“small site of $X$”) whose underlying category is the full subcategory of $S/X$ formed by the objects whose structural morphism is in $M$, endowed with the topology induced (III 3.1) by that of $S$.

1°) For every arrow $u:X\to Y$ of $S$, show that base change by $u$ from $S(Y)$ to $S(X)$ is a continuous functor, hence gives a functor commuting with inductive limits (III)

S(u)^* : S(Y)~ -> S(X)~.

2°) Define an equivalence between the topos $\tilde{S}$ and the category of systems

(F_X)_{X in ob S},    (phi_u)_{u in Fl S}

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formed by objects $F_X\in \mathrm{ob}\widetilde{S(X)}$, and, for every arrow $u:X\to Y$ in $S$, by a morphism

phi_u : S(u)^*(F_Y) -> F_X,

these systems being subject to a transitivity condition for a composite $v\circ u$ of arrows of $S$, and to the condition that $u\in M$ implies that ${\phi }_u$ is an isomorphism.

3°) Define “restriction” and “prolongation” functors

Res_X : (S/X)~ -> S(X)~,
Prol_X : S(X)~ -> (S/X)~.

Show that ${\mathrm{Res}}_X$ commutes with small inductive and projective limits and that ${\mathrm{Prol}}_X$ is fully faithful, its essential image being formed by the sheaves $F$ such that ${\phi }_u$ is an isomorphism for every arrow $u$ of $S/X$.

4°) Define an adjunction morphism making ${\mathrm{Res}}_X$ right adjoint to ${\mathrm{Prol}}_X$. Conclude that there exists a morphism of topoi

f : S(X)~ -> (S/X)~

making $\widetilde{S(X)}$ a subtopos of $\widetilde{(S/X)}$ and such that

f_! = Prol_X,
f* = Res_X.

Show that ${\mathrm{Prol}}_X$ transforms abelian sheaves into abelian sheaves.

5°) Show that if $S(X)$ admits fiber products, ${\mathrm{Prol}}_X$ is exact. Deduce that there then exists a morphism of topoi $g:\widetilde{(S/X)}\to \widetilde{S(X)}$ which is a left retraction of $f$, i.e. such that

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g* = Prol_X,
g_* = Res_X.

(For an example where $S(X)$ does not admit fiber products, take for $S$ the category of schemes endowed with the etale topology, for $M$ the smooth morphisms, and for $X$ a noetherian scheme of dimension $>0$.)

6°) Show that for every abelian sheaf $F$ of $\widetilde{(S/X)}$, one has an isomorphism

H^q(X, F) ~= H^q(X, Res_X F)        for all q.

Show, using for example hypercoverings, that for every abelian sheaf $G$ of $\widetilde{S(X)}$, one has an isomorphism

H^q(X, G) ~= H^q(X, Prol_X G)       for all q.

7°) Buy a chocolate medal for the editor.

5. Induced Topos

5.1. Let $E$ be a topos and let $X$ be an object of $E$. Then the category $E/X$ of objects of $E$ over $X$ is a topos, as follows for example immediately from Giraud’s criterion 1.2 ii). One can also, thanks to 1.2.1, realize $E$ as a category of sheaves $\tilde{C}$, where $C$ is a

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generating subcategory of $E$ which can be chosen so that $X\in \mathrm{ob}C$; then one knows that $E/X=\tilde{C}/X$ is equivalent to $\widetilde{(C/X)}$ (III 5.4), hence it is a topos.

The topos $E/X$ is called the topos induced on the object $X$ of $E$.

5.2. We shall define a canonical morphism of topoi

(5.2.1)    j_X : E/X -> E,

called the inclusion morphism of the induced topos $E/X$ into the ambient topos $E$, or better (cf. 5.7), the localization morphism of $E$ at $X$. This morphism corresponds to a sequence of three adjoint functors (cf. 3.1.3)

(5.2.2)    j_{X!},    j_X^*,    j_{X*},

which can be made explicit as follows:

a) The functor

j_{X!} : E/X -> E

is the “forget the structural arrow” functor from $E/X$ to $E$.

b) The functor

j_X^* : E -> E/X

is defined by

j_X^*(Z) = (X x Z, pr_1),

where ${\mathrm{pr}}_1:X\times Z\to X$ is the first projection. It may also be interpreted as the base change functor relative to the morphism

X -> e_E,

where $e_E$ is the final object of $E$. It is trivial, by definition of the base change functor, that $j_X^{\ast }$ is indeed right adjoint to $j_{X!}$. In particular it commutes with projective limits. It also commutes with

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inductive limits, by virtue of the special exactness properties of topoi (II 4).

c) From this last fact it follows (1.6) that $j_X^{\ast }$ admits a right adjoint

j_{X*} : E/X -> E.

For an object $X^{\prime }$ over $X$, the object $j_{X\ast }(X^{\prime })$ is also sometimes denoted by one of the symbols

prod_{X/e_E}(X'/X),    Hom_{X/e_E}(X, X'),    Res_{X/e_E}(X'/X)

(“Weil restriction”), one or another of which is no doubt already familiar to the learned reader of our modest work.

5.3. One should be careful that the functor $j_{X!}$ commutes with fiber products and transforms monomorphisms into monomorphisms, but it is not in general left exact for all that. It is left exact only if $X$ is a final object of $E$ (because $X=j_{X!}(X,{\mathrm{id}}_X)$, and $(X,{\mathrm{id}}_X)$ is a final object of $E/X$), that is, if and only if $j_X$ is in fact an equivalence of topoi (hence if all the functors (5.2.2) are equivalences). (*)

Similarly, the functor $j_{X\ast }$ is not in general right exact, and does not necessarily transform epimorphisms into epimorphisms.

One concludes from these observations that the direction of the morphism of topoi relating $E/X$ to $E$, for an object $X$ of the topos $E$, is determined without possible ambiguity.