§ 3. Sheaves

1. Sheaves of sets

Let B be a topological space.

In other words (E, III, p. 52), a presheaf $\mathscr{F}$ on B relative to $\mathscr{B}$ is a pair $((\mathscr{F}(U))_{U\in\mathscr{B}},(f_{UV}))$, which is also denoted by $(\mathscr{F}(U), f_{UV})$, where $((\mathscr{F}(U))_{U\in\mathscr{B}}$ is a family of sets having $\mathscr{B}$ as set of indices and where for each pair $(U,V)$ of elements of $\mathscr{B}$ such that $U\subset V,f_{UV}$ is a mapping of $\mathscr{F}(V)$ into $\mathscr{F}(U)$, these mappings satisfying the following conditions:

(PF$_1$) The relations $U\subset V\subset W$ imply $f_{UW}=f_{UV}\circ f_{VW}$;

(PF$_2$) For every open set U $\in \mathscr{B},f_{UU}$ is the identity mapping of

A presheaf on B relative to the set of open subsets of B is simply called a presheaf on B.

Let $\mathscr{F}= (\mathscr{F}(U), f_{UV})$ be a presheaf on B, relative to a basis $\mathscr{B}$ of the topology of B. Let U be an element of the basis $\mathscr{B}$. The elements of $\mathscr{F}(U)$ are called the sections of $\mathscr{F}$ over U. If V is an element of the basis $\mathscr{B}$ containing U and $s$ an element of $\mathscr{F}(V)$, the element $f_{UV}(s)$ of $\mathscr{F}(U)$ is called the restriction of $s$ to U. If there is no danger of ambiguity concerning the mappings $f_{UV}$, the restriction of $s$ to U will be denoted by $s|U$.

Let $B'$ be an open subset of B and $\mathscr{B}'$ a basis of the topology of $B'$ such that $\mathscr{B}'\subset \mathscr{B}$. The presheaf on $B'$, relative to $\mathscr{B}'$, deduced from $\mathscr{F}$ by restriction, is called the restriction of $\mathscr{F}$ to $\mathscr{B}'$ and denoted by $\mathscr{F}|\mathscr{B}'$, the projective system $((\mathscr{F}(U))_{U\in\mathscr{B}'},(f_{UV}))$ deduced from $\mathscr{F}$ by restriction to $\mathscr{B}'$ of the set of indices (loc. cit.). When $\mathscr{F}$ is a presheaf on B and $\mathscr{B}'$ is the set of open subsets of $B'$, the presheaf $\mathscr{F}|\mathscr{B}'$ is also denoted by $\mathscr{F}|B'$ and called the presheaf deduced from $\mathscr{F}$ by restriction to $B'$.

$(F_1)$ The mapping $(f_{U_iU})_{i\in I}:\mathscr{F}(U)\rightarrow \prod_{i\in I}\mathscr{F}(U_i)$ is injective ;

$(F_2)$ For every family $(s_i)\in \prod_{i\in I}\mathscr{F}(U_i)$ such that $f_{(U_i\cap U_j)U_i}(s_i) =$

$f_{(U_i\cap U_j)U_j}(s_j)$ for every pair $(i, j)\in I\times I$, there exists an element $s$

of $\mathscr{F}(U)$ such that for every $i\in I$, one has $f_{U_iU}(s) =s_i$.

2. Subsheaves of a Sheaf

Let B be a topological space. Let $\mathscr{F}= (\mathscr{F}(U), f_{UV})$ be a presheaf on B, relative to a basis $\mathscr{B}$ of the topology of B.

Suppose given, for every open set $U\in \mathscr{B}$, a subset $\mathscr{L}(U)$ of $\mathscr{F}(U)$. If one has $f_{UV}(\mathscr{L}(V))\subset \mathscr{L}(U)$ for every pair $(U,V)$ of elements of $\mathscr{B}$ such that $U\subset V$, the pair $\mathscr{L}= ((\mathscr{L}(U))_{U\in\mathscr{B}},(f'_{UV}))$, where $f'_{UV}:\mathscr{L}(V)\rightarrow \mathscr{L}(U)$ is the map deduced from $f_{UV}$, is a presheaf. Such a presheaf is called a sub-presheaf of $\mathscr{F}$. Since the maps $f'_{UV}$ are determined by the given presheaf $\mathscr{F}$ and the family $(\mathscr{L}(U))_{U\in\mathscr{B}}$, one also says, by abuse of language, that the family $(\mathscr{L}(U))_{U\in\mathscr{B}}$ is a sub-presheaf of $\mathscr{F}$.

Suppose now that $\mathscr{F}$ is a sheaf on B and let, for every open subset U of B$,\mathscr{L}(U)$, a subset of $\mathscr{F}(U)$. In order that $(\mathscr{L}(U))_{U\in\mathscr{B}}$ be a sub-presheaf of $\mathscr{F}$, and that this presheaf be a sheaf, it is necessary and sufficient that the following condition be satisfied:

(F) Let $(U_i)_{i\in I}$ be a family of open sets of B, U its union, and $s$ an

element of $\mathscr{F}(U)$. In order that $s$ belong to $\mathscr{L}(U)$, it is necessary and

sufficient that for every $i$ in I$,f_{U_iU}(s)$ belong to $\mathscr{L}(U_i)$.

Indeed, if condition (F) is satisfied, one has $f_{UV}(\mathscr{L}(V))\subset \mathscr{L}(U)$ for every pair $(U,V)$ of open sets of B such that $U\subset V$ and the properties $(F_1)$ and $(F_2)$ relative to the subpresheaf $(\mathscr{L}(U))$ result from the analogous properties relative to the sheaf $\mathscr{F}$. The converse is immediate.

When condition (F) is satisfied, one says that $(\mathscr{L}(U))$ is a subsheaf of the sheaf $\mathscr{F}$.

3. Examples of sheaves

Let B be a topological space.

1) Sheaves of mappings

Let X be a set. For every open set U of B, denote by $\mathscr{F}(U; X)$ the set of mappings from U into X (E, II, p. 31). For every pair $(U,V)$ of open sets of B such that $U\subset V$, let $r_{UV}:\mathscr{F}(V; X)\rightarrow$ $\mathscr{F}(U; X)$ be the restriction mapping $f\mapsto f|U$. It is clear that the pair $(\mathscr{F}(U; X), r_{UV})$ is a sheaf on B. It is called the sheaf on B of mappings with values in X, and is denoted by $\mathscr{F}(B; X)$.

2) Sheaves of continuous mappings

Let X be a topological space. For every open set U of B, let $\mathscr{C}(U; X)$ be the set of continuous mappings from U into X. Then, $(\mathscr{C}(U; X))$ is a subsheaf of the sheaf $\mathscr{F}(B; X)$ by prop. 4 of TG, I, p. 19. The sheaf thus obtained is denoted by $\mathscr{C}(B; X)$ and called the sheaf on B of continuous mappings with values in X. In the particular case where X is endowed with the discrete topology, the sheaf $\mathscr{C}(B; X)$ takes the name of sheaf on B of locally constant mappings with values in X.

3) Sheaves of continuous sections

Let E be a topological space and let $p: E\rightarrow B$ be a continuous mapping. For every open set U of B, denote by $\mathscr{S}(U;p)$ (or $\mathscr{S}(U; E)$ when there is no possible confusion) the set of continuous sections of $p$ above U. The family $(\mathscr{S}(U;p))$ is a subsheaf of the sheaf $\mathscr{C}(B; E)$. The sheaf thus obtained is denoted by $\mathscr{S}(B; E)$ or simply $\mathscr{S}(E)$ and called the sheaf on B of continuous sections of the B-space $(E, p)$. We shall see in no$^o6$ below that every sheaf on B is isomorphic to the sheaf on B of continuous sections of an étalé B-space.

4) Sheaves of B-morphisms

Let $(E, p)$ and $(E', p')$ be B-spaces. For every open set U of E, denote by $\mathscr{C}_B(U; E')$ the set of B-morphisms from $(U, p|U)$ into $(E', p')$. The family $(\mathscr{C}_B(U; E'))$ is a subsheaf of the sheaf $\mathscr{C}(E; E')$. The sheaf thus obtained is denoted by $\mathscr{C}_B(E; E')$ and called the sheaf on E of B-morphisms with values in $(E', p')$. When $(E, p)$ is equal to $(B$, Id$_B)$, this sheaf is the sheaf $\mathscr{S}(B; E)$ of example 3.

For every open set U of B, let $\mathscr{M}(U)$ be the set of U-morphisms

from $\overset{-1}{p}(U)$ into $(^-{p'}^1)(U)$. For every pair $(U,V)$ of open sets of B such that $U\subset V$, let $m_{UV}:\mathscr{M}(V)\rightarrow \mathscr{M}(U)$ be the mapping which to a V-morphism

$f:\overset{-1}{p}(V)\rightarrow (^-{p'}^1)(V)$ associates the U-morphism from $\overset{-1}{p}(U)$ into $(^-{p'}^1)(U)$ deduced from $f$ by passing to the subsets. Then $(\mathscr{M}(U), m_{UV})$ is a sheaf on B. It is denoted by $\mathscr{M}$or$_B(E; E')$ and is called the sheaf on B of B-morphisms from $(E, p)$ into $(E', p')$.

For every open set U of B, let $\mathscr{I}$s(U) be the subset of $\mathscr{M}(U)$

consisting of the U-isomorphisms of $\overset{-1}{p}(U)$ into $(^-{p'}^1)(U)$. The family $(\mathscr{I}$s(U)) is a sub-sheaf of the sheaf $\mathscr{M}$or$_B(E; E')$ of morphisms from $(E, p)$ into $(E', p')$. The sheaf thus obtained is denoted $\mathscr{I}$som$_B(E; E')$ and is called the sheaf on B of B-isomorphisms from $(E, p)$ into $(E', p')$.

5) Sheaves of mappings of class $C^r$

Let X and Y be manifolds of class $C^r$ over a field K (the conventions concerning K and $r$ being those of VAR, R). For every open set U of X, let $\mathscr{C}^r(U; Y)$ be the set of morphisms of class $C^r$ from U into Y. The family $(\mathscr{C}^r(U; Y))$ is a sub-sheaf of the sheaf $\mathscr{C}(X; Y)$. The sheaf thus obtained is denoted $\mathscr{C}^r(X; Y)$ and is called the sheaf on X of mappings of class $C^r$ with values in Y (cf. VAR, R, 5.4.2).

6) Sheaves of subspaces

If U and V are open sets of B such that $U\subset V$, denote by $i_{UV}:\mathfrak{P}(V)\rightarrow$ $\mathfrak{P}(U)$ the mapping which, to a subset A of V, associates $A\cap U$. The pair $(\mathfrak{P}(U), i_{UV})$ is a sheaf, called the sheaf of subspaces of B and denoted $\mathfrak{P}(B)$. Indeed, if X denotes the set $\{0; 1\}$, the mapping which associates to every subset A of U its characteristic function $\varphi^U_A: U\rightarrow X$ is a bijection of $\mathfrak{P}(U)$ onto $\mathscr{F}(U; X)$ (E, III, p. 38) ; moreover, if U and V are open sets such that $U\subset V$, for every subset A of V$,\varphi^U_{A\cap U}$ is the restriction to U of $\varphi^V_A$ so that $\mathfrak{P}(B)$ is identified with the sheaf on B of mappings with values in X.

Let, for every open set U of B$,\mathscr{L}(U)$ be a subset of $\mathfrak{P}(U)$. In order that $(\mathscr{L}(U))$ be a sub-sheaf of $\mathfrak{P}(B)$, it is necessary and sufficient that the following condition be satisfied:

$(F')$ Let $(U_i)_{i\in I}$ be a family of open sets of B, U its union, and A a

subset of U ; in order that A belong to $\mathscr{L}(U)$, it is necessary and

sufficient that for every $i$ in I, $A\cap U_i$ belong to $\mathscr{L}(U_i)$.

For example, if $\mathscr{L}(U)$ is the set of closed subsets of U, condition $(F')$ is satisfied.

7) Products of sheaves

Let $\mathscr{B}$ be a basis of the topology of B and let I be a set. For every $i\in I$, let $\mathscr{F}_i= (\mathscr{F}_i(U), f_{i,UV})$ be a presheaf on B relative to the basis $\mathscr{B}$. For every open set $U\in \mathscr{B}$, put $\mathscr{F}(U) =\prod_{i\in I}\mathscr{F}_i(U)$, and for every pair $(U,V)$ of elements of $\mathscr{B}$ such that $U\subset V$, denote by $f_{UV}$ the mapping $(f_{i,UV})_{i\in I}:\mathscr{F}(V)\rightarrow \mathscr{F}(U)$. Then $(\mathscr{F}(U), f_{UV})$ is a presheaf on B relative to $\mathscr{B}$ called the product presheaf of the family $(\mathscr{F}_i)$ and denoted $\prod_{i\in I}\mathscr{F}_i$. It is a sheaf if for every $i\in I,\mathscr{F}_i$ is a sheaf.

4. Morphisms of presheaves

In other words (E, III, p. 54), a morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$ is a family $(\varphi_U)_{U\in\mathscr{B}}$ such that :

(MPF$_1$) For every open set U belonging to $\mathscr{B},\varphi_U$ is a mapping

from $\mathscr{F}(U)$ into $\mathscr{G}(U)$ ;

(MPF$_2$) For every pair $(U,V)$ of open sets belonging to $\mathscr{B}$ such that

$U\subset V$, one has $\varphi_U\circ f_{UV}=g_{UV}\circ \varphi_V$.

When $\mathscr{F}$ and $\mathscr{G}$ are sheaves, a morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$ is also called a morphism of sheaves. If $\mathscr{F}$ and $\mathscr{G}$ are presheaves on B relative to $\mathscr{B}$, the morphisms of presheaves from $\mathscr{F}$ to $\mathscr{G}$ constitute a set denoted Mor($\mathscr{F};\mathscr{G}$). Instead of saying : “let $\varphi$ be a morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$”, one will often say “let $\varphi :\mathscr{F}\rightarrow \mathscr{G}$ be a morphism of presheaves”.

Let $\mathscr{F},\mathscr{G},\mathscr{H}$ be presheaves on B relative to $\mathscr{B}$ and let $\varphi :\mathscr{F}\rightarrow \mathscr{G}$, $\psi :\mathscr{G}\rightarrow \mathscr{H}$ be morphisms of presheaves. The family $(\psi_U\circ \varphi_U)_{U\in\mathscr{B}}$ is a morphism of presheaves from $\mathscr{F}$ into $\mathscr{H}$ which is denoted by $\psi \circ \varphi$. The family (Id$_{\mathscr{F}(U)}$)$_{U\in\mathscr{B}}$ is a morphism of presheaves from $\mathscr{F}$ into itself which is denoted by Id$_{\mathscr{F}}$.

For a morphism of presheaves $\varphi = (\varphi_U):\mathscr{F}\rightarrow \mathscr{G}$ to be an isomorphism, it is necessary and sufficient that, for every open subset U of $\mathscr{B},\varphi_U$ be a bijection from $\mathscr{F}(U)$ onto $\mathscr{G}(U)$. It is equivalent to saying that there exists a morphism of presheaves $\psi :\mathscr{G}\rightarrow \mathscr{F}$ such that $\psi \circ \varphi =$ Id$_{\mathscr{F}}$ and $\varphi \circ \psi =$ Id$_{\mathscr{G}}$.

Let $\mathscr{F}$ and $\mathscr{G}$ be presheaves on B, relative to a base $\mathscr{B}$ of the topology of B, let $B'$ be an open subset of B and let $\mathscr{B}'$ be a base of the topology of $B'$ such that $\mathscr{B}'\subset \mathscr{B}$. Let $\varphi = (\varphi_U)_{U\in\mathscr{B}}$ be a morphism of presheaves from $\mathscr{F}$ into $\mathscr{G}$. Then $(\varphi_U)_{U\in\mathscr{B}'}$ is a morphism of presheaves from $\mathscr{F}|\mathscr{B}'$ into $\mathscr{G}|\mathscr{B}'$, which is denoted by $\varphi |\mathscr{B}'$. When $\mathscr{B}$ is the set of open subsets of B and $\mathscr{B}'$ the set of open subsets of $B',\varphi |\mathscr{B}'$ is a morphism of presheaves from $\mathscr{F}|B'$ into $\mathscr{G}|B'$ and is also denoted by $\varphi |B'$.

5. Étale space associated with a presheaf

Let B be a topological space, $\mathscr{B}$ a basis of the topology of B and $\mathscr{F}= (\mathscr{F}(U), r_{UV})$ a presheaf on B relative to the basis $\mathscr{B}$. Let L be the set of couples $(U, s)$ with U $\in \mathscr{B}$ and $s\in \mathscr{F}(U)$. Let $X_{\mathscr{F}}$ denote the sum space of the family $(U)_{(U,s)\in L}$. Thus $X_{\mathscr{F}}$ is the set of triples $(U, s, x)$ where $U\in \mathscr{B},s\in \mathscr{F}(U),x\in U$. Let $R_{\mathscr{F}}$ be the relation in the set $X_{\mathscr{F}}$ defined by $R_{\mathscr{F}}((U, s, x),(U', s', x'))$ if and only if « $x$ = $x'$ and there exists W $\in \mathscr{B}$ such that $x\in W$, W $\subset U\cap U'$ and $r_{WU}(s) =r_{WU'}(s')$ ». The relation $R_{\mathscr{F}}$ is an equivalence relation in $X_{\mathscr{F}}:$ it is, by definition, reflexive and symmetric; let us prove that it is transitive. Let $\xi = (U, s, x),\xi '= (U', s', x')$ and $\xi ''$ = $(U'', s'', x'')$ be elements of $X_{\mathscr{F}}$ such that one has $R_{\mathscr{F}}(\xi , \xi ')$ and $R_{\mathscr{F}}(\xi ', \xi '')$. We then have $x=x'$ = $x''$ and there exist two elements $W'$ and $W''$ of $\mathscr{B}$ containing $x$ such that $W'\subset U\cap U',W''\subset U'\cap U'',r_{W'U}(s) =$ $r_{W'U'}(s'),r_{W''U'}(s') =r_{W''U''}(s'')$. Let W be an element of $\mathscr{B}$ containing $x$ and contained in $W'\cap W''$. We then have $W\subset U\cap U''$,

and, analogously, $r_{WU'}(s') =r_{WU''}(s'')$. Consequently, we have $R_{\mathscr{F}}(\xi , \xi '')$ and the relation $R_{\mathscr{F}}$ is transitive.

Let $E_{\mathscr{F}}$ denote the quotient set $X_{\mathscr{F}}/R_{\mathscr{F}}$ and $[U, s, x]$ the canonical image in $E_{\mathscr{F}}$ of an element $(U, s, x)$ of $X_{\mathscr{F}}$. For $U\in \mathscr{B}$ and $s\in \mathscr{F}(U)$, let $\sigma_{\mathscr{F}}(U, s): U\rightarrow E_{\mathscr{F}}$ denote the mapping $x\mapsto [U, s, x]$. Endow the set $E_{\mathscr{F}}$ with the quotient topology, that is, the finest topology for which the mappings $\sigma_{\mathscr{F}}(U, s)$ for $U\in \mathscr{B}$ and $s\in \mathscr{F}(U)$ are continuous. The mapping pr$_3: X_{\mathscr{F}}\rightarrow B$ defines, after passing to the quotient, a continuous mapping $p: E_{\mathscr{F}}\rightarrow B$ : one has $p([U, s, x]) =x$.

Let $a$ be a point of B. The set $\mathscr{B}(a)$ of open sets $U\in \mathscr{B}$ containing $a$ and ordered by the relation $\supset$ is filtering. From $\mathscr{F}$ one deduces, by restriction of the set of indices to $\mathscr{B}(a)$, an inductive system $((\mathscr{F}(U))_{U\in\mathscr{B}(a)},(r_{UV}))$. By definition (E, III, p. 60), the inductive limit of this system is the quotient of the set of pairs $(U, s)$ such that $a\in U$ and $s\in \mathscr{F}(U)$ by the equivalence relation R defined by $R((U, s),(U', s'))$ if and only if there exists W $\in \mathscr{B}$ containing $a$ and contained in $U\cap U'$ and such that $r_{WU}(s) =r_{WU'}(s')$. This limit is therefore identified with the stalk $\mathscr{F}_a$ of $\mathscr{F}$ at $a$, by definition of inductive limits.

Let $\mathscr{G}$ be a presheaf on B relative to the basis $\mathscr{B}$ and let $\varphi = (\varphi_U)_{U\in\mathscr{B}}$ be a morphism of presheaves from $\mathscr{F}$ into $\mathscr{G}$. The mapping $(U, s, x)\mapsto$ $(U, \varphi_U(s), x)$ from $X_{\mathscr{F}}$ into $X_{\mathscr{G}}$ is compatible with the equivalence relations $R_{\mathscr{F}}$ and $R_{\mathscr{G}}$, by definition of a morphism of presheaves. Let us denote by $E(\varphi ): E_{\mathscr{F}}\rightarrow E_{\mathscr{G}}$ the mapping deduced from it by passing to the quotients. For every $U\in \mathscr{B}$ and every $s\in \mathscr{F}(U)$, we have

consequently, the mapping $E(\varphi )$ is continuous. The mapping $E(\varphi )$ is a B-morphism; it is said to be the B-morphism of $E_{\mathscr{F}}$ into $E_{\mathscr{G}}$ associated with the morphism of presheaves $\varphi$. For every $a\in B$, by restriction to the fibres at $a$, $E(\varphi )$ defines a mapping of the stalk $\mathscr{F}_a$ of $\mathscr{F}$ into the stalk $\mathscr{G}_a$ of $\mathscr{G}$; it is denoted by $\varphi_a$. It is also the inductive limit of the mappings $\varphi_U$ (E, III, p. 63), where U ranges over the set $\mathscr{B}(a)$ of open sets belonging to the base $\mathscr{B}$ and containing $a$.

We have E(Id$_{\mathscr{F}}$) $=$ Id$_{E_{\mathscr{F}}}$.

Let $\mathscr{H}$ be a presheaf on B relative to $\mathscr{B}$ and let $\psi = (\psi_U)$ be a morphism of presheaves from $\mathscr{G}$ into $\mathscr{H}$. For $[U, s, x]\in E_{\mathscr{F}}$, we have

Consequently, we have $E(\psi \circ \varphi ) = E(\psi )\circ E(\varphi )$. In particular, if $a$ is a point of B, $(\psi \circ \varphi )_a=\psi_a\circ \varphi_a$.

If $\varphi$ is an isomorphism, the same is true of $E(\varphi )$.