Definition 4

The spread B-space $(E_{\mathscr{F}}, p)$ defined above is called the spread B-space associated with the presheaf $\mathscr{F}$. For $x\in B$, the fibre of $E_{\mathscr{F}}$ at $x$ is called the stalk of the presheaf $\mathscr{F}$ at $x$ and is denoted by $\mathscr{F}_x$. For every open set $U\in \mathscr{B}$, every section $s\in \mathscr{F}(U)$ of $\mathscr{F}$ over U and every point $x$ of U, the element $[U, s, x]$ of $E_{\mathscr{F}}$ is called the germ at $x$ of the section $s$.

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 $E(\varphi )$.