Scholium

Let B be a topological space. To every sheaf $\mathscr{F}$ on B, one associates an étalé B-space $E_{\mathscr{F}}$ (I, p. 50, def. 4). To every étalé B-space T, one associates the sheaf $\mathscr{S}(T)$ on B of its continuous sections (I, p. 45, example 3). A canonical isomorphism of sheaves $\sigma_{\mathscr{F}}:\mathscr{F}\rightarrow \mathscr{S}(E_{\mathscr{F}})$ has been defined (I, p. 55, cor. 1) and a canonical isomorphism of étalé B-spaces $e_T: E_{\mathscr{S}(T)}\rightarrow T$ (I, p. 52, example 2).

For every pair $(\mathscr{F},\mathscr{G})$ of sheaves on B, a mapping $\varphi \mapsto E(\varphi )$ of the set of morphisms of sheaves from $\mathscr{F}$ into $\mathscr{G}$ into the set of B-morphisms from $E_{\mathscr{F}}$ into $E_{\mathscr{G}}$ has been defined (I, p. 50). One has the relations

E(Id$_{\mathscr{F}}$) $=$ Id$_{E_{\mathscr{F}}},E(\psi \circ \varphi ) = E(\psi )\circ E(\varphi )$.

For every pair $(T,U)$ of étalé spaces over B, a mapping $f\mapsto \mathscr{S}(f)$ from the set of B-morphisms of T into U to the set of sheaf morphisms of $\mathscr{S}(T)$ into $\mathscr{S}(U)$ was defined (I, p. 48, Example 1). One has the relations

$\mathscr{S}$ (Id$_T$) $=$ Id$_{\mathscr{S}(T)},\mathscr{S}(g\circ f) =\mathscr{S}(g)\circ \mathscr{S}(f)$.

With the preceding notation, the following diagrams are commutative:

(4) $\sigma_{_{\mathscr{F}}}\sigma_{_{\mathscr{G}}}$ (5) $e_{_T}e_{_U}$

$\mathscr{S}(E_{\mathscr{F}})^{\mathscr{S}(E(\varphi))}\mathscr{S}(E_{\mathscr{G}})$, T $^fU$.

This follows from I, p. 54, formula (2) for the first, and is an immediate consequence of the definitions for the second. This implies that for every pair $(\mathscr{F},\mathscr{G})$ of sheaves on B and every pair $(T,U)$ of étalé spaces over B, the mappings $\varphi \mapsto E(\varphi )$ and $f\mapsto \mathscr{S}(f)$ considered above are bijective.

These results make it possible to deduce a statement concerning étalé spaces over B from a statement concerning sheaves on B, and conversely.