Example 1

Let B be a topological space, A a subspace of B and $\mathscr{F}= (\mathscr{F}(U), f_{UV})$ a presheaf on A. Let $i: A\rightarrow B$ denote the canonical injection. Then one has $i_*(\mathscr{F}) = (\mathscr{F}'(U), f'_{UV})$ where for every open set U of B$,\mathscr{F}'(U) =\mathscr{F}(U\cap A)$, and for every pair $(U,V)$ of open sets of B with $U\subset V,f'_{UV}=f_{(U\cap A)(V\cap A)}$.

Let now $\mathscr{G}$ be a presheaf on B. The inverse image of the presheaf $\mathscr{G}$ by $u$ is defined to be, and is denoted by $u^*(\mathscr{G})$, the sheaf $\mathscr{C}_B(A; E_{\mathscr{G}})$ on A of B-morphisms with values in the B-space $E_{\mathscr{G}}($I, p. 45, example 4). It is canonically isomorphic to the sheaf on A of sections of the étalé A-space $A\times_BE_{\mathscr{G}}$ (I, p. 9, prop. 3). From this one deduces (I, p. 52, example 2) a canonical isomorphism $\varphi$ of the étalé space associated with $u^*(\mathscr{G})$ onto the A-space $A\times_BE_{\mathscr{G}}$. If moreover $\mathscr{G}$ is a sheaf, for every point $a$ of A there is a canonical bijection $\psi_a:u^*(\mathscr{G})_a\rightarrow \mathscr{G}_{u(a)}:$ for every open neighbourhood U of $a$ in A and every B-morphism $f: U\rightarrow E_{\mathscr{G}}$, one has

Let $\mathscr{G}_1$ and $\mathscr{G}_2$ be presheaves on B and let $\varphi :\mathscr{G}_1\rightarrow \mathscr{G}_2$ be a morphism of presheaves. By base change of the morphism of étalé B-spaces $E(\varphi ): E_{\mathscr{G}_1}\rightarrow E_{\mathscr{G}_2}$ one obtains a morphism of étalé A-spaces $A\times_BE_{\mathscr{G}_1}\rightarrow A\times_BE_{\mathscr{G}_2}$, whence a morphism of sheaves on A$,u^*(\varphi ):u^*(\mathscr{G}_1)\rightarrow u^*(\mathscr{G}_2)$. Let $\mathscr{G}_3$ be a presheaf on B and let $\psi :\mathscr{G}_2\rightarrow \mathscr{G}_3$ be a morphism of presheaves. Then one has the equality $u^*(\psi \circ \varphi ) =u^*(\psi )\circ u^*(\varphi )$.

Let C be a topological space and let $v: B\rightarrow$ C be a continuous mapping. If $\mathscr{G}$ is a presheaf on C, the sheaves $u^*(v^*(\mathscr{G}))$ and $(v\circ u)^*(\mathscr{G})$ are canonically identified (I, p. 5). If $\varphi :\mathscr{G}_1\rightarrow \mathscr{G}_2$ is a morphism of presheaves on C, one has moreover $u^*(v^*(\varphi )) = (v\circ u)^*(\varphi )$.