Example 2

Let B be a topological space and A a subspace of B. Let $i: A\rightarrow B$ denote the canonical injection. For every sheaf $\mathscr{G}$ on B, let $\mathscr{G}_A$ denote the sheaf $i^*(\mathscr{G})$, and say that $\mathscr{G}_A$ is the sheaf on A induced by the sheaf $\mathscr{G}$. The sheaf $\mathscr{G}_A$ is identified with the sheaf $\mathscr{S}(A; (E_{\mathscr{G}})_A)$ of sections of the étalé A-space induced by $E_{\mathscr{G}}$ over A.

Suppose that A is an open subspace of B, and let $\mathscr{G}$ be a sheaf on B. By definition, the sheaf $\mathscr{G}_A$ is the sheaf $\widetilde{\mathscr{G}}|A$ deduced from $\widetilde{\mathscr{G}}$ by restriction to the open set A (I, p. 43). Let $\sigma_{\mathscr{G}}:\mathscr{G}\rightarrow \widetilde{\mathscr{G}}$ be the canonical isomorphism (I, p. 55, cor. 1). Then $\sigma_{\mathscr{G}}|A$ is an isomorphism, called canonical, of the sheaf $\mathscr{G}|A$ onto the sheaf $\mathscr{G}_A$ which is called the canonical isomorphism of $\mathscr{G}|A$ onto $\mathscr{G}_A$.