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The Stacks project

Lemma 6.32.4. Let X be a topological space. Let i : Z \to X be the inclusion of a closed subset. Let (\mathcal{C}, F) be a type of algebraic structure with final object 0. The functor

i_* : \mathop{\mathit{Sh}}\nolimits (Z, \mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C})

is fully faithful. Its essential image consists exactly of those sheaves \mathcal{G} such that \mathcal{G}_ x = 0 for all x \in X \setminus Z.

Proof. Omitted. \square


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