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The Stacks project

Lemma 6.31.11. Let X be a topological space. Let j : U \to X be the inclusion of an open subset. Let (\mathcal{C}, F) be a type of algebraic structure such that \mathcal{C} has an initial object e. The functor

j_! : \mathop{\mathit{Sh}}\nolimits (U, \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 = e for all x \in X \setminus U.

Proof. Omitted. \square


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