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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