The Stacks project

Lemma 6.30.10. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Denote $\mathop{\mathit{Sh}}\nolimits (\mathcal{B}, \mathcal{C})$ the category of sheaves with values in $\mathcal{C}$ on $\mathcal{B}$. There is an equivalence of categories

\[ \mathop{\mathit{Sh}}\nolimits (X, \mathcal{C}) \longrightarrow \mathop{\mathit{Sh}}\nolimits (\mathcal{B}, \mathcal{C}) \]

which assigns to a sheaf on $X$ its restriction to the members of $\mathcal{B}$.

Proof. The inverse functor in given in Lemma 6.30.9 above. Checking the obvious functorialities is left to the reader. $\square$

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