The Stacks project

Lemma 6.33.4. Let $X$ be a topological space. Let $X = \bigcup _{i\in I} U_ i$ be an open covering. The functor which associates to a sheaf of sets $\mathcal{F}$ the following collection of glueing data

\[ (\mathcal{F}|_{U_ i}, (\mathcal{F}|_{U_ i})|_{U_ i \cap U_ j} \to (\mathcal{F}|_{U_ j})|_{U_ i \cap U_ j} ) \]

with respect to the covering $X = \bigcup U_ i$ defines an equivalence of categories between $\mathop{\mathit{Sh}}\nolimits (X)$ and the category of glueing data. A similar statement holds for abelian sheaves, resp. sheaves of algebraic structures, resp. sheaves of $\mathcal{O}$-modules.

Proof. The functor is fully faithful by Lemma 6.33.1 and essentially surjective (via an explicitly given quasi-inverse functor) by Lemma 6.33.2. $\square$

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