The Stacks project

Lemma 20.9.2. Let $X$ be a topological space. Let $\mathcal{F}$ be an abelian presheaf on $X$. The following are equivalent

  1. $\mathcal{F}$ is an abelian sheaf and

  2. for every open covering $\mathcal{U} : U = \bigcup _{i \in I} U_ i$ the natural map

    \[ \mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F}) \]

    is bijective.

Proof. This is true since the sheaf condition is exactly that $\mathcal{F}(U) \to \check{H}^0(\mathcal{U}, \mathcal{F})$ is bijective for every open covering. $\square$

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