Definition 59.13.1. Let $\mathcal{F}$ be a presheaf on the site $\mathcal{C}$ and $\mathcal{U} = \{ U_ i \to U\} \in \text{Cov} (\mathcal{C})$. We define the *zeroth Čech cohomology group* of $\mathcal{F}$ with respect to $\mathcal{U}$ by

\[ \check H^0 (\mathcal{U}, \mathcal{F}) = \left\{ (s_ i)_{i\in I} \in \prod \nolimits _{i\in I }\mathcal{F}(U_ i) \text{ such that } s_ i|_{U_ i \times _ U U_ j} = s_ j |_{U_ i \times _ U U_ j} \right\} . \]

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