Definition 59.18.1. Let $\mathcal{C}$ be a category, $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ a family of morphisms of $\mathcal{C}$ with fixed target, and $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ an abelian presheaf. We define the *Čech complex* $\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})$ by

where the first term is in degree 0, and the maps are the usual ones. Again, it is essential to allow the case $i_0 = i_1$ etc. The *Čech cohomology groups* are defined by

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