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

$\prod _{i_0\in I} \mathcal{F}(U_{i_0}) \to \prod _{i_0, i_1\in I} \mathcal{F}(U_{i_0} \times _ U U_{i_1}) \to \prod _{i_0, i_1, i_2 \in I} \mathcal{F}(U_{i_0} \times _ U U_{i_1} \times _ U U_{i_2}) \to \ldots$

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

$\check{H}^ p(\mathcal{U}, \mathcal{F}) = H^ p(\check{\mathcal{C}}^\bullet (\mathcal{U}, \mathcal{F})).$

Comment #8255 by Haohao Liu on

It seems that the existence of the fiber product $U_{i_0}\times_UU_{i_1}$ is implicitly assumed.

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