The Stacks project

Definition 59.18.1. Let $\mathcal{C}$ be a category. Let $\mathcal{U} = \{ U_ i \to U\} _{i \in I}$ be a family of morphisms of $\mathcal{C}$ with fixed target. Assume that all the fibre products $U_{i_0} \times _ U \ldots \times _ U U_{i_ p}$ exists in $\mathcal{C}$. Let $\mathcal{F} \in \textit{PAb}(\mathcal{C})$ be 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. The Čech cohomology groups are defined by

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

Comments (2)

Comment #8255 by Haohao Liu on

It seems that the existence of the fiber product is implicitly assumed.

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