Example 25.3.4 (Čech hypercoverings). Let \mathcal{C} be a site with fibre products. Let \{ U_ i \to X\} _{i \in I} be a covering of \mathcal{C}. Set K_0 = \{ U_ i \to X\} _{i \in I}. Then K_0 is a 0-truncated simplicial object of \text{SR}(\mathcal{C}, X). Hence we may form
Clearly K passes condition (1) of Definition 25.3.3. Since all the morphisms K_{n + 1} \to (\text{cosk}_ n \text{sk}_ n K)_{n + 1} are isomorphisms by Simplicial, Lemma 14.19.10 it also passes condition (2). Note that the terms K_ n are the usual
A hypercovering of X of this form is called a Čech hypercovering of X.
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