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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