Example 25.3.6 (Čech hypercovering associated to a cover). Let $\mathcal{C}$ be a site with fibre products. Let $U \to X$ be a morphism of $\mathcal{C}$ such that $\{ U \to X\}$ is a covering of $\mathcal{C}$1. Consider the simplical object $K$ of $\text{SR}(\mathcal{C}, X)$ with terms

$K_ n = \{ U \times _ X U \times _ X \ldots \times _ X U \to X\} \quad (n + 1 \text{ factors})$

Then $K$ is a hypercovering of $X$. This example is a special case of both Example 25.3.4 and of Example 25.3.5.

[1] A morphism of $\mathcal{C}$ with this property is sometimes called a “cover”.

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