Definition 25.6.1. Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has equalizers and fibre products. Let $\mathcal{G}$ be a presheaf of sets. A *hypercovering of $\mathcal{G}$* is a simplicial object $K$ of $\text{SR}(\mathcal{C})$ endowed with an augmentation $F(K) \to \mathcal{G}$ such that

$F(K_0) \to \mathcal{G}$ becomes surjective after sheafification,

$F(K_1) \to F(K_0) \times _\mathcal {G} F(K_0)$ becomes surjective after sheafification, and

$F(K_{n + 1}) \longrightarrow F((\text{cosk}_ n \text{sk}_ n K)_{n + 1})$ for $n \geq 1$ becomes surjective after sheafification.

We say that a simplicial object $K$ of $\text{SR}(\mathcal{C})$ is a *hypercovering* if $K$ is a hypercovering of the final object $*$ of $\textit{PSh}(\mathcal{C})$.

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