Definition 25.3.3. Let $\mathcal{C}$ be a site. Assume $\mathcal{C}$ has fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ be an object of $\mathcal{C}$. A *hypercovering of $X$* is a simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ such that

The object $K_0$ is a covering of $X$ for the site $\mathcal{C}$.

For every $n \geq 0$ the canonical morphism

\[ K_{n + 1} \longrightarrow (\text{cosk}_ n \text{sk}_ n K)_{n + 1} \]is a covering in the sense defined above.

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