The Stacks project

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

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

  2. 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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