The Stacks project

Example 25.3.5 (Hypercovering by a simplicial object of the site). Let $\mathcal{C}$ be a site with fibre products. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Let $U$ be a simplicial object of $\mathcal{C}$. As usual we denote $U_ n = U([n])$. Finally, assume given an augmentation

\[ a : U \to X \]

In this situation we can consider the simplicial object $K$ of $\text{SR}(\mathcal{C}, X)$ with terms $K_ n = \{ U_ n \to X\} $. Then $K$ is a hypercovering of $X$ in the sense of Definition 25.3.3 if and only if the following three conditions1 hold:

  1. $\{ U_0 \to X\} $ is a covering of $\mathcal{C}$,

  2. $\{ U_1 \to U_0 \times _ X U_0\} $ is a covering of $\mathcal{C}$,

  3. $\{ U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}\} $ is a covering of $\mathcal{C}$ for $n \geq 1$.

We omit the straightforward verification.

[1] As $\mathcal{C}$ has fibre products, the category $\mathcal{C}/X$ has all finite limits. Hence the required coskeleta exist by Simplicial, Lemma 14.19.2.

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