The Stacks project

Remark 25.8.4. A useful special case of Lemmas 25.8.2 and 25.8.3 is the following. Suppose we have a category $\mathcal{C}$ having fibre products. Let $P \subset \text{Arrows}(\mathcal{C})$ be a subset stable under base change, stable under composition, and containing all isomorphisms. Then one says a $P$-hypercovering is an augmentation $a : U \to X$ from a simplicial object of $\mathcal{C}$ such that

1. $U_0 \to X$ is in $P$,

2. $U_1 \to U_0 \times _ X U_0$ is in $P$,

3. $U_{n + 1} \to (\text{cosk}_ n\text{sk}_ n U)_{n + 1}$ is in $P$ for $n \geq 1$.

The category $\mathcal{C}/X$ has all finite limits, hence the coskeleta used in the formulation above exist (see Categories, Lemma 4.18.4). Then we claim that the morphisms $U_ n \to X$ and $d^ n_ i : U_ n \to U_{n - 1}$ are in $P$. This follows from the aforementioned lemmas by turning $\mathcal{C}$ into a site whose coverings are $\{ f : V \to U\} $ with $f \in P$ and taking $K$ given by $K_ n = \{ U_ n \to X\} $.