The Stacks project

Lemma 25.8.3. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$. Let $K$ be a hypercovering of $X$. Then

  1. $K_ n$ is a covering of $X$ for each $n \geq 0$,

  2. $d^ n_ i : K_ n \to K_{n - 1}$ is a covering for all $n \geq 1$ and $0 \leq i \leq n$.

Proof. Recall that $K_0$ is a covering of $X$ by Definition 25.3.3 and that this is equivalent to saying that $K_0 \to \{ X \to X\} $ is a covering in the sense of Definition 25.3.1. Hence (1) follows from (2) because it will prove that the composition $K_ n \to K_{n - 1} \to \ldots \to K_0 \to \{ X \to X\} $ is a covering by Lemma 25.3.2.

Proof of (2). Observe that $\mathop{\mathrm{Mor}}\nolimits (\Delta [n], K)_0 = K_ n$ by Simplicial, Lemma 14.17.4. Therefore (2) follows from Lemma 25.8.2 applied to the $n + 1$ different inclusions $\Delta [n - 1] \to \Delta [n]$. $\square$

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