Lemma 25.8.2 (01GN)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 25: Hypercoverings
Section 25.8: Adding simplices
Lemma 25.8.2 (cite)

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Lemma 25.8.2. Let $\mathcal{C}$ be a site with fibre products. Let $X$ be an object of $\mathcal{C}$ . Let $K$ be a hypercovering of $X$ . Let $U \subset V$ be simplicial sets, with $U_ n, V_ n$ finite nonempty for all $n$ . Assume that $U$ and $V$ have finitely many nondegenerate simplices. Then the morphism

$\mathop{\mathrm{Hom}}\nolimits (V, K)_0 \longrightarrow \mathop{\mathrm{Hom}}\nolimits (U, K)_0$

of $\text{SR}(\mathcal{C}, X)$ is a covering.

Proof. By Lemma 25.8.1 above, it suffices to prove a simple lemma about inclusions of simplicial sets $U \subset V$ as in the lemma. And this is exactly the result of Simplicial, Lemma 14.21.8. $\square$

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