The Stacks project

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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