Lemma 14.27.2 (019T)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 14: Simplicial Methods
Section 14.27: Homotopies in abelian categories
Lemma 14.27.2 (cite)

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Lemma 14.27.2. Let $\mathcal{A}$ be an additive category. Let $a : U \to V$ be a morphism of simplicial objects of $\mathcal{A}$ . If $a$ is a homotopy equivalence, then $s(a) : s(U) \to s(V)$ is a homotopy equivalence of chain complexes. If in addition $\mathcal{A}$ is abelian, then also $N(a) : N(U) \to N(V)$ is a homotopy equivalence of chain complexes.

Proof. Omitted. See Lemma 14.27.1 above. $\square$

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