Lemma 12.13.11 (0116)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.13: Complexes
Lemma 12.13.11 (cite)

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Lemma 12.13.11. Let $\mathcal{A}$ be an abelian category.

If the maps $f, g : A^\bullet \to B^\bullet$ are homotopic, then the induced maps $H^ i(f)$ and $H^ i(g)$ are equal.
If $f : A^\bullet \to B^\bullet$ is a homotopy equivalence, then $f$ is a quasi-isomorphism.

Proof. Omitted. $\square$

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