The Stacks project

Lemma 12.13.5. Let $\mathcal{A}$ be an abelian category.

  1. 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.

  2. If the map $f : A_\bullet \to B_\bullet $ is a homotopy equivalence, then $f$ is a quasi-isomorphism.

Proof. Omitted. $\square$

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