The Stacks project

Lemma 12.13.11. 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 $f : A^\bullet \to B^\bullet $ is a homotopy equivalence, then $f$ is a quasi-isomorphism.

Proof. Omitted. $\square$

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