Lemma 12.13.1 (010W)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 12: Homological Algebra
Section 12.13: Complexes
Lemma 12.13.1 (cite)

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Hom functors of $\text{Ch}(\mathcal{A})$ respect the homotopy relation.

Lemma 12.13.1. Let $\mathcal{A}$ be an additive category. Let $f, g : B_\bullet \to C_\bullet$ be morphisms of chain complexes. Suppose given morphisms of chain complexes $a : A_\bullet \to B_\bullet$ , and $c : C_\bullet \to D_\bullet$ . If $\{ h_ i : B_ i \to C_{i + 1}\}$ defines a homotopy between $f$ and $g$ , then $\{ c_{i + 1} \circ h_ i \circ a_ i\}$ defines a homotopy between $c \circ f \circ a$ and $c \circ g \circ a$ .

Proof. Omitted. $\square$

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Comment #7383 by Elías Guisado on May 25, 2022 at 08:51

Suggested slogan: Hom functors of $\operatorname{Ch}(\mathcal{A})$ respect the homotopy relation.

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