Lemma 13.9.3. Suppose that $f: K^\bullet \to L^\bullet$ and $g : L^\bullet \to M^\bullet$ are morphisms of complexes such that $g \circ f$ is homotopic to zero. Then

1. $g$ factors through a morphism $C(f)^\bullet \to M^\bullet$, and

2. $f$ factors through a morphism $K^\bullet \to C(g)^\bullet [-1]$.

Proof. The assumptions say that the diagram

$\xymatrix{ K^\bullet \ar[r]_ f \ar[d] & L^\bullet \ar[d]^ g \\ 0 \ar[r] & M^\bullet }$

commutes up to homotopy. Since the cone on $0 \to M^\bullet$ is $M^\bullet$ the map $C(f)^\bullet \to C(0 \to M^\bullet ) = M^\bullet$ of Lemma 13.9.2 is the map in (1). The cone on $K^\bullet \to 0$ is $K^\bullet $ and applying Lemma 13.9.2 gives a map $K^\bullet  \to C(g)^\bullet$. Applying $[-1]$ we obtain the map in (2). $\square$

Comment #291 by arp on

Just a remark: this lemma is a special case of Lemma 13.8.2 (TAG 014F), namely take $f: K^{\bullet} \to L^{\bullet}$ for $f_1$, $g: L^{\bullet} \to M^{\bullet}$ for $b$, and $K^{\bullet}_2 = 0$.

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