Lemma 22.7.4. Let $(A, \text{d})$ be a differential graded algebra. Let $\alpha : K \to L$ be a homomorphism of differential graded $A$-modules. There exists a factorization

$\xymatrix{ K \ar[r]^{\tilde\alpha } \ar@/_1pc/[rr]_\alpha & \tilde L \ar[r]^\pi & L }$

in $\text{Mod}_{(A, \text{d})}$ such that

1. $\tilde\alpha$ is an admissible monomorphism (see Definition 22.7.1),

2. there is a morphism $s : L \to \tilde L$ such that $\pi \circ s = \text{id}_ L$ and such that $s \circ \pi$ is homotopic to $\text{id}_{\tilde L}$.

Proof. The proof is identical to the proof of Derived Categories, Lemma 13.9.6. Namely, we set $\tilde L = L \oplus C(1_ K)$ and we use elementary properties of the cone construction. $\square$

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