The Stacks project

Lemma 22.20.5. In Situation 22.20.2 given a diagram

\[ \xymatrix{x\ar[r]^ f\ar[d]_ a & y\ar[d]^ b\\ z\ar[r]^ g & w} \]

in $\text{Comp}(\mathcal{A})$ commuting up to homotopy. Then

  1. If $f$ is an admissible monomorphism, then $b$ is homotopic to a morphism $b'$ which makes the diagram commute.

  2. If $g$ is an admissible epimorphism, then $a$ is homotopic to a morphism $a'$ which makes the diagram commute.

Proof. To prove (1), observe that the hypothesis implies that there is some $h\in \mathop{\mathrm{Hom}}\nolimits _{\mathcal{A}}(x,w)$ of degree $-1$ such that $bf-ga=d(h)$. Since $f$ is an admissible monomorphism, there is a morphism $\pi : y \to x$ in the category $\mathcal{A}$ of degree $0$. Let $b' = b - d(h\pi )$. Then

\begin{align*} b'f = bf - d(h\pi )f = & bf - d(h\pi f) \quad (\text{since }d(f) = 0) \\ = & bf-d(h) \\ = & ga \end{align*}

as desired. The proof for (2) is omitted. $\square$


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