The Stacks project

Lemma 22.27.7. In Situation 22.27.2 let $x_1 \to x_2 \to \ldots \to x_ n$ be a sequence of composable morphisms in $\text{Comp}(\mathcal{A})$. Then there exists a commutative diagram in $\text{Comp}(\mathcal{A})$:

\[ \xymatrix{x_1\ar[r] & x_2\ar[r] & \ldots \ar[r] & x_ n\\ y_1\ar[r]\ar[u] & y_2\ar[r]\ar[u] & \ldots \ar[r] & y_ n\ar[u]} \]

such that each $y_ i\to y_{i+1}$ is an admissible monomorphism and each $y_ i\to x_ i$ is a homotopy equivalence.

Proof. The case for $n=1$ is trivial: one simply takes $y_1 = x_1$ and the identity morphism on $x_1$ is in particular a homotopy equivalence. The case $n = 2$ is given by Lemma 22.27.6. Suppose we have constructed the diagram up to $x_{n - 1}$. We apply Lemma 22.27.6 to the composition $y_{n - 1} \to x_{n-1} \to x_ n$ to obtain $y_ n$. Then $y_{n - 1} \to y_ n$ will be an admissible monomorphism, and $y_ n \to x_ n$ a homotopy equivalence. $\square$

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