Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Remark 3.4.2.8. Suppose we are given a commutative diagram of simplicial sets

3.49
\begin{equation} \label{diagram:homotopy-pushout-square2} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D. } \end{gathered} \end{equation}

Using Exercise 3.1.7.11, we can factor $f$ as a composition $A \xrightarrow {f'} B' \xrightarrow {w} B$, where $f'$ is a monomorphism and $w$ is a weak homotopy equivalence (in fact, we can even arrange that $w$ is a trivial Kan fibration). Combining Propositions 3.4.2.6 and 3.4.2.4, we conclude that diagram (3.49) is a homotopy pushout square if and only if the induced map $u: C \coprod _{A} B' \rightarrow D$ is a weak homotopy equivalence. In particular, the condition that $u$ is a weak homotopy equivalence does not depend on the choice of factorization $f = w \circ f'$.