Proposition 3.4.2.10 (0117)

Proposition 3.4.2.10. Suppose we are given a commutative diagram of simplicial sets

3.46

\begin{equation} \label{diagram:homotopy-pushout-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ A' \ar [r]^-{f'} & B' } \end{gathered} \end{equation}

where $f$ is a weak homotopy equivalence. Then (3.46) is a homotopy pushout square if and only if $f'$ is a weak homotopy equivalence.

Proof. For every Kan complex $X$, we obtain a commutative diagram of simplicial sets

3.47

\begin{equation} \label{diagram:pushout-lemma} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l]_{u} \\ \operatorname{Fun}(A',X) \ar [u] & \operatorname{Fun}(B',X), \ar [l]_{u'} \ar [u] } \end{gathered} \end{equation}

where $u$ is a homotopy equivalence of Kan complexes (Proposition 3.1.6.17). Applying Corollary 3.4.1.5, we conclude that (3.47) is a homotopy pullback square if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (3.46) is a homotopy pushout square if and only if, for every Kan complex $X$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(B', X) \rightarrow \operatorname{Fun}(A', X)$. By virtue of Proposition 3.1.6.17, this is equivalent to the requirement that $f'$ is a weak homotopy equivalence. $\square$