Proposition 3.4.2.5 (0117)

Proposition 3.4.2.5. Suppose we are given a commutative diagram of simplicial sets

3.44

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

where $f$ is a weak homotopy equivalence. Then (3.44) is homotopy coCartesian if and only if $f'$ is a weak homotopy equivalence.

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

3.45

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

where $u$ is a homotopy equivalence of Kan complexes (Corollary 3.1.6.5). Applying Corollary 3.4.1.3, we conclude that (3.45) is homotopy Cartesian if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (3.44) is homotopy coCartesian if and only if, for every Kan complex $X$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(D, Q) \rightarrow \operatorname{Fun}(C, Q)$. By virtue of Corollary 3.1.6.5), this is equivalent to the requirement that $f'$ is a weak homotopy equivalence. $\square$