Proposition 3.4.2.5. Suppose we are given a commutative diagram of simplicial sets
where $f$ is a weak homotopy equivalence. Then (3.44) is a homotopy pushout square if and only if $f'$ is a weak homotopy equivalence.
Proposition 3.4.2.5. Suppose we are given a commutative diagram of simplicial sets
where $f$ is a weak homotopy equivalence. Then (3.44) 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
where $u$ is a homotopy equivalence of Kan complexes (Corollary 3.1.7.5). Applying Corollary 3.4.1.3, we conclude that (3.45) is a homotopy pullback square if and only if $u$ is a homotopy equivalence of Kan complexes. Consequently, (3.44) is homotopy pushout square if and only if, for every Kan complex $X$, the composition with $f'$ induces a homotopy equivalence $\operatorname{Fun}(D, X) \rightarrow \operatorname{Fun}(C, X)$. By virtue of Corollary 3.1.7.5, this is equivalent to the requirement that $f'$ is a weak homotopy equivalence. $\square$