Kerodon

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

Corollary 7.5.7.9. Suppose we are given a commutative diagram of simplicial sets

7.53
\begin{equation} \begin{gathered}\label{equation:homotopy-pushout-as-homotopy-colimit} \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_{0} \ar [d] \\ A_{1} \ar [r] & A_{01}, } \end{gathered} \end{equation}

which we identify with a functor $\mathscr {F}: [1] \times [1] \rightarrow \operatorname{Set_{\Delta }}$. Then (7.53) is a homotopy pushout square (in the sense of Definition 3.4.2.1) if and only if $\mathscr {F}$ is a homotopy colimit diagram (in the sense of Definition 7.5.7.3).