Kerodon

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

Definition 3.4.2.1. A commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]

is a homotopy pushout square if, for every Kan complex $X$, the diagram of Kan complexes

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}(A_{01}, X) \ar [r] \ar [d] & \operatorname{Fun}(A_{0}, X)\\ \operatorname{Fun}(A_1, X) \ar [r] & \operatorname{Fun}(A, X) } \]

is homotopy pullback square (Definition 3.4.1.1).