Kerodon

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

Definition 3.4.2.1. A commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is homotopy coCartesian if, for every Kan complex $X$, the diagram of Kan complexes

\[ \xymatrix { \operatorname{Fun}(A,X) & \operatorname{Fun}(B,X) \ar [l] \\ \operatorname{Fun}(C, X) \ar [u] & \operatorname{Fun}(D, X) \ar [u] \ar [l] } \]

is homotopy Cartesian (Definition 3.4.1.1).

A homotopy pushout square is a commutative diagram of simplicial sets

\[ \xymatrix { A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

which is homotopy coCartesian.