Kerodon

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

Proposition 3.4.2.4 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets

\[ \xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w_ A} & & B \ar [dd] \ar [dr]^-{ w_{B} } & \\ & A' \ar [rr] \ar [dd] & & B' \ar [dd] \\ C \ar [rr] \ar [dr]^-{ w_ C} & & D \ar [dr]^-{w_ D} & \\ & C' \ar [rr] & & D', } \]

where the morphisms $w_{A}$, $w_{B}$, and $w_ C$ are weak homotopy equivalences. Then any two of the following three conditions imply the third:

$(1)$

The commutative diagram

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

is homotopy coCartesian.

$(2)$

The commutative diagram

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

is homotopy coCartesian.

$(3)$

The morphism $w_{D}$ is a weak homotopy equivalence.