Kerodon

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

Theorem 3.4.3.3 (Mather's Second Cube Theorem [MR402694]). Suppose we are given a cubical diagram of simplicial sets

\[ \xymatrix { \overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd]^{q} \\ A \ar [rr] \ar [dr] & & B \ar [dr] & \\ & C \ar [rr] & & D } \]

having the property that the faces

3.51
\begin{equation} \label{equation:mather-second-cube-statement} \begin{gathered} \xymatrix { \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] & \overline{A} \ar [r] \ar [d] & \overline{C} \ar [d] \\ A \ar [r] & B & A \ar [r] & C \\ \overline{C} \ar [r] \ar [d] & \overline{D} \ar [d] & \overline{B} \ar [r] \ar [d] & \overline{D} \ar [d] \\ C \ar [r] & D & B \ar [r] & D } \end{gathered}\end{equation}

are homotopy Cartesian. If the bottom face

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

is homotopy coCartesian, then the top face

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

is also homotopy coCartesian.

Proof. Using Proposition 3.1.6.1, we can factor $q$ as a composition $\overline{D} \xrightarrow {w} \overline{D}' \xrightarrow {q'} D$, where $w$ is a weak homotopy equivalence and $q$ is a Kan fibration. We then obtain another commutative diagram

3.52
\begin{equation} \label{equation:mather-second-cube3} \begin{gathered} \xymatrix { \overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd]^{q'} \\ A \times _{D} \overline{D}' \ar [rr] \ar [dr] & & B \times _{D} \overline{D}' \ar [dr] & \\ & C \times _{D} \overline{D}' \ar [rr] & & \overline{D}', } \end{gathered} \end{equation}

where the bottom face is homotopy coCartesian by virtue of Proposition 3.4.3.2. Since the diagrams (3.51) are homotopy Cartesian, the vertical arrows in (3.52) are weak homotopy equivalences. Applying Proposition 3.4.2.4, we conclude that the top face

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

is also homotopy coCartesian. $\square$