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

\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd]^{q} \\ A \ar [rr] \ar [dr] & & B \ar [dr] & \\ & C \ar [rr] & & D } \]

having the property that the faces

3.51
\begin{equation} \label{equation:mather-second-cube-statement} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \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 pullback squares. If the bottom face

\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]

is a homotopy pushout square, then the top face

\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] \\ \overline{C} \ar [r] & \overline{D} } \]

is also a homotopy pushout square.

Proof. Using Proposition 3.1.7.1, 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@R =50pt@C=50pt{ \overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd]^{w} \\ 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 a homotopy pushout square by virtue of Proposition 3.4.3.2. Since the diagrams (3.51) are homotopy pullback squares, the vertical arrows in (3.52) are weak homotopy equivalences. Applying Proposition 3.4.2.4, we conclude that the top face

\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] \\ \overline{C} \ar [r] & \overline{D} } \]

is also a homotopy pushout square. $\square$