Our goal in this section is to prove a theorem of Mather (Theorem 3.4.3.3), which asserts that the collection of homotopy pushout squares is stable under the formation of homotopy pullback. This is an analogue (and consequence) of a more elementary statement about sets:
Exercise 3.4.3.1. Suppose we are given a pushout square of sets Then, for every map of sets $\overline{D} \rightarrow D$, the induced diagram is also a pushout square.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D. } \]
\[ \xymatrix@R =50pt@C=50pt{ A \times _{D} \overline{D} \ar [r] \ar [d] & B \times _{D} \overline{D} \ar [d] \\ C \times _{D} \overline{D} \ar [r] & \overline{D} } \]
Since limits and colimits in the category of simplicial sets are computed pointwise, Exercise 3.4.3.1 immediately implies that the collection of pushout squares in the category of simplicial sets is stable under the formation of pullback along any morphism of simplicial sets $q: \overline{D} \rightarrow D$. This statement has an analogue for homotopy pushout diagrams of simplicial sets, provided that we assume that $q$ is a Kan fibration.
Proposition 3.4.3.2. Suppose we are given a homotopy pushout square of simplicial sets and let $q: \overline{D} \rightarrow D$ be a Kan fibration of simplicial sets. Then the induced diagram is also a homotopy pushout square.
3.50
\begin{equation} \label{equation:mather-second-cube} \begin{gathered} \xymatrix@R =50pt@C=50pt{ A \ar [r]^-{f} \ar [d] & B \ar [d] \\ C \ar [r] & D, } \end{gathered} \end{equation}
\[ \xymatrix@R =50pt@C=50pt{ A \times _{D} \overline{D} \ar [r] \ar [d] & B \times _{D} \overline{D} \ar [d] \\ C \times _{D} \overline{D} \ar [r] & \overline{D} } \]
Proof. Choose a factorization of $f$ as a composition $A \xrightarrow {f'} B' \xrightarrow {w} B$, where $f'$ is a monomorphism and $w$ is a weak homotopy equivalence (Exercise 3.1.7.11). Set $D' = B' \coprod _{A} C$. Our assumption that (3.50) is a homotopy pushout square guarantees that the induced map $D' \rightarrow D$ is a weak homotopy equivalence (Proposition 3.4.2.11). We have a commutative diagram of simplicial sets
The left square in this diagram is a pushout square (by virtue of Exercise 3.4.3.1) and the map $A \times _{D} \overline{D} \rightarrow B' \times _{D} \overline{D}$ is a monomorphism, so it is a homotopy pushout square (Example 3.4.2.12). It follows from Corollary 3.3.7.4 that the horizontal maps on the right side of the diagram are weak homotopy equivalences, so the right square is also a homotopy pushout (Proposition 3.4.2.10). Applying Proposition 3.4.2.8, we deduce that the outer rectangle is also a homotopy pushout square, as desired. $\square$
We now formulate a homotopy-invariant version of Proposition 3.4.3.2.
Theorem 3.4.3.3 (Mather's Second Cube Theorem [MR402694]). Suppose we are given a cubical diagram of simplicial sets having the property that the faces are homotopy pullback squares. If the bottom face is a homotopy pushout square, then the top face is also a homotopy pushout square.
\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [rr] \ar [dd] \ar [dr] & & \overline{B} \ar [dd] \ar [dr] & \\ & \overline{C} \ar [rr] \ar [dd] & & \overline{D} \ar [dd]^{q} \\ A \ar [rr] \ar [dr] & & B \ar [dr] & \\ & C \ar [rr] & & D } \]
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}
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & B \ar [d] \\ C \ar [r] & D } \]
\[ \xymatrix@R =50pt@C=50pt{ \overline{A} \ar [r] \ar [d] & \overline{B} \ar [d] \\ \overline{C} \ar [r] & \overline{D} } \]
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
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 3.4.2.9, we conclude that the top face
is also a homotopy pushout square. $\square$