Proposition 3.4.2.9 (Homotopy Invariance). Suppose we are given a commutative diagram of simplicial sets
\[ \xymatrix@C =50pt{ A \ar [rr] \ar [dd] \ar [dr]^-{w} & & A_{0} \ar [dd] \ar [dr]^-{ w_0 } & \\ & B \ar [rr] \ar [dd] & & B_0 \ar [dd] \\ A_{1} \ar [rr] \ar [dr]^-{ w_1} & & A_{01} \ar [dr]^-{w_{01}} & \\ & B_1 \ar [rr] & & B_{01}, } \]
where the morphisms $w$, $w_{0}$, and $w_{1}$ are weak homotopy equivalences. Then any two of the following three conditions imply the third:
- $(1)$
The back face
\[ \xymatrix@R =50pt@C=50pt{ A \ar [r] \ar [d] & A_0 \ar [d] \\ A_1 \ar [r] & A_{01} } \]is a homotopy pushout square.
- $(2)$
The front face
\[ \xymatrix@R =50pt@C=50pt{ B \ar [r] \ar [d] & B_0 \ar [d] \\ B_1 \ar [r] & B_{01} } \]is a homotopy pushout square.
- $(3)$
The morphism $w_{01}$ is a weak homotopy equivalence.