Definition 3.4.1.1. A commutative diagram of simplicial sets
3.36
\begin{equation} \label{diagram:homotopy-pullback-square0} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X } \end{gathered}\end{equation}
is a homotopy pullback square if, for every factorization $q = q' \circ w$ where $w: X_0 \rightarrow X'_0$ is a weak homotopy equivalence and $q': X'_0 \rightarrow X$ is a Kan fibration, the induced map $X_{01} \rightarrow X'_0 \times _{X} X_1$ is a weak homotopy equivalence.