$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 3.4.1.6. Suppose we are given a commutative diagram of simplicial sets
3.41
\begin{equation} \label{diagram:homotopy-pullback-square55} \begin{gathered} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d]^{q'} & X_0 \ar [d]^{q} \\ X_1 \ar [r] & X, } \end{gathered}\end{equation}
where $X$ is a Kan complex. Then (3.41) is a homotopy pullback square if and only if the induced map
\[ \theta : X_{01} \rightarrow X_0 \times _{X} X_1 \hookrightarrow X_{0} \times _{X}^{\mathrm{h}} X_{1} \]
is a weak homotopy equivalence.
Proof.
Using Proposition 3.1.7.1, we can factor $q$ as a composition $X_0 \xrightarrow {w} X'_0 \xrightarrow {q'} X$, where $w$ is a weak homotopy equivalence and $q'$ is Kan fibration. We then have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r]^-{\theta } \ar [d]^{\rho } & X_{0} \times _{X}^{\mathrm{h}} X_{1} \ar [d] \\ X'_{0} \times _{X} X_{1} \ar [r] & X'_{0} \times _{X}^{\mathrm{h}} X_{1}, } \]
where the bottom horizontal map is a weak homotopy equivalence (Proposition 3.4.0.7) and the right vertical map is also a weak homotopy equivalence (Proposition 3.4.0.9). It follows that $\theta $ is a weak homotopy equivalence if and only if $\rho $ is a weak homotopy equivalence. By virtue of Proposition 3.4.1.2, this is equivalent to the requirement that the diagram (3.41) is a homotopy pullback square.
$\square$