Variant 4.5.2.11 (046L)

Variant 4.5.2.11. Suppose we are given a commutative diagram of $\infty $-categories

4.23

\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square55} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d]^{q} \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

where $\operatorname{\mathcal{C}}$ is a Kan complex. If (4.23) is a categorical pullback square, then it is also a homotopy pullback square.

Proof. By assumption, the induced map $\operatorname{\mathcal{C}}_{01} \rightarrow \operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{1}$ is an equivalence of $\infty $-categories, and therefore a weak homotopy equivalence of simplicial sets (Remark 4.5.3.4). The desired result now follows from the criterion of Corollary 3.4.1.6. $\square$