Remark 4.5.2.36 (033W): Categorical Pullback Squares of Simplicial Sets

Remark 4.5.2.36 (Categorical Pullback Squares of Simplicial Sets). Suppose we are given a commutative diagram of simplicial sets

4.29

\begin{equation} \begin{gathered}\label{equation:general-categorical-pullback} \xymatrix@R =50pt@C=50pt{ X_{01} \ar [r] \ar [d] & X_0 \ar [d] \\ X_{1} \ar [r] & X. } \end{gathered} \end{equation}

Applying Proposition 4.1.3.2 repeatedly, we can enlarge 4.29 to a cubical diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{01} \ar [rr] \ar [dd] \ar [dr] & & X_{0} \ar [dd] \ar [dr] & \\ & \operatorname{\mathcal{C}}_{01} \ar [rr] \ar [dd] & & \operatorname{\mathcal{C}}_{0} \ar [dd] \\ X_{1} \ar [rr] \ar [dr] & & X \ar [dr] & \\ & \operatorname{\mathcal{C}}_{1} \ar [rr] & & \operatorname{\mathcal{C}}, } \]

where the diagonal maps are inner anodyne and the front face

4.30

\begin{equation} \begin{gathered}\label{equation:general-categorical-pullback2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_{1} \ar [r] & \operatorname{\mathcal{C}}} \end{gathered} \end{equation}

is a diagram of $\infty $-categories. Let us say that that the diagram of simplicial sets (4.29) is a categorical pullback square if the diagram of $\infty $-categories (4.30) is a categorical pullback square, in the sense of Definition 4.5.2.8. Using Proposition 4.5.2.19, it is not difficult to show that this condition depends only on the original diagram (for a more general statement, see Proposition 7.5.5.13). Beware that this more general notion of categorical pullback diagram can be badly behaved: for example, it does not satisfy the analogue of Proposition 4.5.2.26 (see Warning 4.5.5.12).