Proposition 7.1.9.8 (06AJ)

Proposition 7.1.9.8. Suppose we are given a categorical pullback square of $\infty $-categories

7.11

\begin{equation} \begin{gathered}\label{equation:categorical-pullback-relative-limit-existence} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r]^-{G_0} \ar [d]^{G_1} & \operatorname{\mathcal{C}}_0 \ar [d]^{F_0} \\ \operatorname{\mathcal{C}}_1 \ar [r]^-{F_1} & \operatorname{\mathcal{C}}, } \end{gathered} \end{equation}

and let $q: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram with the following properties:

The diagram $q_0 = G_0 \circ q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0$, which is preserved by the functor $F_0$.
The diagram $q_1 = G_1 \circ q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_1$, which is preserved by the functor $F_1$.

Then $q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_{01}$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if it satisfies the following condition:

$(\ast )$: The compositions $G_0 \circ \overline{q}$ and $G_1 \circ \overline{q}$ are colimit diagrams (in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively).

Proof. Since (7.11) is a categorical pullback square, it induces an equivalence from $\operatorname{\mathcal{C}}_{01}$ to the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. The desired result now follows from Corollary 7.1.9.7. $\square$