Proposition 7.1.9.1 (06AA)

Proposition 7.1.9.1. Let $K$ be a simplicial set, and suppose we are given a categorical pullback square of $\infty $-categories

7.12

\begin{equation} \begin{gathered}\label{equation:categorical-pullback-relative-limit-all-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}

Assume that the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ admit $K$-indexed colimits and that the functors $F_0$ and $F_1$ preserve $K$-indexed colimits. Then:

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed colimits.
A morphism $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{01}$ is a colimit diagram if and only if $G_0 \circ \overline{q}$ and $G_1 \circ \overline{q}$ are colimit diagrams (in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively).

In particular, the functors $G_0$ and $G_1$ preserve $K$-indexed colimits.