Kerodon

Remark 9.1.10.2. In the formulation of Proposition 9.1.10.1, we have implicitly assumed that the filtered $\infty $-category $\operatorname{\mathcal{J}}$ is small (so that the colimit $\varinjlim (\mathscr {F} )$ is well-defined). More generally, suppose that $\lambda $ is an uncountable regular cardinal and $\mathscr {F}: \operatorname{\mathcal{J}}\rightarrow \operatorname{\mathcal{QC}}_{< \lambda }$ is a $\lambda $-small filtered diagram. If each of the $\infty $-categories $\mathscr {F}(J)$ is locally $n$-truncated (for some fixed integer $n$), then the colimit $\varinjlim (\mathscr {F}) \in \operatorname{\mathcal{QC}}_{< \lambda }$ is also locally $n$-truncated. Similar remarks apply to the other results proved in this section.