Kerodon

Remark 9.2.0.8. For any $\infty $-category $\operatorname{\mathcal{C}}_0$, the $\operatorname{Ind}$-completion $\operatorname{Ind}(\operatorname{\mathcal{C}}_0)$ is obtained from $\operatorname{\mathcal{C}}_0$ by (freely) adjoining colimits of diagrams $\operatorname{\mathcal{I}}\rightarrow \operatorname{\mathcal{C}}_0$, where $\operatorname{\mathcal{I}}$ is a small filtered $\infty $-category. For some purposes, it is convenient to consider variants of this construction where we impose additional conditions on the $\infty $-category $\operatorname{\mathcal{I}}$: for example, we can allow only $\infty $-categories $\operatorname{\mathcal{I}}$ which are $\kappa $-filtered and $\lambda $-small, for some pair of regular cardinals $\kappa \leq \lambda $ (Variant 9.2.1.7). With an eye toward future applications, we formulate many of our results in a way that applies to this more general notion of $\operatorname{Ind}$-completion.