Corollary 9.2.6.9. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated.
- $(2)$
There exists a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}'$ (Definition 9.2.1.4).
- $(3)$
There exists a full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ for which the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }$-completion of $\operatorname{\mathcal{C}}_0$.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}_{< \kappa }$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects.