Corollary 9.2.6.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is compactly generated.
- $(2)$
There exists a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}'$ (Definition 9.2.1.1).
- $(3)$
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 $\operatorname{Ind}$-completion of $\operatorname{\mathcal{C}}_0$.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0 = \operatorname{\mathcal{C}}_{< \aleph _0}$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the compact objects.