Proposition 9.2.6.8 (067B)

Proposition 9.2.6.8. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

$(1)$: The $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated.
$(2)$: There exists a functor of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$ which exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}'$ (Variant 9.2.1.7).
$(3)$: 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 $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_0$.

Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0$ to be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $(\kappa ,\lambda )$-compact objects.

Proof. The implication $(3) \Rightarrow (2)$ is immediate, and the implication $(2) \Rightarrow (1)$ follows from Proposition 9.2.3.3 and Corollary 9.2.4.23. To complete the proof, assume that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, and let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. Applying the criterion of Proposition 9.2.3.3, we conclude that the inclusion functor $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}$ exhibits $\operatorname{\mathcal{C}}$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{C}}_0$. $\square$