Example 9.5.5.18. Let $\kappa \trianglelefteq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\lambda $-cocomplete, and let $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denote the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated: that is, every object of $\operatorname{\mathcal{C}}$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram in $\operatorname{\mathcal{C}}_{< \kappa }$ (Variant 9.4.1.7).
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is generated by $\operatorname{\mathcal{C}}_{< \kappa }$ under $\lambda $-small colimits.
The implication $(1) \Rightarrow (2)$ is trivial, and the reverse implication follows by applying Lemma 9.5.5.17 to the inclusion functor $\operatorname{\mathcal{C}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{C}}$.