Variant 9.2.6.5. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated if it satisfies the following conditions:
- $(a_{\kappa ,\lambda })$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small $\kappa $-filtered colimits.
- $(b_{\kappa ,\lambda })$
Every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram $\{ C_{\alpha } \} $, where each $C_{\alpha }$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$.