Remark 9.2.6.11. Let $\kappa \leq \lambda $ be regular cardinals, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $\lambda $-small $\kappa $-filtered colimits. To show that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, it suffices to verify the following (which is a priori weaker than condition $(b_{\kappa ,\lambda })$ of Variant 9.2.6.5) :
- $(b_{\kappa ,\lambda }^{-})$
Let $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by the $(\kappa ,\lambda )$-compact objects. Then $\operatorname{\mathcal{C}}_0$ generated $\operatorname{\mathcal{C}}$ under $\lambda $-small $\kappa $-filtered coliits That is, if $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a replete full subcategory which contains $\operatorname{\mathcal{C}}_0$ and is closed under $\lambda $-small $\kappa $-filtered colimits, then $\operatorname{\mathcal{C}}' = \operatorname{\mathcal{C}}$.
In this case, the recognition principle of Proposition 9.2.3.3 guarantees that $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of the full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$.