Kerodon

Remark 9.2.5.14. Following the convention of Remark 4.9.0.4, a regular cardinal $\kappa $ is small if it satisfies $\kappa < \Omega $, for some fixed strongly inaccessible cardinal $ \Omega $. If $\operatorname{\mathcal{C}}$ is an $\infty $-category which admits small $\kappa $-filtered colimits, then an object $C \in \operatorname{\mathcal{C}}$ is $\kappa $-compact (in the sense of Definition 9.2.5.6) if and only if it is $(\kappa , \Omega )$-compact (in the sense of Definition 9.2.5.12. In particular, if $\operatorname{\mathcal{C}}$ admits small filtered colimits, then an object $C \in \operatorname{\mathcal{C}}$ is compact if and only if it is $(\aleph _0, \Omega )$-compact.