Definition 9.2.6.3. Let $\kappa $ be a small regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is $\kappa $-compactly generated if it satisfies the following conditions:
- $(a_{\kappa })$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits.
- $(b_{\kappa })$
Every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a small $\kappa $-filtered diagram $\{ C_{\alpha } \} $, where each $C_{\alpha }$ is a $\kappa $-compact object of $\operatorname{\mathcal{C}}$.