Definition 9.2.6.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. We say that $\operatorname{\mathcal{C}}$ is compactly generated if it satisfies the following conditions:
- $(a)$
The $\infty $-category $\operatorname{\mathcal{C}}$ admits small filtered colimits.
- $(b)$
Every object $C \in \operatorname{\mathcal{C}}$ can be realized as the colimit of a small filtered diagram $\{ C_{\alpha } \} $, where each $C_{\alpha }$ is a compact object of $\operatorname{\mathcal{C}}$.