Proposition 9.2.8.15. Let $\kappa $ be an uncountable regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small (Definition 4.7.5.1).
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ belongs to the smallest full subcategory of $\operatorname{\mathcal{QC}}$ which contains $\Delta ^1$ and is closed under $\kappa $-small colimits.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{QC}}$.