Proposition 9.2.7.11. Let $\kappa $ be an uncountable regular cardinal and let $X$ be a Kan complex. The following conditions are equivalent:
- $(1)$
The Kan complex $X$ is essentially $\kappa $-small (Definition 4.7.5.1).
- $(2)$
The Kan complex $X$ belongs to the smallest full subcategory of $\operatorname{\mathcal{S}}$ which contains $\Delta ^0$ and is closed under $\kappa $-small colimits.
- $(3)$
The Kan complex $X$ is $\kappa $-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{S}}$.