Definition 9.2.2.6. Let $\kappa $ be a small regular cardinal, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits small $\kappa $-filtered colimits. We say that an object $C \in \operatorname{\mathcal{C}}$ is $\kappa $-compact if the corepresentable functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C, \bullet )$ is $\kappa $-finitary: that is, it preserves small $\kappa $-filtered colimits. We will sometimes write $\operatorname{\mathcal{C}}_{< \kappa }$ for the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects.
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