Warning 9.2.2.7. Let $\kappa $ be an uncountable regular cardinal. We have now assigned two meanings to the notation $\operatorname{\mathcal{C}}_{< \kappa }$ in the cases $\operatorname{\mathcal{C}}= \operatorname{\mathcal{S}}$ and $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}$:
Following the convention of Remark 5.5.4.12 and Variant 5.5.4.13, $\operatorname{\mathcal{C}}_{< \kappa }$ denotes the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects which are $\kappa $-small when viewed as simplicial sets.
Following the convention of Definition 9.2.2.6, $\operatorname{\mathcal{C}}_{< \kappa }$ denotes the full subcategory of $\operatorname{\mathcal{C}}$ spanned by those objects which are $\kappa $-compact.
However, these conventions are almost compatible: we will see later that a Kan complex is $\kappa $-compact (as an object of $\operatorname{\mathcal{S}}$) if and only if it is essentially $\kappa $-small (Proposition 9.2.7.11). Similarly, a small $\infty $-category is $\kappa $-compact (as an object of $\operatorname{\mathcal{QC}}$) if and only if it is essentially $\kappa $-small (Proposition 9.2.8.15).