Remark 5.5.4.12 (Set-Theoretic Conventions). By definition, the objects of the $\infty $-category $\operatorname{\mathcal{QC}}= \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat})$ are small $\infty $-categories. According to the convention of Remark 4.7.0.5, this means that we restrict our attention to essentially $\lambda $-small Kan complexes, where $\lambda $ is some fixed uncountable strongly inaccessible cardinal. In this case, the definitions given in Variant 5.5.4.11 are appropriate only for uncountable cardinals $\kappa < \lambda $. More generally, if $\kappa $ is an arbitrary uncountable cardinal, we can define $\operatorname{\mathcal{QC}}^{< \kappa }$ to be the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}^{< \kappa } )$, where $\operatorname{QCat}^{< \kappa }$ denotes the (simplicially enriched) category of $\kappa $-small $\infty $-categories. We then have three cases:
- $(a)$
If $\kappa < \lambda $, then $\operatorname{\mathcal{QC}}^{< \kappa }$ is a full subcategory of $\operatorname{\mathcal{QC}}$.
- $(b)$
If $\kappa = \lambda $, then $\operatorname{\mathcal{QC}}^{< \kappa }$ coincides with $\operatorname{\mathcal{QC}}$.
- $(c)$
If $\kappa > \lambda $, then $\operatorname{\mathcal{QC}}$ is a full subcategory of $\operatorname{\mathcal{QC}}^{< \kappa }$.
To simplify the exposition, we will often implicitly assume that we are in case $(a)$, as suggested in Variant 5.5.4.11. However, it will be convenient to also allow case $(c)$ when working with $\infty $-categories which are not necessarily small (such as $\operatorname{\mathcal{QC}}$ itself).