Kerodon

Remark 9.4.2.16. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{QC}}^{ \mathrm{ic} }$ denote the full subcategory of $\operatorname{\mathcal{QC}}$ spanned by the idempotent complete $\infty $-categories. If $\lambda $ is small and uncountable, then the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ induces an equivalence from $\operatorname{\mathcal{QC}}^{\mathrm{ic} }$ to the subcategory $\operatorname{\mathcal{QC}}^{ (\kappa ,\lambda )-\mathrm{cg} }$ of Notation 9.4.2.15. More generally, assume that $\lambda $ is uncountable and let $\mu > \lambda $ be a cardinal of exponential cofinality $\geq \lambda $, so that the construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ carries essentially $\mu $-small $\infty $-categories to essentially $\mu $-small $\infty $-categories (see Proposition 9.3.3.7). Then the formation of $\operatorname{Ind}_{\kappa }^{\lambda }$-completions induces an equivalence from $\operatorname{\mathcal{QC}}_{< \mu }^{\mathrm{ic} }$ to the subcategory $\operatorname{\mathcal{QC}}^{ (\kappa ,\lambda )-\mathrm{cg} }_{< \mu } \subseteq \operatorname{\mathcal{QC}}_{< \mu }$. This follows from Proposition 9.4.1.18 and Remark 9.4.2.10.