Kerodon

Example 8.5.4.19. Let $\operatorname{\mathcal{QC}}$ denote the $\infty $-category of (small) $\infty $-categories. Then $\operatorname{\mathcal{QC}}$ is idempotent complete. More generally, for every uncountable cardinal $\kappa $, the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$ of $\kappa $-small $\infty $-categories is idempotent complete. To prove this, we can use Propositions 8.5.4.6 and 8.5.1.15 to reduce to the case where $\kappa $ has uncountable cofinality. In this case, the $\infty $-category $\operatorname{\mathcal{QC}}^{< \kappa }$ admits sequential colimits (Example 7.6.6.8), so the desired result follows from Corollary 8.5.4.17.