Kerodon

Remark 9.4.6.14. Following the convention of Remark 4.9.0.4, an $\infty $-category is essentially small if it is essentially $ \Omega $-small, where $ \Omega $ is some fixed inaccessible cardinal. An accessible $\infty $-category $\operatorname{\mathcal{C}}$ need not satisfy this condition. However, it follows from Corollary 9.3.3.2 that $\operatorname{\mathcal{C}}$ is locally $ \Omega $-small and essentially $ \Omega ^{+}$-small. Moreover, $\operatorname{\mathcal{C}}$ admits small $\kappa $-filtered colimits for some small regular cardinal $\kappa $ (and therefore for every small regular cardinal $\lambda \geq \kappa $). In particular, $\operatorname{\mathcal{C}}$ is idempotent-complete (Example 9.1.1.10).