Kerodon

Remark 9.4.6.8. In the situation of Corollary 9.4.6.7, the $\infty $-category $\operatorname{\mathcal{C}}$ is generally not essentially small. However, it can be written as a directed union $\bigcup _{\lambda \geq \kappa } \operatorname{\mathcal{C}}_{< \lambda }$ of essentially small subcategories $\operatorname{\mathcal{C}}_{< \lambda }$, where $\lambda $ ranges over all small regular cardinals satisfying $\lambda \geq \kappa $. In other words, every object $X \in \operatorname{\mathcal{C}}$ is $\lambda $-compact for some small regular cardinal $\lambda \geq \kappa $. To prove this, we observe that $X$ can be realized as the colimit of small $\kappa $-filtered diagram $\operatorname{\mathcal{K}}\rightarrow \operatorname{\mathcal{C}}_{< \kappa }$; we can then take $\lambda \geq \kappa $ to be any small regular cardinal for which $\operatorname{\mathcal{K}}$ is $\lambda $-small (Proposition 9.2.5.24).