Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 5.4.6.14. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there is a least uncountable cardinal $\kappa $ for which $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. Moreover, $\kappa $ is always a successor cardinal.

Proof. By virtue of Proposition 5.4.6.12, we may assume that $\operatorname{\mathcal{C}}$ is a minimal $\infty $-category. In this case, the desired result follows by combining Corollary 5.4.6.9 with Remark 5.4.4.7. $\square$