Kerodon

Remark 8.1.1.5. Let $\kappa $ be an uncountable cardinal. If $\operatorname{\mathcal{C}}$ is a $\kappa $-small simplicial set, then $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is also $\kappa $-small. To prove this, we may assume without loss of generality that $\kappa $ is the smallest uncountable cardinal for which $\operatorname{\mathcal{C}}$ is $\kappa $-small. In particular, $\kappa $ is regular. It will therefore suffice to show that, for every integer $n$, the set of $n$-simplices of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is $\kappa $-small (Proposition 4.7.4.10). This follows from the $\kappa $-smallness of the set of $(2n+1)$-simplices of $\operatorname{\mathcal{C}}$ (Remark 8.1.1.2).