Kerodon

Remark 4.7.5.10. Let $\kappa $ be an uncountable cardinal. If $K$ and $K'$ are essentially $\kappa $-small simplicial sets, then the join $K \star K'$ is essentially $\kappa $-small. To prove this, choose categorical equivalences $K \rightarrow \operatorname{\mathcal{K}}$ and $K' \rightarrow \operatorname{\mathcal{K}}'$, where $\operatorname{\mathcal{K}}$ and $\operatorname{\mathcal{K}}'$ are $\kappa $-small $\infty $-categories. Then the induced map $K \star K' \rightarrow \operatorname{\mathcal{K}}\star \operatorname{\mathcal{K}}'$ is also a categorical equivalence (Corollary 4.5.8.9), and $\operatorname{\mathcal{K}}\star \operatorname{\mathcal{K}}'$ is a $\kappa $-small $\infty $-category (Corollary 4.7.4.13).