Kerodon

Warning 9.1.7.15. In the statement of Proposition 9.1.7.14, the assumption $\kappa \triangleleft \lambda $ cannot be omitted. For example, let $S$ be a $\lambda $-small set, let $Q$ be the collection of all $\kappa $-small subsets of $S$ (partially ordered by inclusion). Set $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }(Q)$, and let $\operatorname{\mathcal{C}}_0$ be the full subcategory of $\operatorname{\mathcal{C}}$ spanned by the singleton sets $\{ s\} $ for $s \in S$. If $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ is a simplicial subset satisfying the conclusion of Proposition 9.1.7.14, then the collection of vertices of $\operatorname{\mathcal{C}}'$ can be viewed as a $\lambda $-small collection $\{ S_ i \} _{i \in I}$ of $\kappa $-small subsets of $S$, such that every $\kappa $-small subset of $S$ is contained in some $S_ i$.