Example 9.2.3.11. Let $S$ be a set. For every regular cardinal $\kappa $, let $P_{< \kappa }(S)$ denote the nerve of the partially ordered set of $\kappa $-small subsets of $S$. For every regular cardinal $\lambda \geq \kappa $, Proposition 9.2.3.1 and Variant 9.2.2.11 supplies a fully faithful functor
\[ \operatorname{Ind}_{\kappa }^{\lambda }( P_{< \kappa }(S) ) \hookrightarrow P_{ < \lambda }(S). \]
This functor is an equivalence if and only if every $\lambda $-small subset $S' \subseteq S$ can be realized as a $\lambda $-small $\kappa $-filtered union of $\kappa $-small subsets of $S'$ (Remark 9.2.3.4). Assuming that $S$ has cardinality $\geq \lambda $, this is equivalent to the requirement that $\kappa \trianglelefteq \lambda $ (see Definition 9.1.7.5).