Exercise 9.3.7.8. Let $\kappa \leq \lambda $ be regular cardinals, let $S$ be a $\lambda $-small set, and let $\operatorname{Sub}_{< \kappa }(S)$ be (the nerve of) the partially ordered set of $\kappa $-small subsets of $S$. Show that the following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{Sub}_{< \kappa }(S) )$ is $\lambda $-filtered.
- $(2)$
The $\infty $-category $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{Sub}_{< \kappa }(S) )$ has a final object.
- $(3)$
There exists a $\lambda $-small collection of $\kappa $-small subsets $\{ S_{\alpha } \subseteq S \} $ such that every $\kappa $-small subset of $S$ is contained in some $S_{\alpha }$.
In particular, if these conditions are satisfied for every $\lambda $-small set $S$, then $\kappa \trianglelefteq \lambda $.