Variant 9.3.2.17. Let $\kappa \leq \lambda $ be uncountable regular cardinals. Then the inclusion functor $\operatorname{\mathcal{S}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}_{< \lambda }$ induces a fully faithful functor $\operatorname{Ind}_{\kappa }^{\lambda }( \operatorname{\mathcal{S}}_{< \kappa } ) \rightarrow \operatorname{\mathcal{S}}_{ < \lambda }$. If $\kappa \trianglelefteq \lambda $, then this functor is an equivalence of $\infty $-categories: that is, the inclusion exhibits $\operatorname{\mathcal{S}}_{< \lambda }$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{S}}_{< \kappa }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We assume that $\lambda $ is uncountable (otherwise, $\kappa = \lambda $ and there is nothing to prove). Note that $\operatorname{\mathcal{S}}_{< \lambda }$ admits $\lambda $-small colimits (Corollary 7.4.3.10), and that objects of $\operatorname{\mathcal{S}}_{< \kappa }$ are $(\kappa ,\lambda )$-compact when viewed as objects of $\operatorname{\mathcal{S}}_{< \lambda }$ (for $\kappa $ uncountable, this is Proposition 9.2.6.10). The first assertion now follows from Proposition 9.3.2.1. To prove the second, assume that $\kappa \trianglelefteq \lambda $. Then every $\lambda $-small simplicial set $X$ can be realized as the colimit of a $\lambda $-small $\kappa $-filtered diagram $\{ X_{\alpha } \} $, where each $X_{\alpha }$ is $\kappa $-small (see Lemma 9.1.7.18). Applying Proposition 3.3.6.4, we conclude that $\operatorname{Ex}^{\infty }(X)$ can be realized as the colimit of a (small) filtered diagram $\{ \operatorname{Ex}^{\infty }(X_{\alpha } ) \} $ in the category of simplicial sets, and therefore also in the $\infty $-category $\operatorname{\mathcal{S}}_{< \lambda }$ (Variant 9.1.6.4). If $X$ is a Kan complex, then it is homotopy equivalent to $\operatorname{Ex}^{\infty }(X)$ (Proposition 3.3.6.7), and can therefore be realized the colimit of a $\lambda $-small $\kappa $-filtered diagram in $\operatorname{\mathcal{S}}_{< \kappa }$. The desired result now follows from the criterion of Proposition 9.3.2.3. $\square$