Corollary 9.2.7.13. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $. Then the inclusion map $\operatorname{\mathcal{S}}_{< \kappa } \hookrightarrow \operatorname{\mathcal{S}}_{< \lambda }$ exhibits $\operatorname{\mathcal{S}}_{< \lambda }$ as an $\operatorname{Ind}_{\kappa }^{\lambda }$-completion of $\operatorname{\mathcal{S}}_{< \kappa }$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We proceed as in the proof of Corollary 9.2.7.6. 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 9.2.7.12). Our assumption that $\kappa \trianglelefteq \lambda $ guarantees that 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 of Kan complexes which are essentially $\kappa $-small (or essentially finite, in the case $\kappa = \aleph _0$). The desired result now follows from the criterion of Proposition 9.2.3.3. $\square$