Corollary 9.2.7.6. The inclusion functor $\iota : \operatorname{\mathcal{S}}_{\mathrm{fin}} \hookrightarrow \operatorname{\mathcal{S}}$ exhibits $\operatorname{\mathcal{S}}$ as an $\operatorname{Ind}$-completion of the $\infty $-category $\operatorname{\mathcal{S}}_{\mathrm{fin}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. The $\infty $-category $\operatorname{\mathcal{S}}$ admits small filtered colimits (Corollary 7.4.3.10), and every object of $\operatorname{\mathcal{S}}_{\mathrm{fin}}$ is compact when viewed as an object of $\operatorname{\mathcal{S}}$ (Corollary 9.2.7.5). Recall that every (small) simplicial set $X$ can be realized as the colimit of a (small) filtered diagram $ \{ X_{\alpha } \} $, where each $X_{\alpha }$ is a finite simplicial set (Remark 3.6.1.8). 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 $\operatorname{\mathcal{S}}$ (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 as the colimit of a small filtered diagram of essentially finite Kan complexes. The desired result now follows from the criterion of Corollary 9.2.3.6. $\square$