Kerodon

Proof. If $\kappa = \aleph _0$, this follows from Variant 9.2.0.5. We may therefore assume without loss of generality that $\kappa $ is uncountable. Assume first that $X$ is a $\kappa $-small simplicial set, and let $\operatorname{{\bf \Delta }}_{X}$ be the category of simplices of $X$ (Construction 1.1.3.9). Then $X$ can be realized as a colimit $\varinjlim _{ ([n],\sigma ) \in \operatorname{{\bf \Delta }}_{X} } \Delta ^ n$ (see the proof of Proposition 1.1.3.11, or Theorem 8.4.2.1). Each of the standard simplices $\Delta ^ n$ is a $\kappa $-compact object of the category $\operatorname{Set_{\Delta }}$, since the evaluation functor $(Y_{\bullet } \in \operatorname{Set_{\Delta }}) \mapsto Y_{n}$ preserves colimits (Remark 1.1.0.8). Our assumption that $X$ is $\kappa $-small guarantees that the category $\operatorname{{\bf \Delta }}_{X}$ is $\kappa $-small (Proposition 4.7.4.10), so that $X$ is $\kappa $-compact by virtue of Proposition 9.2.2.21.

We now prove the converse. As in Example 9.2.2.9, we can realize $X$ as a $\kappa $-filtered colimit $\varinjlim _{\alpha \in A} X_{\alpha }$, where $\{ X_{\alpha } \} _{\alpha \in A}$ is the collection of all $\kappa $-small simplicial subsets of $X$. If $X$ is $\kappa $-compact, then the identity map $\operatorname{id}_{X}$ factors through some $X_{\alpha }$, so that $X = X_{\alpha }$ is $\kappa $-small. $\square$