Example 9.2.2.9. Let $\kappa $ be a (small) regular cardinal. For every (small) set $S$, the following conditions are equivalent:
- $(1)$
The set $S$ is $\kappa $-small: that is, it has cardinality $< \kappa $.
- $(2)$
The set $S$ is $\kappa $-compact when viewed as an object of (the nerve of) the category of sets.
To prove the implication $(1) \Rightarrow (2)$, we can use the decomposition $S \simeq \coprod _{s \in S} \{ s\} $ to reduce to the case where $S$ consists of a single element (Corollary 9.2.2.22). In this case, $S$ corepresents the identity functor $\operatorname{id}: \operatorname{Set}\rightarrow \operatorname{Set}$, which preserves all colimits. For the converse, we observe that $S$ can be realized as a $\kappa $-filtered colimit of $\kappa $-small subsets of itself. Consequently, if $S$ is $\kappa $-compact, then the identity function $\operatorname{id}: S \rightarrow S$ factors through some $\kappa $-small subset $S_0 \subseteq S$. It follows that $S = S_0$ is $\kappa $-small.