Proposition 4.7.4.9. Let $\kappa $ be an infinite cardinal and $S_{\bullet }$ be a simplicial set. Assume that the cofinality of $\kappa $ is larger than $\aleph _0$ (this condition is satisfied, for example, if $\kappa $ is uncountable and regular). The following conditions are equivalent:
- $(1)$
The simplicial set $S_{\bullet }$ is $\kappa $-small.
- $(2)$
For every integer $n \geq 0$, the set $S_{n}$ is $\kappa $-small.
- $(3)$
For every finite simplicial set $K$, the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K,S_{\bullet })$ is $\kappa $-small.
Proof.
We first show that $(1)$ implies $(2)$. Assume that $S_{\bullet }$ is $\kappa $-small and let $n \geq 0$ be an integer. For each integer $m \geq 0$, let $S_{m}^{\mathrm{nd}}$ denote the set of nondegenerate $m$-simplices of $S_{\bullet }$. Using Proposition 1.1.3.8, we can identify $S_ n$ with the coproduct ${\coprod }_{ \alpha : [n] \twoheadrightarrow [m] } S_{m}^{\mathrm{nd}}$, where $\alpha $ ranges over all surjective maps of linearly ordered sets $[n] \twoheadrightarrow [m]$. Our assumption that $S_{\bullet }$ is $\kappa $-small guarantees that each of the sets $S_{m}^{\mathrm{nd}}$ is $\kappa $-small, so that $S_{n}$ is also $\kappa $-small (Corollary 4.7.3.6).
We now show that $(2)$ implies $(1)$. Assume that, for each $n \geq 0$, the set $S_{n}$ is $\kappa $-small. Since $\kappa $ has cofinality $> \aleph _0$ it follows that the coproduct ${\coprod }_{n \geq 0} S_{n}$ is also $\kappa $-small. In particular, the coproduct ${\coprod }_{n \geq 0} S_{n}^{\mathrm{nd}}$ is $\kappa $-small: that is, the simplicial set $S_{\bullet }$ is $\kappa $-small.
The implication $(3) \Rightarrow (2)$ is immediate from the definition. We will complete the proof by showing that $(2) \Rightarrow (3)$. Assume that, for each $n \geq 0$, the set $S_{n}$ is $\kappa $-small, and let $K$ be a finite simplicial set. By virtue of Proposition 3.6.1.7, there exists an epimorphism $f: K' \twoheadrightarrow K$, where $K' = {\coprod }_{i \in I} \Delta ^{n_ i}$ is a disjoint union of finitely many standard simplices. Then precomposition with $f$ induces a monomorphism
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, S_{\bullet } ) \hookrightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K', S_{\bullet }) \simeq {\prod }_{i \in I} S_{n_ i}. \]
Since the collection of $\kappa $-small sets is closed under finite products and passage to subsets (Proposition 4.7.3.5 and Remark 4.7.3.4), it follows that the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, S_{\bullet } )$ is also $\kappa $-small.
$\square$