Proposition 4.7.7.1. Let $X$ be a Kan complex and let $\kappa $ be an uncountable regular cardinal. Then $X$ is essentially $\kappa $-small if and only if it satisfies the following pair of conditions:
The set $\pi _0(X)$ is $\kappa $-small.
For each vertex $x \in X$ and each integer $n > 0$, the homotopy group $\pi _{n}(X,x)$ is $\kappa $-small.
Proof. By virtue of Proposition 4.7.6.15, we may assume without loss of generality that the Kan complex $X$ is minimal. If $X$ is essentially $\kappa $-small, then it is $\kappa $-small (Corollary 4.7.6.12), so that conditions $(1)$ and $(2)$ follow immediately from the definitions. Conversely, suppose that $(1)$ and $(2)$ are satisfied; we wish to show that $X$ is $\kappa $-small. By virtue of Proposition 4.7.4.10, it will suffice to show that the collection of $n$-simplices of $X$ is $\kappa $-small, for each $n \geq 0$. Our proof proceeds by induction on $n$. Using our inductive hypothesis (together with Remark 4.7.3.4 and Proposition 4.7.3.5), we see that the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ n, X)$ is $\kappa $-small. Since $\kappa $ is regular, it will suffice to show that each fiber of the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ n, X)$ is $\kappa $-small.
Set $E = \operatorname{Fun}( \Delta ^{n}, X)$ and $B = \operatorname{Fun}( \operatorname{\partial \Delta }^{n}, X)$, so that the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$ induces a Kan fibration $q: E \rightarrow B$ (Corollary 3.1.3.3) For each vertex $b \in B$, let $E_{b}$ denote the fiber $\{ b\} \times _{B} E$; we wish to show that the set of vertices of $E_{b}$ is $\kappa $-small. Since the Kan complex $X$ is minimal, each vertex of $E_{b}$ belongs to a different connected component. It will therefore suffice to show that the set $\pi _{0}(E_ b)$ is $\kappa $-small. If $n=0$, this follows from condition $(1)$. Let us therefore assume that $n > 0$, and identify $b$ with a morphism of simplicial sets $\operatorname{\partial \Delta }^{n} \rightarrow X$. If this morphism is not nullhomotopic, then the Kan complex $E_{b}$ is empty and there is nothing to prove. We may therefore assume that there is a homotopy from $b$ to a constant map $b': \operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \hookrightarrow X$. In this case, Proposition 5.2.2.19 supplies a homotopy equivalence of $E_{b}$ with $E_{b'}$. We are therefore reduced to proving that the set $\pi _0( E_{b'} ) \simeq \pi _{n}(X,x)$ is $\kappa $-small, which follows from condition $(2)$. $\square$