Proposition 3.5.5.16. Let $X$ be a Kan complex and let $n \geq 0$ be an integer. The following conditions are equivalent:
The Kan complex $X$ is isomorphic to an Eilenberg-MacLane space $\mathrm{K}(G,n)$. Here $G$ is a set if $n=0$, a group if $n =1$, and an abelian group if $n \geq 2$ (see Construction 2.5.6.9).
The Kan complex $X$ is an $n$-groupoid having a single $m$-simplex for each $m < n$.
Proof. If $n = 0$, the desired result follows from Proposition 3.5.5.7. If $n = 1$, then $X$ is an $n$-groupoid if and only if it is isomorphic to the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, where $\operatorname{\mathcal{C}}$ is a groupoid (Proposition 3.5.5.8). In this case, the assumption that $X$ has a single vertex is equivalent to the requirement that the category $\operatorname{\mathcal{C}}$ contains a single object $C$, in which case we can identify $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ with the classifying simplicial set $\mathrm{K}(G,1) =B_{\bullet }G$ for $G = \operatorname{Aut}_{\operatorname{\mathcal{C}}}(C)$. We may therefore assume that $n \geq 2$. The implication $(1) \Rightarrow (2)$ follows from Proposition 3.5.5.9. For the converse, assume that $X$ is an $n$-groupoid having a single $m$-simplex for each $m < n$. Let $x$ be the unique vertex of $X$ and set $G = \pi _{n}(X,x)$. For every $n$-simplex $\sigma $ of $X$, the restriction $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map taking the value $x$, so the homotopy class $[\sigma ]$ can be regarded as an element of the group $G$. Our assumption that $X$ is an $n$-groupoid guarantees that the assignment $\sigma \mapsto [\sigma ]$ determines a bijection from the collection of $n$-simplices of $X$ to the group $G$, which determines an isomorphism $f_0$ from the $n$-skeleton of $X$ to the $n$-skeleton of $\mathrm{K}(G,n)$. Invoking Theorem 3.2.2.10, we see that a morphism $\tau _0: \operatorname{\partial \Delta }^{n+1} \rightarrow X$ can be extended to an $(n+1)$-simplex of $X$ if and only if the composite map
can be extended to an $(n+1)$-simplex of $\mathrm{K}(G,n)$. Since $\mathrm{K}(G,n)$ is weakly $n$-coskeletal, it follows that $f_0$ extends uniquely to a morphism of simplicial sets $f: X \rightarrow \mathrm{K}(G,n)$ (Proposition 3.5.4.12) which is surjective on $(n+1)$-simplices. In particular, $f$ exhibits $\mathrm{K}(G,n)$ as a weak $n$-coskeleton of $X$ (Definition 3.5.4.14). Since $X$ is weakly $n$-coskeletal (Proposition 3.5.5.10), we conclude that $f$ is an isomorphism. $\square$