Proposition 3.5.5.9. Let $n$ be a nonnegative integer, let $A_{\ast }$ be a chain complex of abelian groups, and let $X = \mathrm{K}( A_{\ast } )$ denote the associated Eilenberg-MacLane space (Construction 2.5.6.3). Then $X$ is an $n$-groupoid if and only if the abelian groups $A_{m}$ vanish for $m > n$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Fix a pair of integers $0 \leq i \leq m$ with $m > n$. Let $\sigma : \Delta ^{m} \rightarrow \Delta ^{m}$ be the identity map, which we identify with its image in the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^{m}; \operatorname{\mathbf{Z}})$. Then the chain complex $\mathrm{N}_{\ast }( \Delta ^{m}; \operatorname{\mathbf{Z}})$ splits as a direct sum of $\mathrm{N}_{\ast }( \Lambda ^{m}_{i}; \operatorname{\mathbf{Z}})$ with the subcomplex $C_{\ast }$ spanned by the $\sigma $ and $\partial (\sigma )$. We therefore obtain a canonical bijection
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) & \simeq & \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( \mathrm{N}_{\ast }( \Delta ^ m; \operatorname{\mathbf{Z}}), A_{\ast } ) \\ & \simeq & \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( \mathrm{N}_{\ast }( \Lambda ^ m_{i}; \operatorname{\mathbf{Z}}), A_{\ast } ) \times \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( C_{\ast }, A_{\ast } ) \\ & \simeq & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{i}, X) \times A_{m}. \end{eqnarray*}
It follows that the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Lambda ^{m}_{i}, X)$ is a bijection if and only if the abelian group $A_ m$ vanishes. The desired result follows by allowing the integers $0 \leq i \leq m$ to vary. $\square$