Kerodon

Example 2.5.6.8. Let $G$ be an abelian group and let $G[1]$ denote the chain complex consisting of the single abelian group $G$, concentrated in degree $1$. To supply an $n$-simplex of the simplicial set $\mathrm{K}(G[1])$, one must give a chain map $\sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow G[1]$. By definition, a homomorphism of graded abelian groups from $\mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}})$ to $G[1]$ is given by a system $\{ a_{i,j} \} _{0 \leq i < j \leq n}$ of elements of $G$, indexed by the set of all nondegenerate edges of $\Delta ^ n$. Under this identification, the chain maps can be identified with those systems $\{ g_{i,j} \} _{0 \leq i < j \leq n}$ satisfying $g_{i,j} + g_{j,k} = g_{i,k}$ for $0 \leq i < j < k \leq n$. It follows that the Eilenberg-MacLane space $\mathrm{K}(G[1])$ can be identified with the classifying simplicial set $B_{\bullet }G$ of Construction 1.3.2.5.