Kerodon

Example 2.5.6.7. Let $G$ be an abelian group and let $G[1]$ denote the chain complex given by the single group $G$, concentrated in degree $0$. To supply an $n$-simplex of the simplicial set $\mathrm{K}( G[0] )$, one must give a chain map $\sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow G[0]$. By definition, a homomorphism of graded abelian groups from $\mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}})$ to $G[0]$ is given by a tuple $\{ g_ i \} _{0 \leq i \leq n}$ of elements of $G$, indexed by the set $[n] = \{ 0 < 1 < \cdots < n \} $ of vertices of $\Delta ^ n$. Under this identification, the chain maps can be identified with those tuples $\{ g_ i \} _{0 \leq i \leq n}$ which are constant: that is, which satisfy $g_ i = g_ j$ for all $i,j \in [n]$. It follows that the Eilenberg-MacLane space $\mathrm{K}( G[0] )$ can be identified with the constant simplicial abelian group $\underline{G}$.