Kerodon

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Example 2.5.6.9. Let $A$ be an abelian group. To supply an $n$-simplex of the simplicial set $\mathrm{K}(A,0)$, one must give a chain map $\sigma : \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}}) \rightarrow A[0]$. By definition, a homomorphism of graded abelian groups from $\mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}})$ to $A[0]$ is given by a tuple $\{ a_ i \} _{0 \leq i \leq n}$ of elements of $A$, 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 $\{ a_ i \} _{0 \leq i \leq n}$ which are constant: that is, which satisfy $a_ i = a_ j$ for all $i,j \in [n]$. It follows that the Eilenberg-MacLane space $\mathrm{K}(A,0)$ can be identified with the constant simplicial abelian group taking the value $A$.