Kerodon

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Construction 2.5.6.3 (The Eilenberg-MacLane Functor). Let $n$ be a nonnegative integer and let $\mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}})$ denote the normalized chain complex of the standard $n$-simplex (Construction 2.5.5.9). For every chain complex $M_{\ast }$, we let $\mathrm{K}_{n}( M_{\ast } )$ denote the collection of chain maps from $\mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}})$ into $M_{\ast }$ (which we regard as an abelian group under addition). Note that the construction $[n] \mapsto \mathrm{N}_{\ast }( \Delta ^ n; \operatorname{\mathbf{Z}})$ determines a functor from the simplex category $\operatorname{{\bf \Delta }}$ to the category of chain complexes, so we can regard $[n] \mapsto \mathrm{K}_{n}( M_{\ast } )$ as a functor from $\operatorname{{\bf \Delta }}^{\operatorname{op}}$ to the category of abelian groups. We denote this simplicial abelian group by $\mathrm{K}( M_{\ast } )$, and refer to it as the Eilenberg-MacLane space associated to $M_{\ast }$.