Example 3.1.6.11 (03DY)

Example 3.1.6.11. Let $S$ be a simplicial set and let $\mathrm{N}_{\ast }(S;\operatorname{\mathbf{Z}})$ for the normalized chain complex of $S$ (Construction 2.5.5.9). Let $M_{\ast }$ be a chain complex of abelian groups, let $\mathrm{K}( M_{\ast } )$ denote the associated (generalized) Eilenberg-MacLane space, and let

\[ H_{\ast } = \operatorname{Hom}_{ \operatorname{Ch}(\operatorname{\mathbf{Z}}) }( \mathrm{N}_{\ast }(S, \operatorname{\mathbf{Z}}), M_{\ast } )_{\ast } \]

denote the chain complex of maps from $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}})$ to $M_{\ast }$. Then there is a map of Kan complexes

\[ \lambda : \mathrm{K}( H_{\ast } ) \rightarrow \operatorname{Fun}(S, \mathrm{K}( M_{\ast } )), \]

which classifies the map of chain complexes

\begin{eqnarray*} \mathrm{N}_{\ast }( S \times \mathrm{K}(H_{\ast }); \operatorname{\mathbf{Z}}) & \xrightarrow { \mathrm{AW} } & \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }( \mathrm{K}( H_{\ast } ); \operatorname{\mathbf{Z}}) \\ & \rightarrow & \mathrm{N}_{\ast }(S) \boxtimes H_{\ast } \\ & \xrightarrow {\operatorname{ev}} & M_{\ast } \end{eqnarray*}

where $\mathrm{AW}$ is the Alexander-Whitney map (see Construction 2.5.8.6). The morphism $\lambda $ is a homotopy equivalence of Kan complexes. To prove this, it will suffice to show that for every simplicial set $T$, composition with $\lambda $ induces a bijection

\[ \lambda _{T}: \pi _0( \operatorname{Fun}(T, \mathrm{K}( H_{\ast } ) ) \rightarrow \pi _0( \operatorname{Fun}(S \times T, \mathrm{K}(M_{\ast } ) ) ). \]

Using Example 3.1.5.8 (and the definition of the chain complex $H_{\ast }$), we can identify the source of $\lambda _{T}$ with the set of chain homotopy classes of maps the tensor product $\mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T;\operatorname{\mathbf{Z}})$ into $M_{\ast }$, and the target of $\lambda _{T}$ with the set of chain homotopy classes of maps from $\mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}})$ into $M_{\ast }$. Under these identifications, we see that $\lambda _{T}$ is induced by precomposition with the Alexander-Whitney map

\[ \mathrm{AW}: \mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }(S ;\operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T;\operatorname{\mathbf{Z}}). \]

This map is a quasi-isomorphism (Corollary 2.5.8.11), and therefore admit a chain homotopy inverse (since the source and target of $\mathrm{AW}$ are nonnegatively graded complexes of free abelian groups; see Remark ).