Kerodon

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Construction 2.5.8.6 (The Alexander-Whitney Construction: Normalized Version). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups. It follows from Proposition 2.5.8.5 that there is a unique chain map $\mathrm{AW}: \mathrm{N}_{\ast }(A \otimes B) \rightarrow \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B)$ for which the diagram

\[ \xymatrix { \mathrm{C}_{\ast }(A \otimes B) \ar [r]^-{\overline{\mathrm{AW}}} \ar@ {->>}[d] & \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) \ar@ {->>}[d] \\ \mathrm{N}_{\ast }(A \otimes B) \ar [r]^-{ \mathrm{AW} } & \mathrm{N}_{\ast }(A) \boxtimes \mathrm{N}_{\ast }(B). } \]

We will refer to $\mathrm{AW}$ as the Alexander-Whitney homomorphism.