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Construction 2.5.8.2 (The Alexander-Whitney Construction: Unnormalized Version). Let $A_{\bullet }$ and $B_{\bullet }$ be simplicial abelian groups with Moore complexes $\mathrm{C}_{\ast }(A)$ and $\mathrm{C}_{\ast }(B)$, respectively. We define a map of graded abelian groups $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ by the formula

$\overline{\mathrm{AW}}( a \otimes b) = \sum _{0 \leq p \leq n} \iota _{\leq p}^{\ast }(a) \boxtimes \iota _{\geq p}^{\ast }(b)$

for $a \in A_ n$ and $b \in B_ n$. We will refer to $\overline{\mathrm{AW}}$ as the unnormalized Alexander-Whitney homomorphism.