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Proposition 2.5.8.4. The collection of unnormalized Alexander-Whitney homomorphisms $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ determine a colax monoidal structure on the Moore complex functor $\mathrm{C}_{\ast }: \operatorname{ Ab }_{\Delta } \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$ (see Variant 2.1.5.11).

Proof. We first show that the unnormalized Alexander-Whitney homomorphisms determine a nonunital colax monoidal structure on the functor $\mathrm{C}_{\ast }$ (Variant 2.1.4.16). By construction, the homomorphism $\overline{\mathrm{AW}}: \mathrm{C}_{\ast }(A \otimes B) \rightarrow \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B)$ is natural in $A_{\bullet }$ and $B_{\bullet }$. It will therefore suffice to show that, for every triple of simplicial abelian groups $A_{\bullet }$, $B_{\bullet }$, and $C_{\bullet }$, the diagram of chain complexes

\[ \xymatrix { \mathrm{C}_{\ast }( A \otimes (B \otimes C) ) \ar [r]^-{\sim } \ar [d]^{ \overline{\mathrm{AW}} } & \mathrm{C}_{\ast }( (A \otimes B) \otimes C) \ar [d]^{\overline{\mathrm{AW}}} \\ \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B \otimes C) \ar [d]^{\operatorname{id}\boxtimes \overline{\mathrm{AW}}} & \mathrm{C}_{\ast }(A \otimes B) \boxtimes \mathrm{C}_{\ast }(C) \ar [d]^{\overline{\mathrm{AW}} \boxtimes \operatorname{id}} \\ \mathrm{C}_{\ast }(A) \boxtimes ( \mathrm{C}_{\ast }(B) \boxtimes \mathrm{C}_{\ast }(C) ) \ar [r]^-{\sim } & (\mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(B) ) \boxtimes \mathrm{C}_{\ast }(C) } \]

commutes, where the horizontal maps are given by the associativity constraints of the monoidal categories $\operatorname{ Ab }_{\Delta }$ and $\operatorname{Ch}(\operatorname{\mathbf{Z}})$, respectively. Unwinding the definitions, we see that both the clockwise and counterclockwise composition are given by the construction

\[ a \otimes (b \otimes c) \mapsto \sum _{0 \leq p \leq q \leq n} ( \iota _{\leq p}^{\ast }(a) \boxtimes \rho ^{\ast }(b)) \boxtimes \iota _{\geq q}^{\ast }(c) \]

for $a \in A_ n$, $b \in B_ n$, and $c \in C_ n$, where $\rho $ denotes the nondecreasing map $[q-p] \hookrightarrow [n]$ given by $\rho (i) = i + p$.

Note that the unit object of the category of simplicial abelian groups is the constant functor $\operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{ Ab }$ taking the value $\operatorname{\mathbf{Z}}$, which we can identify with the free simplicial abelian group $\operatorname{\mathbf{Z}}[ \Delta ^0]$ generated by the simplicial set $\Delta ^0$. The image of this object under the functor $\overline{\mathrm{AW}}$ is the unnormalized chain complex $\mathrm{C}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$. On the other hand, the unit object of $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ is the chain complex $\operatorname{\mathbf{Z}}[0]$, which we will identify with the normalized chain complex $\mathrm{N}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$. We will complete the proof of Proposition 2.5.8.4 by showing that the quotient map $\epsilon : \mathrm{C}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}}) \twoheadrightarrow \mathrm{N}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$ is a counit for the nonunital colax monoidal structure constructed above (in the sense of Variant 2.1.5.11). To prove this, we must show that for every simplicial abelian group $A_{\bullet }$, both of the composite maps

\[ \mathrm{C}_{\ast }(A) \simeq \mathrm{C}_{\ast }(A \otimes \operatorname{\mathbf{Z}}[ \Delta ^0] ) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(A) \boxtimes \mathrm{C}_{\ast }(\Delta ^0; \operatorname{\mathbf{Z}}) \xrightarrow {\operatorname{id}\boxtimes \epsilon } \mathrm{C}_{\ast }(A) \boxtimes \operatorname{\mathbf{Z}}[0] \simeq \mathrm{C}_{\ast }(A) \]

\[ \mathrm{C}_{\ast }(A) \simeq \mathrm{C}_{\ast }(\operatorname{\mathbf{Z}}[ \Delta ^0] \otimes A) \xrightarrow { \overline{\mathrm{AW}} } \mathrm{C}_{\ast }(\Delta ^0;\operatorname{\mathbf{Z}}) \boxtimes \mathrm{C}_{\ast }(A) \xrightarrow {\epsilon \boxtimes \operatorname{id}} \operatorname{\mathbf{Z}}[0] \boxtimes \mathrm{C}_{\ast }(A) \simeq \mathrm{C}_{\ast }(A) \]

are equal to the identity. This follows immediately from the construction (using the fact that $\epsilon $ vanishes on every element of $\mathrm{C}_{\ast }( \Delta ^0; \operatorname{\mathbf{Z}})$ of positive degree). $\square$