Kerodon

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

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

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

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). $\square$