$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Example Let $S_{\bullet }$ and $T_{\bullet }$ be simplicial sets, and let $\operatorname{\mathbf{Z}}[ S_{\bullet } ]$ and $\operatorname{\mathbf{Z}}[ T_{\bullet } ]$ denote the free simplicial abelian groups generated by $S_{\bullet }$ and $T_{\bullet }$, respectively. Then the tensor product $\operatorname{\mathbf{Z}}[ S_{\bullet }] \otimes \operatorname{\mathbf{Z}}[ T_{\bullet } ]$ can be identified with the free simplicial abelian group $\operatorname{\mathbf{Z}}[ S_{\bullet } \times T_{\bullet } ]$ generated by the cartesian product $S_{\bullet } \times T_{\bullet }$. Invoking Construction, we obtain shuffle product maps

\[ \triangledown : \mathrm{N}_{p}(S; \operatorname{\mathbf{Z}}) \times \mathrm{N}_{q}(T; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{p+q}(S \times T; \operatorname{\mathbf{Z}}) \]

which induce a map of chain complexes $\mathrm{EZ}: \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}) \boxtimes \mathrm{N}_{\ast }(T; \operatorname{\mathbf{Z}}) \rightarrow \mathrm{N}_{\ast }(S \times T; \operatorname{\mathbf{Z}})$. Allowing $S_{\bullet }$ and $T_{\bullet }$ to vary, these chain maps furnish a lax (symmetric) monoidal structure on the functor

\[ \mathrm{N}_{\ast }( -; \operatorname{\mathbf{Z}}): \operatorname{Set_{\Delta }}\rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}}) \quad \quad S_{\bullet } \mapsto \mathrm{N}_{\ast }(S; \operatorname{\mathbf{Z}}). \]