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

Remark Let $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ be chain complexes. The tensor product chain complex $(C_{\ast } \boxtimes D_{\ast }, \partial )$ of Proposition is characterized up to (unique) isomorphism by the universal property of Exercise However, the construction of this tensor product complex (and, by extension, the monoidal structure on $\operatorname{Ch}(\operatorname{\mathbf{Z}})$) depends on auxiliary choices. These choices are ultimately irrelevant in the sense that they do not change the isomorphism class of the monoidal category $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ or, equivalently, of the classifying simplicial set $B_{\bullet } \operatorname{Ch}(\operatorname{\mathbf{Z}})$ of Example This simplicial set can be described concretely (without auxiliary choices): its $n$-simplices can be identified with systems of chain complexes $\{ C(j,i)_{\ast } \} _{0 \leq i < j \leq n}$ together with bilinear maps

\[ C(k,j)_{q} \times C(j,i)_{p} \rightarrow C(k,i)_{q+p} \quad \quad (y,z) \mapsto yz \]

for $0 \leq i < j < k \leq n$ which satisfy the Leibniz rule $\partial (yz) = (\partial y)z + (-1)^{q} y(\partial z)$ together with the associative law $x(yz) = (xy)z$ for $x \in C(\ell ,k)_{r}$, $y \in C(k,j)_{q}$, $z \in C(j,i)_{p}$ with $0 \leq i < j < k < \ell \leq n$.