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Construction (The Monoidal Structure on Chain Complexes). Let $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ denote the category of chain complexes of abelian groups (Definition We define a monoidal structure on $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ as follows:

  • The tensor product functor $\boxtimes : \operatorname{Ch}(\operatorname{\mathbf{Z}}) \times \operatorname{Ch}(\operatorname{\mathbf{Z}}) \rightarrow \operatorname{Ch}(\operatorname{\mathbf{Z}})$ carries each pair of chain complexes $(C_{\ast }, \partial )$ and $(D_{\ast }, \partial )$ to the tensor product chain complex $( C_{\ast } \boxtimes D_{\ast }, \partial )$ of Proposition, and carries a pair of chain maps $f: C_{\ast } \rightarrow C'_{\ast }$, $g: D_{\ast } \rightarrow D'_{\ast }$ to the tensor product map

    \[ (f \boxtimes g): C_{\ast } \boxtimes D_{\ast } \rightarrow C'_{\ast } \boxtimes D'_{\ast } \quad \quad (f \boxtimes g)(x \boxtimes y) = f(x) \boxtimes g(y). \]
  • For every triple of chain complexes $C = (C_{\ast }, \partial )$, $D = (D_{\ast }, \partial )$, and $E = (E_{\ast }, \partial )$, the associativity constraint

    \[ \alpha _{C,D,E}: C_{\ast } \boxtimes (D_{\ast } \boxtimes E_{\ast }) \simeq (C_{\ast } \boxtimes D_{\ast }) \boxtimes E_{\ast } \]

    is the isomorphism of Remark

  • The unit object of $\operatorname{Ch}(\operatorname{\mathbf{Z}})$ is the chain complex $\operatorname{\mathbf{Z}}[0]$ of Example, and the unit constraint $\upsilon : \operatorname{\mathbf{Z}}[0] \boxtimes \operatorname{\mathbf{Z}}[0] \simeq \operatorname{\mathbf{Z}}[0]$ is the isomorphism classified by the bilinear map

    \[ \operatorname{\mathbf{Z}}\times \operatorname{\mathbf{Z}}\rightarrow \operatorname{\mathbf{Z}}\quad \quad (m,n) \mapsto mn. \]