Tensor product of chain complexes

Definition 6.6.23 (Tensor product of chain complexes). The tensor product of chain complexes \(C_{\bullet }, C'_{\bullet } \in \Ch (\Aa )\) is the chain complex \((C \otimes C')_{\bullet }\) defined by \[ (C \otimes C')_n \,:=\, \bigoplus _{p + q = n} C_p \otimes C'_q \] whose differential acts on the summand \(C_p \otimes C'_q\) as the sum \(d^C \otimes \id _{C'} + (-1)^p \id _C \otimes d^{C'}\) of the two canonical composite maps. This makes \(\Ch (\Aa )\) into a symmetric monoidal category. Moreover, tensor products preserve chain homotopies: if \(f \simeq g\colon C_{\bullet } \to D_{\bullet }\) via a homotopy \(h\), then \(f \otimes \id _{C'} \simeq g \otimes \id _{C'}\) via \(h \otimes \id _{C'}\).