For any \(k\)-linear category \(\mathcal C\) with zero object, we write \(\mathrm{Ch}^b(\mathcal C)\) for the \(k\)-linear category of bounded (on both sides) chain complexes in \(\mathcal C\), with chain maps as morphisms. If \(\mathcal C\) is additive (and thus has a zero object) or equipped with a \(\mathbb{Z}\)-action, then so is \(\mathrm{Ch}^b(\mathcal C)\). If \(\mathcal C\) is additive and equipped with a monoidal structure compatible with \(\oplus\), then this is inherited by \(\mathrm{Ch}^b(\mathcal C)\). There is a natural notion of homotopy between chain maps and the nullhomotopic chain maps form a \(k\)-linear (monoidal) ideal.
The quotient of \(\mathrm{Ch}^b(\mathcal C)\) by the nullhomotopic chain maps is the chain homotopy category \(\mathrm{K}^b(\mathcal C)\). An isomorphism between objects of \(\mathrm{K}^b(\mathcal C)\) is called a chain homotopy equivalence. 13