Chain complexes

Definition 6.1.7. A chain complex in \(\Aa \) is a pair \((C_{\bullet },d_{\bullet })\) consisting of a \(\Z \)-graded object \(C_{\bullet } = (C_n)_{n \in \Z }\), \(C_n \in \Aa \), and a collection of ‘boundary maps’ \(d_{\bullet } = (d_n\colon C_n \to C_{n-1})_{n \in \Z }\) satisfying the property that \(d_{n-1} \circ d_n = 0\) for all \(n\). We will often denote the chain complex simply by \(C_{\bullet }\) or \(C\).

A chain map (or morphism of chain complexes) \(f_{\bullet }\colon C_{\bullet } \to D_{\bullet }\) consists of a collection of morphisms \(f_n\colon C_n \to D_n\) that commute with the boundary maps, in the sense that \(d^D_n \circ f_n = f_{n-1} \circ d_n^C\) for all \(n\). There is a clear way to compose these, resulting in a 1-category \(\Ch (\Aa )\) of chain complexes in \(\Aa \).