Homology of a chain complex

Definition 6.1.12. Given a chain complex \(C_{\bullet }\) and an integer \(n\), the \(n\)-th homology of \(C_{\bullet }\) is the object \[ H_n(C) \quad := \quad \ker (d_n) \, / \, \im (d_{n+1}) \qin \Aa . \] Any chain map \(f_{\bullet }\colon C_{\bullet } \to D_{\bullet }\) induces a map \(H_n(f)\colon H_n(C) \to H_n(D)\) on \(n\)-th homology, resulting in a functor \(H_n(-)\colon \Ch (\Aa ) \to \Aa \).