Quasi-isomorphisms and chain homotopies

Definition 6.1.13. A chain map \(f_{\bullet }\) is called a quasi-isomorphism if the induced map \(H_n(f)\colon H_n(C) \to H_n(D)\) is an isomorphism for all \(n\).

Given two chain maps \(f,g\colon C_{\bullet } \to D_{\bullet }\), a chain homotopy between \(f\) and \(g\) is a family of maps \((H_n\colon C_n \to D_{n+1})_{n \in \Z }\) satisfying \(f_n - g_n = \partial ^D \circ H_n + H_{n-1} \circ \partial ^C\). We say \(f\) is a chain homotopy equivalence if there exists a chain map \(g\colon D_{\bullet } \to C_{\bullet }\) and chain homotopies \(g \circ f \sim \id _{C_{\bullet }}\) and \(f \circ g \sim \id _{D_{\bullet }}\).