Homological algebra and homotopical equality

Example 1.1.2 (Homological algebra). The phenomenon of ‘equality as structure’ also naturally appears in homological algebra, where it leads us to think about chain complexes not as rigid algebraic objects but as representatives of more flexible homotopical entities:

• Instead of asking whether two chain maps \(f,g\colon C_{\bullet } \to D_{\bullet }\) between chain complexes are equal, we ask whether they are chain homotopic, meaning that there exists a family of maps \(\phi _n\colon C_n \to D_{n+1}\) satisfying \(f_n - g_n = \partial ^D \circ \phi _n + \phi _{n-1} \circ \partial ^C\). Likewise, a chain map \(C_{\bullet } \to D_{\bullet }\) is regarded as an equivalence if it admits an inverse up to chain homotopy.
• In practice, one often wants to go further and even identify two chain complexes whenever they are connected by a quasi-isomorphism, that is, a chain map inducing isomorphisms on all homology groups. Forcing the quasi-isomorphisms to become actual isomorphisms results in the derived category of chain complexes.