Derived Homology of Triangulated Categories: A Canonical Abelianization

This paper introduces a novel theory of derived homology for triangulated categories, offering a systematic approach to extracting abelian information from these highly structured, yet non-abelian, algebraic settings. While triangulated categories provide a powerful framework for studying derived functors in various contexts such as algebraic geometry, representation theory, and K-theory, their non-abelian nature often obscures underlying homological invariants that are naturally expressible in abelian categories. We construct a functorial derived homology theory that associates to each object in a triangulated category a sequence of abelian groups, satisfying properties analogous to classical homology theories, including long exact sequences. Furthermore, we define a "canonical abelianization" functor from a triangulated category to an abelian category, which is shown to be universal among exact functors to abelian categories. We demonstrate that our derived homology groups can be naturally recovered as the classical homology groups of objects within this canonical abelianization. This work provides a crucial bridge between triangulated and abelian categories, enhancing our ability to apply traditional homological methods to derived settings and offering new perspectives on the structure of triangulated categories.