Abelian Signatures of Triangulated Categories: Derived Reconstruction Theorems

This paper introduces the concept of "abelian signatures" for triangulated categories as novel invariants aimed at providing deeper insights into their structure and enabling derived reconstruction. Triangulated categories, fundamental in modern algebraic geometry, representation theory, and topology, often conceal the underlying geometric or algebraic objects from which they are derived. While existing reconstruction theorems, such as those by Bondal-Orlov and Matsui, utilize spectral methods derived from tensor or general triangulated categories, this work proposes a new approach leveraging the intrinsic homological properties that link triangulated categories back to abelian categories. We define an abelian signature as a collection of invariants capturing the "abelian-like" behavior within a triangulated category, often expressed through functorial constructions into or from suitable abelian categories. We demonstrate how these signatures can distinguish non-equivalent triangulated categories and, crucially, how they facilitate the reconstruction of associated abelian categories or even geometric spaces. As an application, we provide an alternative framework for understanding and proving derived reconstruction theorems, particularly highlighting their utility in contexts where tensor structures may be absent or less prominent. The proposed framework offers a complementary perspective to spectral methods, potentially simplifying reconstruction arguments in certain settings and opening new avenues for classification and characterization within higher category theory.