Universal Semisimple Reconstruction and Obstruction Theory

Abstract

We develop a semisimple obstruction theory for finite-length abelian categories. To each object \(X\) we associate a canonical obstruction subobject \(\mathcal O(X)\), consisting of the part invisible to all morphisms into semisimple objects. The quotient \(X/\mathcal O(X)\) is semisimple and universal among such morphisms, yielding a canonical semisimplification reflector and characterizing semisimplicity by the vanishing of obstruction. Iterating the construction gives an obstruction filtration which, in finite-length module categories, recovers the classical radical filtration and Loewy theory. Obstruction is also heart-dependent: under Happel--Reiten--Smalø tilting, extension data can appear as new obstruction layers, leading to obstruction shadows and visible torsion. Thus classical radical theory arises as the fixed-heart module-theoretic instance of a categorical framework for semisimple reconstruction, persistence, and variation of heart.