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 𝒪(X), consisting of the part invisible to all morphisms into semisimple objects. The quotient X/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.
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BibTeX
@misc{gildea2026universal,
title = {{Universal Semisimple Reconstruction and Obstruction Theory}},
author = {Joe Gildea},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11497},
eprint = {2610.11497},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11497},
}
RIS
TY - UNPB
TI - Universal Semisimple Reconstruction and Obstruction Theory
AU - Joe Gildea
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11497
UR - https://arxiv.org/abs/2610.11497
AB - We develop a semisimple obstruction theory for finite-length abelian categories. To each object X we associate a canonical obstruction subobject 𝒪(X), consisting of the part invisible to all morphisms into semisimple objects. The quotient X/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.
ER -