Torsion Theories in a non-pointed Context II: iterated torsion theories

Abstract

Generalizing a known result in the context of homological categories, we show that, in a regular protomodular category 𝒞 equipped with a posetal, monocoreflective subcategory whose coreflector inverts monomorphisms, every torsion theory, in a non-pointed sense, satisfying suitable properties gives rise to a torsion theory in the category Arr(𝒞) of arrows in 𝒞. Moreover, we show that the induced torsion theory in Arr(𝒞) satisfies the same good properties of the original one in 𝒞, allowing to iterate the process. Examples of this situation can be found in protomodular varieties of universal algebras with more than one constant or in duals of elementary toposes.

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BibTeX

@misc{cappelletti2026torsion,
  title = {{Torsion Theories in a non-pointed Context II: iterated torsion theories}},
  author = {Andrea Cappelletti and Andrea Montoli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37956},
  eprint = {2610.37956},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37956},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Torsion Theories in a non-pointed Context II: iterated torsion theories
AU  - Andrea Cappelletti
AU  - Andrea Montoli
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37956
UR  - https://arxiv.org/abs/2610.37956
AB  - Generalizing a known result in the context of homological categories, we show that, in a regular protomodular category 𝒞 equipped with a posetal, monocoreflective subcategory whose coreflector inverts monomorphisms, every torsion theory, in a non-pointed sense, satisfying suitable properties gives rise to a torsion theory in the category Arr(𝒞) of arrows in 𝒞. Moreover, we show that the induced torsion theory in Arr(𝒞) satisfies the same good properties of the original one in 𝒞, allowing to iterate the process. Examples of this situation can be found in protomodular varieties of universal algebras with more than one constant or in duals of elementary toposes.
ER  -

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