Quasi-pointed exactness and torsion
Abstract
We show that, under some mild assumptions, every quasi-pointed category C carries a canonical near-torsion theory (T,F), where T consists of the 0-supported objects, that is, those admitting a morphism to the initial object 0, and F consists of the subterminal objects. The coreflection and the reflection of an object A are then given by the projection A × 0 → A and by the strong-epimorphic part of A → 1, respectively. Using the general construction of the "largest" non-pointed torsion theory associated with a near-torsion theory, we prove that (T,F) is a (non-pointed) torsion theory in the full subcategory of those objects A for which the canonical sequence t(A) → A → f(A) is short exact, in the sense that it is both a kernel and a cokernel diagram. This subcategory is the whole of C whenever C is sequentiable, that is, a regular protomodular and quasi-pointed category. We describe the resulting torsion theories in the category of Lie-Rinehart algebras and in the categories of crossed modules over a fixed group. We also show that (T,F) is a torsion theory in the (non-protomodular) category of monoids over a group G all of whose elements have finite order.
Related
- Torsion Theories in a non-pointed Context II: iterated torsion theories
- The moduli space of torsion-free sheaves with quasi-maximal third Chern class
- Derived category of moduli of pointed curves: II
- Eventual Exactness and Rigidity for Abelian Covers of Groups
- Exactness and Drinfeld-Sokolov Realization of the McRae-Yang Tensor Functors
BibTeX
@misc{gran2026quasipointed,
title = {{Quasi-pointed exactness and torsion}},
author = {Marino Gran and George Janelidze},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05309},
eprint = {2610.05309},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05309},
}
RIS
TY - UNPB
TI - Quasi-pointed exactness and torsion
AU - Marino Gran
AU - George Janelidze
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05309
UR - https://arxiv.org/abs/2610.05309
AB - We show that, under some mild assumptions, every quasi-pointed category C carries a canonical near-torsion theory (T,F), where T consists of the 0-supported objects, that is, those admitting a morphism to the initial object 0, and F consists of the subterminal objects. The coreflection and the reflection of an object A are then given by the projection A × 0 → A and by the strong-epimorphic part of A → 1, respectively. Using the general construction of the "largest" non-pointed torsion theory associated with a near-torsion theory, we prove that (T,F) is a (non-pointed) torsion theory in the full subcategory of those objects A for which the canonical sequence t(A) → A → f(A) is short exact, in the sense that it is both a kernel and a cokernel diagram. This subcategory is the whole of C whenever C is sequentiable, that is, a regular protomodular and quasi-pointed category. We describe the resulting torsion theories in the category of Lie-Rinehart algebras and in the categories of crossed modules over a fixed group. We also show that (T,F) is a torsion theory in the (non-protomodular) category of monoids over a group G all of whose elements have finite order.
ER -