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arxiv: 2510.07040 · v2 · submitted 2025-10-08 · 🧮 math.CT

Pretorsion theories in prenormal categories

Sandra Mantovani , Mariano Messora This is my paper

Pith reviewed 2026-05-18 09:32 UTC · model grok-4.3

classification 🧮 math.CT
keywords pretorsion theoriesprenormal categoriestorsion theoriesclosure operatorsfactorization systemshereditary torsion theoriesnon-pointed categoriescategorical algebra
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The pith

Pretorsion theories extend classical torsion pair results to non-pointed settings by using kernels and cokernels relative to trivial objects in prenormal categories.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops non-pointed versions of torsion theories, called pretorsion theories, inside prenormal categories. These theories rely on a fixed class of trivial objects to define relative kernels and cokernels instead of requiring a zero object. The authors recover characterizations of the torsion and torsion-free classes, along with their links to closure operators and stable factorization systems. They also treat hereditary cases and produce fresh examples. A reader would care if this machinery lets torsion-theoretic ideas apply in categories that lack a natural zero object.

Core claim

In prenormal categories a pretorsion theory consists of a pair of full subcategories (T, F) such that T comprises precisely the objects whose cokernels relative to the trivial class lie in F, and F comprises the objects whose kernels relative to the trivial class lie in T; under this definition the classical bijections with certain closure operators and with stable factorization systems continue to hold, and the same pattern specializes to hereditary pretorsion theories.

What carries the argument

Pretorsion theory, a pair of full subcategories defined by mutual closure under relative kernels and cokernels with respect to a fixed class of trivial objects.

If this is right

  • Characterizations of torsion and torsion-free subcategories carry over directly.
  • The classical correspondence between torsion theories and closure operators remains valid.
  • The correspondence between torsion theories and stable factorization systems extends to the non-pointed case.
  • Hereditary pretorsion theories inherit the same structural results.
  • New examples of pretorsion theories arise in categories that satisfy the prenormal axioms.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same relative-kernel construction could be tried in ordinary categories of sets or graphs to produce torsion-like decompositions without forcing a zero object.
  • Connections between pretorsion theories and factorization systems in non-additive settings might become easier to state once the prenormal language is adopted.
  • Explicit computation of the trivial class and the induced closure operator in a known prenormal category would give a direct test of the recovered correspondences.

Load-bearing premise

The ambient category must admit kernels and cokernels relative to a fixed class of trivial objects, i.e., it must be prenormal.

What would settle it

A concrete prenormal category together with a candidate pair of subcategories that satisfies the kernel-cokernel definition yet fails to correspond to any closure operator or stable factorization system.

read the original abstract

In this paper we extend several classical results on pointed torsion theories -- also known as torsion pairs -- to the setting of non-pointed torsion theories defined via kernels and cokernels relative to a fixed class of trivial objects (often referred to as pretorsion theories). Our results are developed in the recently introduced framework of (non-pointed) prenormal categories and other related contexts. Within these settings, we recover some characterisations of torsion and torsion-free subcategories, as well as the classical correspondences between torsion theories and closure operators. We also suitably extend a correspondence between torsion theories and (stable) factorisation systems on the ambient category, known in the homological case. Some of these results are then further specialised to an appropriate notion of hereditary torsion theory. Finally, we apply the developed theory to construct new examples of pretorsion theories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper extends classical results on pointed torsion theories (torsion pairs) to non-pointed pretorsion theories in prenormal categories. Pretorsion theories are defined via kernels and cokernels relative to a fixed class of trivial objects. The work recovers characterizations of torsion and torsion-free subcategories, establishes correspondences with closure operators and stable factorization systems, specializes some results to hereditary pretorsion theories, and constructs new examples in the framework.

Significance. If the derivations hold, the paper offers a coherent generalization of torsion-theoretic tools to non-pointed categorical settings, which may prove useful for homological algebra beyond pointed categories. The recovery of classical correspondences (characterizations, closure operators, factorization systems) indicates that the prenormal axioms support the expected structure without introducing visible inconsistencies. The construction of new examples provides concrete illustrations of the theory.

minor comments (3)
  1. The abstract states that results are 'further specialised to an appropriate notion of hereditary torsion theory'; a brief pointer to the relevant section or theorem number would help readers locate this specialization quickly.
  2. In the introduction or §2, the comparison with the pointed case could include one or two explicit citations to the classical statements being generalized (e.g., the characterization of torsion pairs or the correspondence with factorization systems) to make the extension more immediately verifiable.
  3. Notation for the class of trivial objects and the associated kernels/cokernels is introduced early; ensuring consistent use of this notation throughout the proofs of the main correspondences would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and significance assessment of our work extending torsion-theoretic results to pretorsion theories in prenormal categories. We appreciate the recommendation for minor revision and will prepare an updated manuscript accordingly.

Circularity Check

0 steps flagged

No significant circularity; derivations are self-contained extensions of external axioms

full rationale

The paper extends classical pointed torsion theory results to pretorsion theories in prenormal categories by defining them via kernels and cokernels relative to a fixed class of trivial objects. All characterizations of torsion/torsion-free classes, correspondences with closure operators, and extensions to factorization systems are derived directly from the prenormal category axioms and standard categorical constructions. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the framework is invoked as an external setting, and proofs recover expected properties without circular reduction. The development is conditioned on the stated axioms without smuggling in ansatzes or renaming known results as new derivations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the framework of prenormal categories and the definition of pretorsion theories via relative kernels and cokernels; no free parameters or invented entities are evident from the abstract.

axioms (1)
  • domain assumption The category is prenormal, allowing kernels and cokernels relative to a fixed class of trivial objects.
    Invoked throughout the extension of results from pointed to non-pointed settings.

pith-pipeline@v0.9.0 · 5659 in / 1094 out tokens · 27411 ms · 2026-05-18T09:32:39.561718+00:00 · methodology

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

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