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Comonadic approach to pretorsion theories

Elena Caviglia, Zurab Janelidze, Luca Mesiti

TL;DR

Using a comonadic lens, the paper shows that pretorsion theories in semiexact categories can be captured as pseudo-coalgebras for a pseudo-comonad on a 2-category of categories with closed ideals. Bihereditary pretorsion theories correspond to pseudo-coalgebras for $\Omega$, and all pretorsion theories correspond to pseudo-coalgebras for the extended $\Omega^{\operatorname{ex}}$, with generalized pretorsion theories arising when the coalgebra map does not preserve a short exact sequence in its image. The construction yields a cofree bihereditary pretorsion theory on any base category and clarifies how pretorsion-theory morphisms lift to coalgebra morphisms, opening avenues for five-adjoint analyses and further study of generalized notions. This framework unifies torsion-like structures in semiexact settings and provides a tool for systematic exploration of examples and further generalizations.

Abstract

We present a comonadic approach to pretorsion theories on semiexact categories, i.e. categories equipped with a closed ideal of null morphisms that admits all kernels and all cokernels. We first prove that bihereditary pretorsion theories are comonadic in a 2-dimensional sense over the 2-category of semiexact categories with naturally chosen 1-cells. We then extend the built pseudo-comonad to guarantee that all pretorsion theories are pseudo-coalgebras. But interestingly, not all pseudo-coalgebras are pretorsion theories. Rather, pseudo-coalgebras give a generalized notion of pretorsion theory.

Comonadic approach to pretorsion theories

TL;DR

Using a comonadic lens, the paper shows that pretorsion theories in semiexact categories can be captured as pseudo-coalgebras for a pseudo-comonad on a 2-category of categories with closed ideals. Bihereditary pretorsion theories correspond to pseudo-coalgebras for , and all pretorsion theories correspond to pseudo-coalgebras for the extended , with generalized pretorsion theories arising when the coalgebra map does not preserve a short exact sequence in its image. The construction yields a cofree bihereditary pretorsion theory on any base category and clarifies how pretorsion-theory morphisms lift to coalgebra morphisms, opening avenues for five-adjoint analyses and further study of generalized notions. This framework unifies torsion-like structures in semiexact settings and provides a tool for systematic exploration of examples and further generalizations.

Abstract

We present a comonadic approach to pretorsion theories on semiexact categories, i.e. categories equipped with a closed ideal of null morphisms that admits all kernels and all cokernels. We first prove that bihereditary pretorsion theories are comonadic in a 2-dimensional sense over the 2-category of semiexact categories with naturally chosen 1-cells. We then extend the built pseudo-comonad to guarantee that all pretorsion theories are pseudo-coalgebras. But interestingly, not all pseudo-coalgebras are pretorsion theories. Rather, pseudo-coalgebras give a generalized notion of pretorsion theory.
Paper Structure (4 sections, 13 theorems, 39 equations)

This paper contains 4 sections, 13 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. The category of short exact sequences
  3. A comonad for bihereditary pretorsion theories
  4. A comonad for generalized pretorsion theories

Key Result

Lemma 2.5

Let $\mathbb{C}$ be a category with a closed ideal of null morphisms $\mathcal{N}$. The following fact hold: We thus have the following canonical short exact sequences associated to any object $Y \in \mathbb{C}$:

Theorems & Definitions (37)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • proof
  • Proposition 2.8
  • ...and 27 more