Comonadic approach to pretorsion theories

TL;DR

This paper introduces a comonadic framework for pretorsion theories in semiexact categories, revealing a 2-dimensional comonadic structure and extending to a generalized notion of pretorsion theories.

Contribution

It establishes a comonadic approach to pretorsion theories, showing bihereditary theories are comonadic and extending the framework to pseudo-coalgebras for a broader concept.

Findings

Bihereditary pretorsion theories are comonadic in a 2-dimensional sense.

The pseudo-comonad extension ensures all pretorsion theories are pseudo-coalgebras.

Pseudo-coalgebras form a generalized notion of pretorsion theories.

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.