A 2-dimensional torsion theory on symmetric monoidal categories

TL;DR

This paper develops a homotopy torsion theory in symmetric monoidal categories, introducing 2-dimensional properties and generalizing classical torsion theories from abelian groups and monoids.

Contribution

It introduces a novel 2-dimensional homotopy torsion theory in symmetric monoidal categories, extending classical torsion theories to higher categorical structures.

Findings

Homotopy torsion theory in symmetric monoidal categories

Use of natural isomorphisms for nullhomotopy structure

Generalization of classical torsion theories to 2-categories

Abstract

In this paper we describe a homotopy torsion theory in the category of small symmetric monoidal categories. Thanks to the use of natural isomorphisms as basis for the nullhomotopy structure, this homotopy torsion theory enjoys some interesting 2-dimensional properties which may be the starting point for a definition of "2-dimensional torsion theory". As torsion objects we take symmetric 2-groups, thus generalising a known pointed torsion theory in the category of commutative monoids where abelian groups play the part of torsion objects. In the last part of the paper we carry out an analogous generalisation for the classical torsion theory in the category of abelian groups given by torsion and torsion-free groups.