A 2-dimensional torsion theory on symmetric monoidal categories

In this paper, we describe a homotopy torsion theory on the category of small symmetric monoidal categories. By using natural isomorphisms as the basis for the nullhomotopy structure, this homotopy torsion theory exhibits some interesting 2-dimensional properties which could be the foundation for a definition of “2-dimensional torsion theory”. We choose symmetric 2-groups as torsion objects, thereby generalising a known pointed torsion theory in the category of commutative monoids where abelian groups are taken as torsion objects. In the final 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 abelian groups.