A 2-dimensional torsion theory on symmetric monoidal categories
Mariano Messora
TL;DR
This work develops a 2-dimensional analogue of torsion theory by placing torsion theory in the 2-category of symmetric monoidal categories using a nullhomotopy framework. It identifies symmetric 2-groups as the torsion objects and symmetric purely monoidal categories as the torsion-free objects, and builds a canonical HTT with inc and prj maps yielding a Θ-exact sequence, while highlighting intriguing 2-dimensional features and a counterexample showing it is not a pretorsion theory. A parallel construction for abelian groups is presented, and homotopy invariants are introduced to relate 1-dimensional and 2-dimensional torsion theories, including connections to the Picard 2-group and endomorphism-based invariants. The paper also extends the approach by formulating a second, parallel 2D torsion theory on symmetric 2-groups, mirroring the abelian-group case and suggesting a robust 2-dimensional theory with rich categorical and homotopical structure. Overall, the work lays foundational steps toward a 2-dimensional torsion theory with potential developments in biadjunctions, bilimits, and broader 2-category homotopy theory.
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.