Torsion aspects of varieties of simplicial groups

TL;DR

This paper explores the structure of torsion theories within simplicial groups, focusing on their restriction to subcategories and revealing new semi-abelian properties and examples of torsion theories.

Contribution

It demonstrates that certain subcategories of simplicial groups are semi-abelian and introduces new examples of torsion and pretorsion theories in this context.

Findings

Categories of 2-crossed modules and crossed complexes are semi-abelian.

New examples of torsion and pretorsion theories are identified.

Torsion theories relate to truncated Moore complexes in simplicial groups.

Abstract

There is a lattice of torsion theories in simplicial groups such that the torsion/torsion-free categories are given by simplicial groups with truncated Moore complex below/above a certain degree. We study the restriction of these torsion theories to certain subcategories of simplicial groups. In particular, we prove that the categories of D.Conduch\'{e}'s 2-crossed modules and Ashley's crossed complexes in groups are semi-abelian and we give some descriptions of their torsion theories. These examples of torsion theories also give rise to new examples of pretorsion theories in the sense of A. Facchini and C. Finocchiaro, as well as examples of torsion torsion-free theories (TTF theories).