Torsion theories of simplicial groups with truncated Moore complex

TL;DR

This paper introduces a new lattice of torsion theories for simplicial groups based on Moore complexes, linking algebraic torsion concepts with homotopy group calculations.

Contribution

It defines a novel lattice of torsion theories in simplicial groups using Moore complexes, extending existing theories in internal groupoids.

Findings

Lattice of torsion theories in simplicial groups is established.

Connections between torsion theories and homotopy groups are demonstrated.

Torsion subobjects relate to homotopy group calculations.

Abstract

We introduce a linearly ordered lattice $\mu(Grp)$ of torsion theories in simplicial groups. The torsion theories are defined where the torsion/torsion-free subcategories are given by the simplicial groups with bounded above/below Moore complex, respectively. These torsion theories extend naturally the torsion theories in internal groupoids in groups. Connections of this lattice with the homotopy groups are established since the homotopy groups of a simplicial group can be calculated as the quotients of torsion subojects.