Torsion theories in simplicial groups and homology
(2022)
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- Authors
- Supervisors
- Gran, Marino
- Abstract
- Torsion theories were originally introduced by D. Dickson to study abelian categories, and in particular categories of modules over rings. More recently torsion theories have been studied in many different non-abelian settings, with different applications in homological algebra. The objective of this thesis is to apply some techniques of torsion theories and preradicals, well known in the abelian context, to the semi-abelian category of simplicial groups. First, we introduce a family of examples of torsion theories in simplicial groups which are defined by simplicial groups with a trivial Moore complex below or above a certain degree. These examples extend the examples of torsion theories in internal groupoids in groups in a natural way. Moreover, we prove that the category of internal groupoids in groups is itself a torsion-free subcategory of simplicial groups. We observe that these torsion theories constitute a linearly ordered lattice, and from the relations between the homotopy groups and the homology of the Moore complex, we can study the homotopy groups of a simplicial group as suitable quotients of torsion subobjects. Since simplicial groups can be studied through their Moore complexes with operations, we find easier descriptions of torsion theories in certain subcategories of simplicial groups. In particular, we consider torsion theories in D. Conduché’s 2-crossed modules and N. Ashley’s reduced crossed complexes which expand on the known examples of torsion theories in the category of crossed modules.
- Affiliations
Citations
López Cafaggi, G. A. (2022). Torsion theories in simplicial groups and homology. https://hdl.handle.net/2078.5/103378