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. Conduché'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).
Keywords: semi-abelian categories, torsion theories, simplicial groups, internal groupoids, crossed modules, 2-crossed modules, crossed complexes, non-abelian homological algebra
2020 MSC: 18E13 18E40 18N50 18D40 18G45 18G35 18G50
Theory and Applications of Categories, Vol. 41, 2024, No. 32, pp 1044-1076.
Published 2024-08-30.
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