Categories of skew left braces and trifactorised groups
A. Ballester-Bolinches, R. Esteban-Romero, P. Pérez-Altarriba, V. Pérez-Calabuig
TL;DR
The paper develops a unified framework connecting skew left braces to trifactorised groups, introducing a generalised category $\mathbf{3factGrp}$ that encapsulates large, small, and new trifactorised group constructions arising from braces. It proves that isomorphism classes of trifactorised groups associated with a brace correspond to automorphism-orbits on a kernel of the lambda action, and that every trifactorised group is a quotient of the large construction with the small one as a further quotient. A key result is the equivalence between the brace category $\mathbf{SKB}$ and the category of large trifactorised groups $\mathbf{L3factGrp}$, extended via a functor $\mathsf{B}$, and a detailed correspondence between brace substructures/ideals and trifactorised subgroups/normal subgroups. The work also provides precise characterisations of substructures and quotients in the trifactorised framework, offering tools to transfer brace problems to group-theoretic questions and enabling potential computational benefits. Overall, the paper advances the categorical and structural understanding of braces through a robust trifactorised-group perspective, with implications for brace representations and constructions.
Abstract
The main objective of this paper is to deepen the relationship between skew left braces and trifactorised groups that encodes the information about skew left braces, their structure, their quotients, and their homomorphisms.