Algebraic Structures in Cat1-Groups: Normality, Quotients, Coverings, and Actions via Crossed Modules

This paper develops a systematic framework for coverings, actions, and quotient structures in cat1-groups. While cat1-groups are categorically equivalent to crossed modules and 2-groups, their explicit treatment via defining maps has remained limited. We introduce coverings and actions directly within the cat1-group structure and establish categorical equivalences between coverings, actions, and liftings of crossed modules. Constructive definitions of normal subcat1-groups and quotient cat1-groups are provided by translating properties of normal subcrossed modules through the equivalence functor. Our aim is to present a coherent algebraic framework that connects quotient structures of cat1-groups to coverings and crossed modules, thereby filling a gap in the literature and offering new tools for homological algebra and exact sequences. In particular, we show that the quotient structure of a cat1-group simultaneously generates its covering morphism and aligns with lifting constructions of crossed modules.