Algebraic Structures in Cat<SUP>1</SUP>-Groups: Normality, Quotients, Coverings, and Actions via Crossed Modules
JOURNAL OF MATHEMATICS, cilt.2026, sa.1, 2026 (SCI-Expanded)
- Yayın Türü: Makale / Tam Makale
- Cilt numarası: 2026 Sayı: 1
- Basım Tarihi: 2026
- Doi Numarası: 10.1155/jom/5519446
- Dergi Adı: JOURNAL OF MATHEMATICS
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED)
- Recep Tayyip Erdoğan Üniversitesi Adresli: Evet
Özet
This paper develops a systematic framework for coverings, actions, and quotient structures in cat(1)-groups. While cat(1)-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 cat(1)-group structure and establish categorical equivalences between coverings, actions, and liftings of crossed modules. Constructive definitions of normal subcat(1)-groups and quotient cat(1)-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 cat(1)-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 cat(1)-group simultaneously generates its covering morphism and aligns with lifting constructions of crossed modules.