Akademik Veri Yönetim Sistemi
AIMS Mathematics, cilt.11, sa.3, ss.6141-6161, 2026 (SCI-Expanded, Scopus)
In this paper, we introduce the notions of cat1-2-groups, which are defined as group objects in the category of cat1-categories (or cat1-groupoids). We investigated their structure and established fundamental categorical equivalences that connect them with classical algebraic constructs. Central to our work was the introduction of cat1-crossed modules over groups, which were demonstrated to be an equivalent and more manageable algebraic model for cat1-2-groups. We established categorical equivalences between cat1-2-groups, crossed modules over 2-groups, and cat1-crossed modules. We also obtained cat1-2-groups as internal cat1-categories in the category of groups. Furthermore, we explored simplicial 2-groups whose Moore complex was of length one, proving their equivalence with cat1-2-groups. Our findings were extended to define catn-2-groups, generalizing the theory to higher dimensions. These results offer algebraic tractability and deepen the structural understanding within higher-dimensional categorical algebra.