The theory of cat¹-2-groups among higher categorical models

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 ${\text{cat}}^n$-2-groups, generalizing the theory to higher dimensions. These results offer algebraic tractability and deepen the structural understanding within higher-dimensional categorical algebra.