TY  - JOUR
AU  - Temel, Sedat
PY  - 2026
TI  - The theory of cat1-2-groups among higher categorical models
JO  - AIMS Mathematics
SP  - 6141
EP  - 6161
VL  - 11
IS  - 3
AB  - 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        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.
UR  - https://doi.org/10.3934/math.2026254
DO  - 10.3934/math.2026254