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AIMS Mathematics

2026, Volume 11, Issue 3: 6141-6161. doi: 10.3934/math.2026254
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Research article

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

  • Department of Mathematics, Recep Tayyip Erdogan University, Rize 53100, TÜRKİYE
  • Published: 11 March 2026

  • MSC : 18N25, 18G50, 18G55, 18G30, 20J15

  • 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.

    Citation: Sedat Temel. The theory of cat1-2-groups among higher categorical models[J]. AIMS Mathematics, 2026, 11(3): 6141-6161. doi: 10.3934/math.2026254

    Related Papers:

  • 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.



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