Whiskered Groupoids and Crossed Modules with Diagrams

Year 2024, Issue: 46 , 23 - 39 , 29.03.2024

Zeynep Güler , Erdal Ulualan

https://doi.org/10.53570/jnt.1403803

https://izlik.org/JA42WZ65SA

Abstract

In this study, we investigate the relationships between the category of crossed modules of groups and the category of whiskered groupoids. Our first aim is to construct a crossed module structure over groups from a whiskered groupoid with the objects set - a group (regular groupoid) - using the usual functor between the categories of crossed modules and cat groups. Conversely, the second aim is to construct a whiskered groupoid structure with the objects set, which is a group, from a crossed module of groups. While establishing this relationship, we frequently used arrow diagrams representing morphisms to make the axioms more comprehensible. We provide the conditions for the bimorphisms in a whiskered groupoid and give the relations between this structure and internal groupoids in the category of whiskered groupoids with the objects set as a group.

Keywords

Groupoid , crossed module , whiskered categories , bimorphism

References

• R. Brown, P. J. Higgins, On the connection between the second relative homotopy groups of some related spaces, Proceedings of the London Mathematical Society s3-36 (2) (1978) 193-212.
• H. J. Baues, Combinatorial homotopy and 4-dimenional complexes, Walter de Gruyter, Berlin, 1991.
• H. J. Baues, A. Tonks, On the twisted cobar construction, Mathematical Proceedings of the Cambridge Philosophical Society 121 (2) (1997) 229-245.
• R. Brown, Possible connections between Whiskered categories and groupoids, Leibniz algebras, automorphism structures and local-to-global questions, Journal of Homotopy and Related Structures 5 (1) (2010) 305-318.
• J. H. C. Whitehead, Combinatorial homotopy II, Bulletin of the American Mathematical Society 55 (5) (1949) 453-496.
• R. Brown, C. B. Spencer, $\mathcal{G}$-groupoids, crossed modules and the fundamental groupoid of a topological group, Indagationes Mathematicae (Proceedings) 79 (4) (1976) 296-302.
• R. Brown, O. Mucuk, Covering groups of non-connected topological groups revisited, Mathematical Proceedings of the Cambridge Philosophical Society 115 (1) (1994) 97-110.
• R. Brown, N. D. Gilbert, Algebraic models of 3-types and automorphism structures for crossed modules, Proceedings of the London Mathematical Society 59 (3) (1989) 51-73.
• Z. Arvasi, E. Ulualan, 3-types of simplicial groups and braided regular crossed modules, Homology, Homotopy and Applications 9 (1) (2007) 139-161.
• A. Fernández-Fariña, M. Ladra, Universal central extensions of braided crossed modules of Lie algebras, Hacettepe Journal of Mathematics and Statistics 51 (4) (2022) 1013-1028.
• A. Fernández-Fariña, M. Ladra, Braiding for categorical algebras and crossed modules of algebras I: Associative and Lie algebras, Journal of Algebra and Its Applications 19 (09) (2020) 2050176 24 pages.
• Ö. Gürmen Alansal, Ideal and factor conditions for crossed modules and algebra-algebroids, Hacettepe Journal of Mathematics and Statistics 52 (3) (2023) 698-707.
• R. Brown, From groups to groupoids, Bulletin of the American Mathematical Society 19 (2) (1987) 113-134.
• R. Brown, Topology and groupoids, 3rd Edition, BookSurge Publishing, Charleston, 2006.
• J. L. Loday, Spaces with finitely many non-trivial homotopy groups, Journal of Pure and Applied Algebra 24 (2) (1982) 179-202.
• A. Mutlu, T. Porter, Applications of Peiffer pairings in the Moore complex of a simplicial group, Theory and Applications of Categories 4 (7) (1998) 148-173.
• Ö. Gürmen Alansal, E. Ulualan, Peiffer pairings in multi simplicial groups and crossed $n$-cubes and applications for bisimplicial groups, Turkish Journal of Mathematics 45 (1) (2021) 360-386.
• Ö. Gürmen Alansal, E. Ulualan, Bisimplicial commutative algebras and crossed squares, Fundamental Journal of Mathematics and Applications 6 (3) (2023) 177-187.
• N. D. Gilbert, Monoid presentations and associated groupoids, International Journal of Algebra and Computation 8 (2) (1998) 141-152.
• O. Avcıoğlu, İ. Akça, Free modules and crossed modules of $R$-algebroids, Turkish Journal of Mathematics 42 (6) (2018) 2863-2875.
• O. Mucuk, T. Şahan, N. Alemdar, Normality and quotients in crossed modules and group-groupoids, Applied Categorical Structures 23 (3) (2015) 415-428.
• T. Porter, Some categorical results in the theory of crossed modules in commutative algebras, Journal of Algebra 109 (2) (1987) 415-429.
• E. Soylu Yılmaz, U. Ege Arslan, Internal cat-1 and xmod, Journal of New Theory (38) (2022) 79-87.
• E. Soylu Yılmaz, K. Yılmaz, On relations among quadratic modules, Mathematical Methods in the Applied Sciences 45 (18) (2022) 12231-12244.