4-Dimensional 2-Crossed Modules
Abstract
In this work, we defined a new category called 4-Dimensional 2-crossed modules. We identified the subobjects and ideals in this category. The notion of the subobject is a generalization of ideas like subsets from set theory, subspaces from topology, and subgroups from group theory. We then exemplified subobjects and ideals in the category of 4-Dimensional 2-crossed modules. A quotient object is the dual concept of a subobject. Concepts like quotient sets, spaces, groups, graphs, etc. are generalized with the notion of a quotient object. Using the ideal, we obtain the quotient of two subobjects and prove that the intersection of finite ideals is also an ideal in this category.
Keywords
References
- D. Conduché, Modules Croisés Généralisés de Longueur 2, Journal of Pure and Applied Algebra 34 (2-3) (1984) 155–178.
- A. R. Grandjeán, M. J. Vale, 2-Modulos Cruzados en la Cohomologia de André-Quillen, Memorias de la Real Academia de Ciencias 22 (1986) 1–28.
- G. Ellis, Crossed Squares and Combinatorial Homotopy, Mathematische Zeitschrift 214 (1) (1993) 93–110.
- H. J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes, Berlin, De Gruyter, 2011.
- R. Brown, N. D. Gilbert, Algebraic Models of 3-Types and Automorphism Structures for Crossed Modules, In Proceedings of the London Mathematical Society 59 (1) (1989) 51–73.
- K. H. Kamps, T. Porter, 2-Groupoid Enrichments in Homotopy Theory and Algebra, K-theory 25 (4) (2002) 373–409.
- J. F. Martins, The Fundamental 2-Crossed Complex of a Reduced CW-Complex, Homology, Homotopy and Applications 13 (2) (2011) 129–157.
- H. J. Baues, B. Bleile, Presentation of Homotopy Types Under a Space, ArXiv Preprint, ArXiv:1005.4810, 2010.
- U. E. Arslan, S. Kaplan, On Quasi 2-Crossed Modules for Lie Algebras and Functorial Relations, Ikonion Journal of Mathematics 4 (1) (2022) 17–26.
- R. Brown, I. I，cen, Homotopies and Automorphisms of Crossed Modules of Groupoids, Applied Categorical Structures 11 (2) (2003) 185–206.