A Note on 4-Dimensional 2-Crossed Modules
Abstract
The study presents the direct product of two objects in the category of 4-dimensional 2-crossed modules. The structures of the domain, kernel, image, and codomain can be related using isomorphism theorems by defining the kernel and image of a morphism in a category. It then establishes the kernel and image of a morphism in the category of 4-dimensional 2-crossed modules to apply isomorphism theorems. These isomorphism theorems provide a powerful tool to understand the properties of this category. Moreover, isomorphism theorems in 4-dimensional 2-crossed modules allow us to establish connections between different algebraic structures and simplify complicated computations. Lastly, the present research inquires whether additional studies should be conducted.
Keywords
References
- J. H. C. Whitehead, Combinatorial Homotopy II, Bulletin of American Mathematical Society 55 (5) (1949) 453–496.
- T. Porter, Homology of Commutative Algebras and an Invariant of Simis and Vasconcelos, Journal of Algebra 99 (2) (1986) 458–465.
- D. Conduch´e, Modules Crois´es G´en´eralis´es de Longueur 2, Journal of Pure and Applied Algebra 34 (2-3) (1984) 155–178.
- A. R. Grandje´an, M. J. Vale, 2-Modulos Cruzados en la Cohomologia de Andr´e-Quillen, Memorias de la Real Academia de Ciencias 22 (1986) 1–28.
- H. J. Baues, Combinatorial Homotopy and 4-dimensional Complexes, Walter de Gruyter, Berlin, 1991.
- Z. Arvasi and E. Ulualan, On Algebraic Models for Homotopy 3-types, Journal of Homotopy and Related Structures 1 (1) (2006) 1–27.
- Z. Arvasi, Crossed Squares and 2-Crossed Modules of Commutative Algebras, Theory and Applications of Categories 3 (1997) 160–181.
- G. J. Ellis, Higher Dimensional Crossed Modules of Algebras, Journal of Pure and Applied Algebra 52 (3) (1988) 277–282.
- E. S. Yılmaz, (Co)Limit Calculations in the Category of 2-crossed R-modules, Turkish Journal of Mathematics 46 (7) (2022) 2902–2915.
- A. Mutlu, T. Porter, Freeness Conditions for 2-crossed Modules and Complexes, Theory and Applications of Categories 4 (8) (1998) 174–194.