Research Article
Year 2023,
Volume: 5 Issue: 1, 1 - 19, 22.08.2023
Abstract
This paper introduces a categorification of k-algebras called 2-algebras, where k is a commutative ring. We define the 2-algebras as a 2-category with single object in which collections of all 1-morphisms and all 2-morphisms are k-algebras. It is shown that the category of 2-algebras is equivalent to the category of crossed modules in commutative k-algebras.
Keywords
References
- Reference1 Arvasi Z, Ege U. Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures, 2003; 11: 487-506.
- Reference2 Baez J.C, Crans A.S. Higher Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories, 2004; 12: (15), 492-538.
- Reference3 Brown R. Spencer C. G-groupoids, Crossed modules and the Fundamental Groupoid of a Topological Group, Proc. Kon. Ned. Akad.v.Wet, 1976; 79: 296-302.
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- Reference5 Ehresmann C. Categories structures, Ann. Ec. Normale Sup. 1963; 80.
- Reference6 Lue A.S.T. Semi-Complete Crossed Modules and Holomorphs of Groups, Bull. London Math. Soc., 1979; 11; 8-16.
- Reference7 Mac Lane S. Extension and Obstructures for Rings, Illinois Journal of Math. 1958; 121; 316-345.
- Reference8 Norrie K.J. Actions and Automorphisms of Crossed Modules, Bull. Soc. Math. France. 1990; 118; 129-146.
- Reference9 Porter T. Some Categorical Results in the Theory of Crossed Modules in Commutative Algebras, Journal of Algebra, 1987; 109; 415-429.
- Reference10 Porter T. The Crossed Menagerie: An Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology, http://ncatlab.org/timporter/files/menagerie10.pdf.
- Reference11 Whitehead JHC. Combinatorial Homotopy I and II, Bull. Amer. Math. Soc., 1949: 55; 231-245 and 453-456.
Year 2023,
Volume: 5 Issue: 1, 1 - 19, 22.08.2023
Abstract
References
- Reference1 Arvasi Z, Ege U. Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures, 2003; 11: 487-506.
- Reference2 Baez J.C, Crans A.S. Higher Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories, 2004; 12: (15), 492-538.
- Reference3 Brown R. Spencer C. G-groupoids, Crossed modules and the Fundamental Groupoid of a Topological Group, Proc. Kon. Ned. Akad.v.Wet, 1976; 79: 296-302.
- Refernce4 Borceux F. Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge, Cambridge U. Press, 1994.
- Reference5 Ehresmann C. Categories structures, Ann. Ec. Normale Sup. 1963; 80.
- Reference6 Lue A.S.T. Semi-Complete Crossed Modules and Holomorphs of Groups, Bull. London Math. Soc., 1979; 11; 8-16.
- Reference7 Mac Lane S. Extension and Obstructures for Rings, Illinois Journal of Math. 1958; 121; 316-345.
- Reference8 Norrie K.J. Actions and Automorphisms of Crossed Modules, Bull. Soc. Math. France. 1990; 118; 129-146.
- Reference9 Porter T. Some Categorical Results in the Theory of Crossed Modules in Commutative Algebras, Journal of Algebra, 1987; 109; 415-429.
- Reference10 Porter T. The Crossed Menagerie: An Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology, http://ncatlab.org/timporter/files/menagerie10.pdf.
- Reference11 Whitehead JHC. Combinatorial Homotopy I and II, Bull. Amer. Math. Soc., 1949: 55; 231-245 and 453-456.
There are 11 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Acceptance Date | December 23, 2022 |
| Publication Date | August 22, 2023 |
| DOI | https://doi.org/10.54286/ikjm.1183609 |
| IZ | https://izlik.org/JA67XB25US |
| Published in Issue | Year 2023 Volume: 5 Issue: 1 |