Categorification of Algebras: 2-Algebras

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

• 2-categories
• Crossed modules
• Homotopy.

References

1. Reference1 Arvasi Z, Ege U. Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures, 2003; 11: 487-506.
2. Reference2 Baez J.C, Crans A.S. Higher Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories, 2004; 12: (15), 492-538.
3. 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.
4. Refernce4 Borceux F. Handbook of Categorical Algebra 1: Basic Category Theory, Cambridge, Cambridge U. Press, 1994.
5. Reference5 Ehresmann C. Categories structures, Ann. Ec. Normale Sup. 1963; 80.
6. Reference6 Lue A.S.T. Semi-Complete Crossed Modules and Holomorphs of Groups, Bull. London Math. Soc., 1979; 11; 8-16.
7. Reference7 Mac Lane S. Extension and Obstructures for Rings, Illinois Journal of Math. 1958; 121; 316-345.
8. Reference8 Norrie K.J. Actions and Automorphisms of Crossed Modules, Bull. Soc. Math. France. 1990; 118; 129-146.

9. Reference9 Porter T. Some Categorical Results in the Theory of Crossed Modules in Commutative Algebras, Journal of Algebra, 1987; 109; 415-429.
10. Reference10 Porter T. The Crossed Menagerie: An Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology, http://ncatlab.org/timporter/files/menagerie10.pdf.
11. Reference11 Whitehead JHC. Combinatorial Homotopy I and II, Bull. Amer. Math. Soc., 1949: 55; 231-245 and 453-456.