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Journals / Ikonion Journal of Mathematics / 2023 / Cilt: 5 Sayı: 1

Categorification of Algebras: 2-Algebras

İbrahim AKÇA
Ummahan EGE ARSLAN
Pages
1–19
Publication Date
2023-08-22
DOI
—
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

It is shown that the category of 2-algebras is equivalent to the category of crossed modules in commutative k-algebras.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:This paper introduces a categorification of k-algebras called 2-algebras, where k is a commutative ring.
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Key Findings & Data: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.
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Conclusion & Impact:It is shown that the category of 2-algebras is equivalent to the category of crossed modules in commutative k-algebras.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:2-categories · Crossed modules · Homotopy.

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.

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Cite This Article

AKÇA, İ., ARSLAN, U. E. (2023). Categorification of Algebras: 2-Algebras. Ikonion Journal of Mathematics, Cilt: 5 Sayı: 1, 1–19. https://ajindex.com/articles/959501

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References (10)

  1. Arvasi Z, Ege U. Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures, 2003; 11: 487-506
  2. Baez J.C, Crans A.S. Higher Dimensional Algebra VI: Lie 2-Algebras, Theory and Applications of Categories, 2004; 12: (15), 492-538
  3. 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
  4. Ehresmann C. Categories structures, Ann. Ec. Normale Sup. 1963
  5. Lue A.S.T. Semi-Complete Crossed Modules and Holomorphs of Groups, Bull. London Math. Soc., 1979; 11; 8-16
  6. Mac Lane S. Extension and Obstructures for Rings, Illinois Journal of Math. 1958; 121; 316-345
  7. Norrie K.J. Actions and Automorphisms of Crossed Modules, Bull. Soc. Math. France. 1990; 118; 129-146
  8. Porter T. Some Categorical Results in the Theory of Crossed Modules in Commutative Algebras, Journal of Algebra, 1987; 109; 415-429
  9. Porter T. The Crossed Menagerie: An Introduction to Crossed Gadgetry and Cohomology in Algebra and Topology,
  10. Whitehead JHC. Combinatorial Homotopy I and II, Bull. Amer. Math. Soc., 1949: 55; 231-245 and 453-456.