SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2019 / Cilt: 48 - Sayı: 5

Group-2-groupoids and 2G-crossed modules

Sedat Temel
Nazmiye Alemdar
Pages
1388–1397
Publication Date
2022-10-18
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

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

Then we prove that the category Gp2Gd of group-2-groupoids and the category 2GXMod of 2G-crossed modules are equivalent.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:In this paper, we introduce the notion of a group-2-groupoid as a group object in the category of 2-groupoids.
📊
Key Findings & Data:We also obtain a 2G-crossed module by using the structure of a group-2-groupoid.
💡
Conclusion & Impact:Then we prove that the category Gp2Gd of group-2-groupoids and the category 2GXMod of 2G-crossed modules are equivalent.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:group-2-groupoid · 2G-crossed module

Abstract

In this paper, we introduce the notion of a group-2-groupoid as a group object in the category of 2-groupoids. We also obtain a 2G-crossed module by using the structure of a group-2-groupoid. Then we prove that the category Gp2Gd of group-2-groupoids and the category 2GXMod of 2G-crossed modules are equivalent.

🔖

Cite This Article

Temel, S., Alemdar, N. (2022). Group-2-groupoids and 2G-crossed modules. Hacettepe Journal of Mathematics and Statistics, Cilt: 48 - Sayı: 5, 1388–1397. https://ajindex.com/articles/530135

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (17)

  1. N. Alemdar and O. Mucuk, The Liftings of R-Modules to Covering Groupoid, Hacet. J. Math. Stat. 41 (6), 813-822, 2012
  2. J.C. Baez, An Introduction to n-Categories, in: Moggi E., Rosolini G. (eds) Category Theory and Computer Science, Lecture Notes in Computer Science 1290, Springer, Berlin, Heidelberg, 1997,
  3. J.C. Baez, A. Baratin, L. Freidel and D.K. Wise, Infinite-Dimensional Representa- tions of 2-Groups, Mem. Amer. Math. Soc. 219, Number 1032, 2012
  4. J.C. Baez and A.D. Lauda, Higher-dimensional Algebra. V. 2-groups, Theory Appl. Categ. 12, 423-491, 2004
  5. J.C. Baez and D. Stevenson The Classifying Space of a Topological 2-Group, in: Al- gebraic Topology. Abel Symposia 4, Springer, Berlin, Heidelberg, 2009
  6. R. Brown, From Groups to Groupoids: A Brief Survey, Bull. London Math. Soc. 19, 113-134, 1987
  7. R. Brown and C.B. Spencer, G-groupoids, Crossed Modules and The Fundamental Groupoid of a Topological Group, Proc. Konn. Ned. Akad. v. Wet. 79, 296-302 1976
  8. M. Forrester-Barker, Group objects and internal categories, Preprint arXiv: math/0212065v1, 2002
  9. P.J. Higgins, Categories and Groupoids, Van Nostrand, New York, 1971
  10. S. Maclane, Categories for the Working Mathematician, Graduate Text in Mathemat- ics, Vol. 5, Springer-Verlag, New York, 1971
  11. O. Mucuk, Coverings and ring-groupoids, Geor. Math. J. 5, 475-482, 1998
  12. O. Mucuk, B. Bagriyanik, N. Alemdar and T. Şahan, Group-Groupoids And Mon- odromy Groupoids, Topology Appl. 158 (15), 2034-2042, 2011
  13. O. Mucuk, T. Şahan and N. Alemdar, Normality and Quotients in Crossed Modules and Group-groupoids, Appl. Categor. Struct. 23 (3), 415-428, 2015
  14. B. Noohi, Notes on 2-Groupoids, 2-Groups and Crossed Modules, Homology Homo- topy Appl. 9 (1), 75-106, 2007
  15. T. Porter, Extensions, Crossed Modules and Internal Categories in Categories of Groups With Operations, Proc. Edinb. Math. Soc. 30, 371-381, 1987
  16. J.H.C. Whitehead, Note on a previous paper entitled "On adding relations to homo- topy group", Ann. Math. 47, 806-810, 1946
  17. J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949.