Group-2-groupoids and 2G-crossed modules

Year 2019, Volume: 48 Issue: 5, 1388 - 1397, 08.10.2019

Nazmiye Alemdar , Sedat Temel

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.

Keywords

group-2-groupoid, 2G-crossed module

References

• [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, https://doi.org/10.1007/BFb0026978.
• [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.