Korean J. Math. Vol. 27 No. 1 (2019) pp.151-163
DOI: https://doi.org/10.11568/kjm.2019.27.1.151

Normality and quotient in crossed modules over groupoids and 2-Groupoids

Main Article Content

Sedat Temel


The aim of this paper is to consider the categorical equivalence between crossed modules within groupoids and 2-groupoids; and then relate normality and quotient in these two categories.

Article Details


[1] J.C. Baez and A.D. Lauda, Higher Dimensional Algebra V: 2-Groups, Theory and Applications of Categories 12 (14) (2004), 423–491. Google Scholar

[2] J.C. Baez, A. Baratin, L. Freidel, and D.K. Wise, Infinite-Dimensional Representations of 2-Groups, Memoirs of the American Mathematical Society, 219 (1032) (2012). Google Scholar

[3] J. Be nabou, Introduction to bicategories, Reports of the Midwest Category Seminar Lecture Notes in Mathematics 47 (1967), 1-77. Google Scholar

[4] R. Brown and C.B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proc. Konn. Ned. Akad. 1976; 79: 296–302. Google Scholar

[5] R. Brown, Higher dimensional group theory, in Low Dimensional Topology, London Math Soc. Lect. Notes, Cambridge Univ. Press; 48 (1982), 215–238. Google Scholar

[6] R. Brown, P.J. Higgins, and R. Sivera, Nonabelian Algebraic Topology: Filtered spaces, crossed complexes, cubical homotopy groupoids, European Mathematical Society Tracts in Mathematics 15 (2011). Google Scholar

[7] R. Brown, Topology and Groupoids. BookSurge LLC North Carolina, (2006). Google Scholar

[8] R. Brown and P.J. Higgins, Crossed Complexes and non-Abelian Extensions, Georgian Mathematical Journal 962 (6) (1981), 39–50. Google Scholar

[9] R. Brown and P.J. Higgins, Tensor Products and Homotopies for ω−groupoids and crossed complexes, J. Pure and Appl. Algebra 47 (1987), 1–33. Google Scholar

[10] R. Brown, I. Icen, Homotopies and Automorphisms of Crossed Module Over Groupoids, Appl. Categorical Structure 11 (2003), 185–206. Google Scholar

[11] P.J. Higgins, Categories and Groupoids, Reprints in Theory and Applications of Categories 7 (2005), 1–195. Google Scholar

[12] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helvetici; 55: 302–314 (1980). Google Scholar

[13] I. Icen, The Equivalence of 2-Groupoids and Crossed Modules, Commun. Fac. Sci. Univ. Ankara Series A1 49 (2000), 39–48. Google Scholar

[14] G. Janelidze, L. Marki, and W. Tholen, Semi-abelian categories, Journal of Pure and Appliying Algebra 168 (2002), 367–386. Google Scholar

[15] G.M. Kelly and R. Street, Review of the Elements of 2-Categories, (eds) Category Seminar. Lecture Notes in Mathematics, vol 420. Springer, Berlin, Heidelberg Google Scholar

[16] J.-L. Loday, Cohomologie et groupes de Steinberg relatifs, J. Algebra, t. 54 (1978), 178–202. Google Scholar

[17] K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry. London Mathematical Society Lecture Note Series. 124, Cambridge Uni. Press New York, (1987). Google Scholar

[18] S. Maclane, Categories for the Working Mathematician, Graduate Text in Mathematics, Volume 5. Springer-Verlag, New York (1971). Google Scholar

[19] O. Mucuk, T. Sahan, and N. Alemdar, Normality and Quoutients in Crossed Modules and Group-Groupoids, Applied Categorical Structures 23 (3) (2015), 415–428. Google Scholar

[20] O. Mucuk and S. Demir, Normality and quotient in crossed modules over groupoids and double groupoids, Turk J Math 42 (2018), 2336– 2347. Google Scholar

[21] B. Noohi, Notes on 2-Groupoids, 2-Groups and Crossed Modules, Homology Homotopy Appl. 9 (1) (2007), 75–106. Google Scholar

[22] K. Norrie, Actions and Automorphisms of Crossed Modules, Bull. Soc. Math. France 118 (1990), 129–146. Google Scholar

[23] T. Porter, Extensions, Crossed Modules and Internal Categories in Categories of Groups With Operations, Proceedings of the Edinburgh Mathematical Society 30 (1987), 371–381. Google Scholar

[24] T. Porter, Crossed Modules in Cat and a Brown-Spencer Theorem for 2-Categories, Cahiers de Topologie et Geometrie Differentielle Categoriques, Vol. XXVI-4 (1985). Google Scholar

[25] J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55 (1949), 453–496. Google Scholar

[26] J.H.C. Whitehead, Note on a previous paper entitled " On adding relations to homotopy group" , Ann. Math. 47 (1946), 806–810. Google Scholar