Actions of Double Group-groupoids and Covering Morphism

Year 2020, , 810 - 819, 01.12.2020

Serap Demir , Osman Mucuk

https://doi.org/10.35378/gujs.604849

Abstract

The purpose of the paper is to consider the covering morphism and the action of a double groupoid on a groupoid; and then characterize these notions for double group- groupoids. We also give a categorical equivalence between the actions and covering morphisms of double group-groupoids.                                                                                                                                                                                                                         

Keywords

Group-groupoid, Double groupoid, Action, Covering morphism

References

• Brown, R., Mucuk, O., “Covering groups of nonconnected topological groups revisited”,Math. Proc. Cambridge Philos. Soc. 115(1): 97-110, (1994).
• Brown, R., Danesh-Naruie, G. and Hardy, J. P. L., “Topological groupoids. II. Covering morphisms and G-spaces” Math. Nachr. 74: 143-156, (1976).
• Ehresmann, C., “Cat´egories doubles et cat´egories structur´ees” (French) C.R. Acad. Sci. Paris 256: 1198-1201, (1963).
• Ehresmann, C., “Cat´egories structur´ees” (French) Ann. Sci. École Norm. Sup. 80 (3): 349-426, (1963).
• Brown, R., Mackenzie, K. C. H., “ Determination of a double Lie groupoid by its core diagram” J. Pure Appl. Algebra 80 ( 3): 237-272, (1992).
• Akız, H. F., Alemdar, N., Mucuk, O. and S¸ahan, T., “ Coverings of internal groupoids and crossed modules in the category of groups with operations” Georgian Math. J. 20 ( 2) : 223-238, (2013).
• Baez, J. C. and Lauda, A. D., “Higher-dimensional algebra V. 2-groups.” Theory Appl. Categ. 12 : 423-491, (2004).
• Brown, R., Higgins, P. J. and Sivera, R., Nonabelian algebraic topology. Filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics, 15. European Mathematical Society (EMS), Zu¨rich xxxvi+668 pp. ISBN: 978-3-03719-083-8, (2011).
• Brown, R. and Spencer, C. B., “Double groupoids and crossed modules” Cahiers Topologie G´eom. Diff´erentielle, 17(4) : 343-362, (1976).
• Brown, R. and Spencer, C. B., “G -groupoids, crossed modules and the fundamental groupoid of a topological group” , Nederl. Akad. Wetensch. Proc. Ser. A 38(4): 296-302, (1976).
• Brown, R., Topology and groupoids, BookSurge, LLC, Charleston, SC, xxvi+512 pp. ISBN:1-4196-2722-8, (2006).
• Mackenzie, K., Lie groupoids and Lie algebroids in differential geometry, London Math- ematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, xvi+327 pp. ISBN: 0-521-34882-X, (1987).
• Mucuk, O. and Demir, S., “Normality and quotient in crossed modules over groupoids and double groupoids” Turkish J. Math. 42 (5) : 2336-2347, (2018).
• Temel S., Şahan T. and Mucuk O., “Crossed modules, double group-groupoids and crossed squares”, arXiv:1802.03978, submitted for publication, (2018).
• Mucuk, O. and Şahan, T., “Covering groupoids of categorical groups” Hacet. J. Math. Stat. 42(4): 419-430, (2013).
• Mucuk, O. and Demir, S., “Covering groupoids of categorical rings”, Filomat, 29 (1): 39-49, (2015).
• Higgins, P. J., Categories and groupoids, Repr. Theory Appl. Categ. No. 7, 1-178, (2005).
• Şahan, T. and Erciyes, A., “Actions of internal groupoids in the category of Leibniz algebras”, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68 (1): 619–632, (2019).
• Whitehead, J. H. C., “Note on a previous paper entitled On adding relations to homotopy groups” Ann. of Math. 47(2): 806-810, (1946).
• Whitehead, J. H. C., “Combinatorial homotopy II” Bull. Amer. Math. Soc. 55: 453-496, (1949).