Topological Group-Groupoids and Equivalent Categories

Year 2022, , 667 - 673, 25.12.2022

Abdullah Fatih Özcan , İlhan İçen

https://doi.org/10.53433/yyufbed.1137668

Abstract

The concept of groupoid was offered by Brandt (1926). The structure of the topological groupoid was given by Ehresmann (1958). A groupoid action is a significant appliance in algebraic topology offered by Ehresmann. Another algebraic notion is a covering given by Brown (1988). The topological group-groupoids (Γ-groupoid) were first put forward by Icen & Ozcan (2001). The definition of coverings of topological Γ groupoid and actions of topological Γ-groupoid were also presented by Icen et al. (2005). In this paper, we are going to create a category TΓGpdCov(Γ) of covering morphisms of TΓ-groupoid and a category TΓGpdOp(Γ) of actions of TΓ-groupoid. We will then prove that these categories are equivalent.

Keywords

Action, Covering, Groupoid

Thanks

This paper was presented in 4th International Conference on Pure and Applied Mathematics (ICPAM - VAN 2022), Van-Turkey, June 22-23, 2022.

Topolojik Grup-Grupoidler ve Denk Kategoriler

Abstract

Grupoid kavramı Brandt (1926) tarafından tanımlandı. Topolojik grupoidin tanımı Ehresmann (1958) tarafından verilmiştir. Grupoid etki, cebirsel topolojide Ehresmann tarafından sunulan çok önemli bir araçtır. Diğer bir cebirsel kavram ise Brown (1988) tarafından verilen bir örtü kavramıdır. Topolojik grup-grupoidler (Γ-grupoid) ilk olarak Icen & Ozcan (2001) tarafından tanımlanmıştır. Topolojik Γ-grupoidinin örtülerinin ve topolojik Γ-grupoidin etkisinin tanımı Icen ve ark. (2005) tarafından verilmiştir. Bu çalışmada, topolojik Γ-groupoid örtülerinin bir TΓGpdCov(Γ) kategorisi ve topolojik Γ-groupoid etkilerinin bir TΓGpdOp(Γ) kategorisi tanımlandı. Daha sonra bu kategorilerin denk olduğunu ispatlandı.

Keywords

Etki, Grupoidler, Örtü

References

• Brandt, H. (1926). Uber ein verallgemeinerung des Gruppen begriffes. Mathematische Annalen, 96, 360-366. doi: 10.1007/BF01209171
• Brown, R., & Danesh-Naruie, G. (1975). The fundamental groupoid as a topological groupoid. Proceedings of the Edinburgh Mathematical Society, 19-2(3), 237-244. doi: 10.1017/S0013091500015509
• Brown, R., Danesh-Naruie, G., & Hardy, J. P. L. (1976). Topological groupoids II: Covering morphism and G-space. Mathematische Nachrichten, 74, 143-145. doi: 10.1002/mana.3210740110
• Brown, R., & Spencer, C. B. (1976). G-groupoids, crossed modules and the fundamental groupoid of a topological group. Proceedings of the Koninklijke Nedderlandse Akademie van Wetenschappen, 79, 196-302. doi: 10.1016/1385-7258(76)90068-8
• Brown, R. (1988). Topology: A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid. Chichester, England: Ellis Horwood.
• Ehresmann, C. (1958). Categories topologiques et categories differentiables. Colloque de Geometrie Differentielle Globale, Cenrtre Belge de Recherches Mathematiques, Bruxelles, 137-150.
• Icen, I., & Ozcan, A. F. (2001). Topological crossed modules and G groupoids. Algebras, Groups and Geometries, 18, 401-410.
• Icen, I., Ozcan, A. F., & Gursoy, M. H. (2005). Topological group-groupoids and their coverings. Indian Journal of Pure and Applied Mathematics, 36(9), 493-502.