Journals / Gazi University Journal of Science / 2020 / Volume: 33 Issue: 4

Actions of Double Group-groupoids and Covering Morphism

Serap Demir(Erciyes Üniversitesi)0000-0001-6493-2710
Osman Mucuk(Erciyes Üniveristesi)0000-0001-7411-2871
Pages
—
Publication Date
2020-12-01
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AI Summary & Key Takeaways

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

We also give a categorical equivalence between the actions and covering morphisms of double group-groupoids.

📌 3 Key Takeaways

🎯
Objective & Hypothesis: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.
📊
Key Findings & Data: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.
💡
Conclusion & Impact:We also give a categorical equivalence between the actions and covering morphisms of double group-groupoids.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Group-groupoid · Double groupoid · Action

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.

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Cite This Article

Demir, S., Mucuk, O. (2020). Actions of Double Group-groupoids and Covering Morphism. Gazi University Journal of Science, Volume: 33 Issue: 4. https://doi.org/10.35378/gujs.604849

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References (25)

  1. Brown, R., Mucuk, O., “Covering groups of nonconnected topological groups revisited”,Math. Proc. Cambridge Philos. Soc. 115(1): 97-110, (1994).
  2. Brown, R., Danesh-Naruie, G. and Hardy, J. P. L., “Topological groupoids. II. Covering morphisms and G-spaces” Math. Nachr. 74: 143-156, (1976).
  3. Ehresmann, C., “Cat´egories doubles et cat´egories structur´ees” (French) C.R. Acad. Sci. Paris 256: 1198-1201, (1963).
  4. Ehresmann, C., “Cat´egories structur´ees” (French) Ann. Sci. École Norm. Sup. 80 (3): 349-426, (1963).
  5. 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).
  6. 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).
  7. Baez, J. C. and Lauda, A. D., “Higher-dimensional algebra V. 2-groups.” Theory Appl. Categ. 12: 423-491, (2004).
  8. 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).
  9. Brown, R. and Spencer, C. B., “Double groupoids and crossed modules” Cahiers Topologie G´eom. Diff´erentielle, 17(4): 343-362, (1976).
  10. 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).
  11. Brown, R., Topology and groupoids, BookSurge, LLC, Charleston, SC, xxvi+512 pp. ISBN:1-4196-2722-8, (2006).
  12. 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).
  13. Mucuk, O. and Demir, S., “Normality and quotient in crossed modules over groupoids and double groupoids” Turkish J. Math. 42 (5): 2336-2347, (2018).
  14. Temel S., Şahan T. and Mucuk O., “Crossed modules, double group-groupoids and crossed squares”, arXiv:1802.03978, submitted for publication, (2018).
  15. Mucuk, O. and Şahan, T., “Covering groupoids of categorical groups” Hacet. J. Math. Stat. 42(4): 419-430, (2013).
  16. Mucuk, O. and Demir, S., “Covering groupoids of categorical rings”, Filomat, 29 (1): 39-49, (2015).
  17. Higgins, P. J., Categories and groupoids, Repr. Theory Appl. Categ. No. 7, 1-178, (2005).
  18. 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).
  19. Whitehead, J. H. C., “Note on a previous paper entitled On adding relations to homotopy groups” Ann. of Math. 47(2): 806-810, (1946).
  20. Whitehead, J. H. C., “Combinatorial homotopy II” Bull. Amer. Math. Soc. 55: 453-496, (1949).
  21. 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).
  22. Baez, J. C. and Lauda, A. D., “Higher-dimensional algebra V. 2-groups.” Theory Appl. Categ. 12: 423-491, (2004).
  23. Brown, R. and Spencer, C. B., “Double groupoids and crossed modules” Cahiers Topologie G´eom. Diff´erentielle, 17(4): 343-362, (1976).
  24. 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).
  25. Mucuk, O. and Demir, S., “Normality and quotient in crossed modules over groupoids and double groupoids” Turkish J. Math. 42 (5): 2336-2347, (2018).