SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2026 / Volume: 49 Issue: 3

Covering morphisms of topological internal groupoids

Osman Mucuk(ERCIYES UNIVERSITY)0000-0001-7411-2871
Hürmet Fulya Akız(Yozgat Bozok University)0000-0002-8547-2175
Pages
—
Publication Date
2020-06-02
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

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

We also prove that for a topological internal groupoid $G$, the category $\text{Cov}_{Gpd(TC)}/G$ of covering morphisms of topological internal groupoids based on $G$ and the category $\text{ACT}_{Gpd(TC)}/G$ of topological internal groupoid actions of $G$ on topological groups with operations are equivalent.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Let $X$ be a topological group with operations whose underlying space has a universal cover.
📊
Key Findings & Data:Then the fundamental groupoid $\pi X$ becomes a topological internal groupoid, i.e., an internal groupoid in the category of topological groups.
💡
Conclusion & Impact:We also prove that for a topological internal groupoid $G$, the category $\text{Cov}_{Gpd(TC)}/G$ of covering morphisms of topological internal groupoids based on $G$ and the category $\text{ACT}_{Gpd(TC)}/G$ of topological internal groupoid actions of $G$ on topological groups with operations are equivalent.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Covering groupoid · internal groupoids · topological internal groupoids

Abstract

Let $X$ be a topological group with operations whose underlying space has a universal cover. Then the fundamental groupoid $\pi X$ becomes a topological internal groupoid, i.e., an internal groupoid in the category of topological groups. In this paper, we prove that the slice category $\text{Cov}_{sTC}/X$ of covering morphisms $p:\tilde{X}\rightarrow X$ of topological groups with operations in which $\tilde{X}$ has also a universal cover and the category $\text{Cov}_{Gpd(TC)}/\pi X$ of covering morphisms $q:\tilde{G}\rightarrow \pi X $ of topological internal groupoids based on $\pi X$ are equivalent. We also prove that for a topological internal groupoid $G$, the category $\text{Cov}_{Gpd(TC)}/G$ of covering morphisms of topological internal groupoids based on $G$ and the category $\text{ACT}_{Gpd(TC)}/G$ of topological internal groupoid actions of $G$ on topological groups with operations are equivalent.

🔖

Cite This Article

Mucuk, O., Akız, H. F. (2020). Covering morphisms of topological internal groupoids. Hacettepe Journal of Mathematics and Statistics, Volume: 49 Issue: 3. https://doi.org/10.15672/hujms.467559

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (25)

  1. H.F. Akız, Covering Morphisms of Local Topological Group-Groupoids, Proc. Nat. Acad. Sci. India Sect. A, 88 (4), 2018.
  2. H.F. Akız, N. Alemdar, O. Mucuk and T. Şahan, Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J. 20 (2), 223–238, 2013.
  3. N. Alemdar and O. Mucuk, The liftings of R-modules to covering groupoids, Hacet. J. Math. Stat. 41 (6), 813–822, 2012.
  4. J.C. Baez and A.D. Lauda, Higher-dimensional algebra. V. 2-groups. Theory Appl. Categ. 12 423–491, 2004.
  5. R. Brown, Topology and groupoids, Booksurge PLC, 2006.
  6. R. Brown and G. Danesh-Naruie, The Fundamental Groupoid as a Topological Groupoid, Proc. Edinb. Math. Soc. 19 (2), 237–244, 1975.
  7. R. Brown and J.P.L. Hardy, Topological Groupoids I: Universal Constructions, Math. Nachr. 71, 273–286, 1976.
  8. R. Brown and O. Mucuk, Covering groups of non-connected topological groups revis- ited, Math. Proc. Camb. Phill. Soc. 115, 97–110, 1994.
  9. 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.
  10. R. Brown, G. Danesh-Naruie and J.P.L. Hardy, Topological groupoids II: Covering morphisms and G-spaces, Math. Nachr. 74, 143–156, 1976.
  11. T. Datuashvili, Categorical, homological and homotopical properties of algebraic ob- jects, Dissertation, Georgian Academy of Science, Tbilisi, 2006.
  12. J.P. Hardy, Topological Groupoids: Coverings and Universal Constractions, Univer- sity Colloge of North., 1974.
  13. P.J. Higgins, Groups with multiple operators, Proc. London Math. Soc. 3 (6), 366–416, 1956.
  14. İ. İçen, A.F. Özcan and M.H. Gürsoy, Topological group-groupoids and their coverings, Indian J. Pure Appl. Math. 36 (9), 493–502, 2005.
  15. K.C.H. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, London Math. Soc. Lecture Note Series 124, Cambridge University Press, 1987.
  16. O. Mucuk, Covering groups of non-connected topological groups and the monodromy groupoid of a topological groupoid, PhD Thesis, University of Wales, 1993.
  17. O. Mucuk, Coverings and ring-groupoids, Georgian Math. J. 5, 475–482, 1998.
  18. O. Mucuk and H.F. Akız, Monodromy Groupoid of an Internal Groupoid in Topolog- ical Groups with Operations, Filomat, 29, 2355–2366, 2015.
  19. O. Mucuk and T. Şahan, Coverings and crossed modules of topological groups with operations, Turkish J. Math. 38, 833–845, 2014.
  20. O. Mucuk, T. Şahan and N. Alemdar, Normality and quotients in crossed modules and group-groupoids, Appl. Categor. Struct. 23, 415–428, 2015.
  21. O. Mucuk, B. Kılıçarslan, T. Şahan and N. Alemdar, Group-groupoid and monodromy groupoid, Topology Appl. 158, 2034–2042, 2011.
  22. G. Orzech, Obstruction theory in algebraic categories I and II, J. Pure. Appl. Algebra 2, 287–314 and 315–340, 1972.
  23. T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373–381, 1987.
  24. R. Brown and J.P.L. Hardy, Topological Groupoids I: Universal Constructions, Math. Nachr. 71, 273–286, 1976.
  25. 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.