Year 2020, Volume 49 , Issue 3, Pages 1020 - 1029 2020-06-02

Covering morphisms of topological internal groupoids

Osman MUCUK [1] , Hürmet Fulya AKIZ [2]


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.
Covering groupoid, internal groupoids, topological internal groupoids
  • [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.
Primary Language en
Subjects Mathematics
Journal Section Mathematics
Authors

Orcid: 0000-0001-7411-2871
Author: Osman MUCUK
Institution: ERCIYES UNIVERSITY
Country: Turkey


Orcid: 0000-0002-8547-2175
Author: Hürmet Fulya AKIZ (Primary Author)
Institution: Yozgat Bozok University
Country: Turkey


Dates

Publication Date : June 2, 2020

Bibtex @research article { hujms467559, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe University}, year = {2020}, volume = {49}, pages = {1020 - 1029}, doi = {10.15672/hujms.467559}, title = {Covering morphisms of topological internal groupoids}, key = {cite}, author = {Mucuk, Osman and Akız, Hürmet Fulya} }
APA Mucuk, O , Akız, H . (2020). Covering morphisms of topological internal groupoids . Hacettepe Journal of Mathematics and Statistics , 49 (3) , 1020-1029 . DOI: 10.15672/hujms.467559
MLA Mucuk, O , Akız, H . "Covering morphisms of topological internal groupoids" . Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1020-1029 <https://dergipark.org.tr/en/pub/hujms/issue/54699/467559>
Chicago Mucuk, O , Akız, H . "Covering morphisms of topological internal groupoids". Hacettepe Journal of Mathematics and Statistics 49 (2020 ): 1020-1029
RIS TY - JOUR T1 - Covering morphisms of topological internal groupoids AU - Osman Mucuk , Hürmet Fulya Akız Y1 - 2020 PY - 2020 N1 - doi: 10.15672/hujms.467559 DO - 10.15672/hujms.467559 T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1020 EP - 1029 VL - 49 IS - 3 SN - 2651-477X-2651-477X M3 - doi: 10.15672/hujms.467559 UR - https://doi.org/10.15672/hujms.467559 Y2 - 2019 ER -
EndNote %0 Hacettepe Journal of Mathematics and Statistics Covering morphisms of topological internal groupoids %A Osman Mucuk , Hürmet Fulya Akız %T Covering morphisms of topological internal groupoids %D 2020 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 49 %N 3 %R doi: 10.15672/hujms.467559 %U 10.15672/hujms.467559
ISNAD Mucuk, Osman , Akız, Hürmet Fulya . "Covering morphisms of topological internal groupoids". Hacettepe Journal of Mathematics and Statistics 49 / 3 (June 2020): 1020-1029 . https://doi.org/10.15672/hujms.467559
AMA Mucuk O , Akız H . Covering morphisms of topological internal groupoids. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1020-1029.
Vancouver Mucuk O , Akız H . Covering morphisms of topological internal groupoids. Hacettepe Journal of Mathematics and Statistics. 2020; 49(3): 1020-1029.