Covering morphisms of topological internal groupoids

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.

Keywords:

Covering groupoid internal groupoids, topological internal groupoids,

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[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.

Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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