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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

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Group-groupoid actions and liftings of crossed modules

Osman MucukORCID iD: http://orcid.org/0000-0001-7411-2871 / Tunçar ŞahanORCID iD: http://orcid.org/0000-0002-6552-4695
Published Online: 2018-02-07 | DOI: https://doi.org/10.1515/gmj-2018-0001


The aim of this paper is to define the notion of lifting via a group morphism for a crossed module and give some properties of this type of liftings. Further, we obtain a criterion for a crossed module to have a lifting crossed module. We also prove that the category of the lifting crossed modules of a certain crossed module is equivalent to the category of group-groupoid actions on groups, where the group-groupoid corresponds to the crossed module.

Keywords: Group-groupoid; action groupoid; lifting crossed module; covering groupoid

MSC 2010: 18D35; 20L05; 22A05; 57M10


  • [1]

    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 (2013), no. 2, 223–238. Web of ScienceGoogle Scholar

  • [2]

    J. C. Baez and A. D. Lauda, Higher-dimensional algebra. V. 2-groups, Theory Appl. Categ. 12 (2004), 423–491. Google Scholar

  • [3]

    R. Brown, Topology and Groupoids, BookSurge, Charleston, 2006. Google Scholar

  • [4]

    R. Brown, G. Danesh-Naruie and J. P. L. Hardy, Topological groupoids. II. Covering morphisms and G-spaces, Math. Nachr. 74 (1976), 143–156. Google Scholar

  • [5]

    R. Brown, P. J. Higgins and R. Sivera, Nonabelian Algebraic Topology, EMS Tracts Math. 15, European Mathematical Society, Zürich, 2011. Google Scholar

  • [6]

    R. Brown and O. Mucuk, Covering groups of nonconnected topological groups revisited, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 97–110. CrossrefGoogle Scholar

  • [7]

    R. Brown and C. B. Spencer, G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indag. Math. 38 (1976), no. 4, 296–302. Google Scholar

  • [8]

    T. Datuashvili, Cohomology of internal categories in categories of groups with operations, Categorical Topology and its Relation to Analysis, Algebra and Combinatorics (Prague 1988), World Scientific Publisher, Teaneck (1989), 270–283. Google Scholar

  • [9]

    T. Datuashvili, Cohomologically trivial internal categories in categories of groups with operations, Appl. Categ. Structures 3 (1995), no. 3, 221–237. CrossrefGoogle Scholar

  • [10]

    T. Datuashvili, Whitehead homotopy equivalence and internal category equivalence of crossed modules in categories of groups with operations, Proc. A. Razmadze Math. Inst. 113 (1995), 3–30. Google Scholar

  • [11]

    T. Datuashvili, Kan extensions of internal functors. Nonconnected case, J. Pure Appl. Algebra 167 (2002), no. 2–3, 195–202. CrossrefGoogle Scholar

  • [12]

    P. J. Higgins, Notes on Categories and Groupoids, Van Nostrand Rienhold Math. Stud. 32, Van Nostrand Reinhold, London, 1971. Google Scholar

  • [13]

    J.-L. Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982), no. 2, 179–202. CrossrefGoogle Scholar

  • [14]

    O. Mucuk and H. F. Akız, Monodromy groupoid of an internal groupoid in topological groups with operations, Filomat 29 (2015), no. 10, 2355–2366. Web of ScienceCrossrefGoogle Scholar

  • [15]

    O. Mucuk, B. Kılıçarslan, T. Şahan and N. Alemdar, Group-groupoids and monodromy groupoids, Topology Appl. 158 (2011), no. 15, 2034–2042. CrossrefWeb of ScienceGoogle Scholar

  • [16]

    O. Mucuk and T. Şahan, Coverings and crossed modules of topological groups with operations, Turkish J. Math. 38 (2014), no. 5, 833–845. CrossrefWeb of ScienceGoogle Scholar

  • [17]

    O. Mucuk, T. Şahan and N. Alemdar, Normality and quotients in crossed modules and group-groupoids, Appl. Categ. Structures 23 (2015), no. 3, 415–428. CrossrefWeb of ScienceGoogle Scholar

  • [18]

    G. Orzech, Obstruction theory in algebraic categories. I, J. Pure Appl. Algebra 2 (1972), 287–314. CrossrefGoogle Scholar

  • [19]

    G. Orzech, Obstruction theory in algebraic categories. II, J. Pure Appl. Algebra 2 (1972), 315–340. CrossrefGoogle Scholar

  • [20]

    T. Porter, Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinburgh Math. Soc. (2) 30 (1987), no. 3, 373–381. CrossrefGoogle Scholar

  • [21]

    J. H. C. Whitehead, Note on a previous paper entitled “On adding relations to homotopy groups”, Ann. of Math. (2) 47 (1946), 806–810. CrossrefGoogle Scholar

  • [22]

    J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949), 453–496. CrossrefGoogle Scholar

About the article

Received: 2016-01-10

Revised: 2016-08-02

Accepted: 2016-09-19

Published Online: 2018-02-07

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2018-0001.

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