Coverings and liftings of generalized crossed modules

Document Type : Research Paper


Department of Mathematics, Aksaray University, P. O. Box 68100, Aksaray, Turkey.


In the theory of crossed modules, considering arbitrary selfactions instead of conjugation allows for the extension of the concept of crossed modules and thus the notion of generalized crossed module emerges. In this paper we give a precise definition for generalized cat1-groups and obtain a functor from the category of generalized cat1-groups to generalized crossed modules. Further, we introduce the notions of coverings and liftings for generalized crossed modules and investigate properties of these structures. Main objective of this study is to obtain an equivalence between the category of coverings and the category of liftings of a given generalized crossed module (A,B, α).


[1] Akız, H.F., Alemdar, N., Mucuk, O., and Şahan, T., Coverings of internal groupoids and crossed modules in the category of groups with operations, Georgian Math. J. 20(2) (2013), 223-238.
[2] Brown, R., Higgins, P.J., and Sivera, R., “Nonabelian Algebraic Topology: Filtered Spaces, Crossed Complexes, Cubical Homotopy Groupoids”, EMS series of lectures in mathematics, European Mathematical Society, 2011.
[3] Brown, R. and Huebschmann, J., “Identities among relations”, in: Low-Dimensional Topology, Cambridge University Press, 1982.
[4] Brown, R. and Mucuk, O., Covering groups of non-connected topological groups revisited, Math. Proc. Camb. Phil. Soc. 115(1), (1994), 97-110.
[5] Brown, R. and Spencer, C.B., G-groupoids, crossed modules and the fundamental groupoid of a topological group, Indagat. Math. 79(4) (1976), 296-302.
[6] Datuashvili, T., Central series for groups with action and leibniz algebras, Georgian Math. J. 9(4) (2002), 671-682.
[7] Datuashvili, T., Witt’s theorem for groups with action and free leibniz algebras, Georgian Math. J. 11(4) (2004), 691-712.
[8] Huebschmann, J., Crossed n-folds extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), 302-313.
[9] Karakaş , S.D. and Mucuk, O., Liftings and covering morphisms of crossed modules in group-groupoids, Turk. J. Math. 45(3) (2021), 1407-1417.
[10] Loday, J.L., Cohomologie et groupe de Steinberg relatifs, J. Algebra 54(1) (1978), 178-202.
[11] Loday, J.L., Spaces with finitely many non-trivial homotopy groups, J. Pure. Appl. Algebra 24(2) (1982), 179-202.
[12] Lue, A.S.T., Semi-complete crossed modules and holomorphs of groups, B. Lond. Math. Soc. 11(1) (1979), 8-16.
[13] Mucuk, O. and Akız, H.F., Covering morphisms of topological internal groupoids, Hacet. J. Math. Stat. 49 (2020), 1020-1029.
[14] Mucuk, O. and Şahan, T., Coverings and crossed modules of topological groups with operations, Turk. J. Math. 38(5) (2014), 833-845.
[15] Mucuk, O. and Şahan, T., Group-groupoid actions and liftings of crossed modules, Georgian Math. J. 26(3), (2019), 437-447.
[16] Mucuk, O., Şahan, T. and Alemdar, N., Normality and quotients in crossed modules and group-groupoids, Appl. Categor. Struct. 23(3) (2015), 415-428.
[17] Orzech, G., Obstruction theory in algebraic categories, I, J. Pure. Appl. Algebra 2(4) (1972), 287-314.
[18] Orzech, G., Obstruction theory in algebraic categories, II, J. Pure. Appl. Algebra 2(4) (1972), 315-340.
[19] Porter, T., Extensions, crossed modules and internal categories in categories of groups with operations, P. Edinburgh Math. Soc. 30(3) (1987), 373-381.
[20] Şahan, T., Further remarks on liftings of crossed modules, Hacet. J. Math. Stat. 48(3) (2019), 743-752.
[21] Temel, S., Crossed semimodules and cat1-monoids, Korean J. Math. 27(2) (2019), 535-545.
[22] Temel, S., Crossed squares, crossed modules over groupoids and cat1-2-groupoids, Categ. Gen. Algebr. Struct. Appl. 13(1) (2020), 125-142.
[23] Temel, S., Şahan, T. and Mucuk, O., Crossed modules, double group-groupoids and crossed squares, Filomat 34(6) (2020), 1755-1769.
[24] Temel, S., Some notes on crossed semimodules, Turk. J. Math. 46 (2022), 768-784.
[25] Whitehead, J.H.C., Note on a previous paper entitled “On adding relations to homotopy groups”, Ann. Math. 47(4) (1946), 806-810.
[26] Whitehead, J.H.C., Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55(5) (1949), 453-496.
[27] Yavari, M. and Salemkar, A., The category of generalized crossed modules, Categ. Gen. Algebr. Struct. Appl. 10(1) (2019), 157-171.