Further remarks on liftings of crossed modules

Şahan, Tunçar

doi:10.15672/HJMS.2018.554

Further remarks on liftings of crossed modules

Tunçar Şahan (Aksaray Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Aksaray, Türkiye)

Hacettepe Journal of Mathematics and Statistics

2 0

Yıl: 2019 Cilt: 48 Sayı: 3 Sayfa Aralığı: 743 - 752 Metin Dili: İngilizce DOI: 10.15672/HJMS.2018.554 İndeks Tarihi: 21-10-2020

Further remarks on liftings of crossed modules

Öz:

In this paper we define the notion of pullback lifting of a lifting crossed module overa crossed module morphism and interpret this notion in the category of group-groupoidactions as pullback action. Moreover, we give a criterion for the lifting of homotopic crossedmodule morphisms to be homotopic, which will be called homotopy lifting property forcrossed module morphisms. Finally, we investigate some properties of derivations of liftingcrossed modules according to base crossed module derivations.

Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık

[1]H.F. Akız, N. Alemdar, O. Mucuk and T. Şahan,Coverings of internal groupoids andcrossed modules in the category of groups with operations, Georgian Math. J.20(2),223-238, 2013.
[2]R. Brown,Higher dimensional group theory, In: Low Dimensional Topology, LondonMath. Soc. Lect. Notes48, 215-238. Cambridge Univ. Press, 1982.
[3]R. Brown,Topology and Groupoids, BookSurge LLC, North Carolina, 2006.
[4]R. Brown, G. Danesh-Naruie and J.P.L. Hardy,Topological Groupoids: II. CoveringMorphisms and G-Spaces, Math. Nachr.74, 143-156, 1976.
[5]R. Brown and P.J. Higgins,On the connection between the second relative homotopygroups of some related space, Proc. London Math. Soc.36, 193-212, 1978.
[6]R. Brown, P.J. Higgins and R. Sivera,Nonabelian Algebraic Topology: filtered spaces,crossed complexes, cubical homotopy groupoids, EMS Tracts Math.15, 2011.
[7]R. Brown and J. Huebschmann,Identities among relations. In: Low DimentionalTopology, London Math. Soc. Lect. Notes48, 153-202. Cambridge Univ. Press, 1982.
[8]R. Brown and O. Mucuk,Covering groups of non-connected topological groups revis-ited, Math. Proc. Camb. Phil. Soc.115, 97-110, 1994.
[9]R. Brown and C.B. Spencer,G-groupoids, crossed modules and the fundamentalgroupoid of a topological group, Proc. Konn. Ned. Akad. v. Wet.79, 296-302, 1976.
[10]W.H. Cockcroft,On the homomorphisms of sequences, Proc. Cambridge. Phil. Soc.45, 521-532, 1952.
[11]P.J. Higgins,Categories and groupoids, Van Nostrand, New York, 1971.
[12]J. Huebschmann,Crossedn-fold extensions of groups and cohomology, Comment.Math. Helvetici55, 302-314, 1980.
[13]J.-L. Loday,Cohomologie et groupes de Steinberg relatifs, J. Algebra54, 178-202,1978.
[14]J.-L. Loday,Spaces with finitely many non-trivial homotopy groups, J. Pure Appl.Algebra24, 179-202, 1982.
[15]A.S.T. Lue,Cohomology of groups relative to a variety, J. Algebra69, 155-174, 1981.
[16]J.F. Martins,The fundamental 2-crossed complex of a reduced CW-complex, Homol-ogy Homotopy Appl.13(2), 129-157, 2011.
[17]O. Mucuk,Covering groups of non-connected topological groups and the monodromygroupoid of a topological groupoid, PhD Thesis, University of Wales, Bangor, 1993.
[18]O. Mucuk and T. Şahan,Covering groupoids of categorical groups, Hacet. J. Math.Stat.42(4), 419-430, 2013.
[19]O. Mucuk and T. Şahan,Coverings and crossed modules of topological groups withoperations, Turk. J. Math.38(5), 833-845, 2014.
[20]O. Mucuk and T. Şahan,Group-groupoid actions and liftings of crossed modules,Georgian Math. J., doi:10.1515/gmj-2018-0001, 2017.
[21]O. Mucuk, T. Şahan and N. Alemdar,Normality and Quotients in Crossed Modulesand Group-groupoids, Appl. Categor. Struct.23, 415-428, 2015.
[22]K. Norrie,Actions and Automorphisms of crossed modules, Bull. Soc. Math. France118, 129-146, 1990.
[23]J.H.C. Whitehead,Note on a previous paper entitled “On adding relations to homo-topy group", Ann. Math.47, 806-810, 1946.
[24]J.H.C. Whitehead,On operators in relative homotopy groups, Ann. Math.49, 610-640, 1948.
[25]J.H.C. Whitehead,Combinatorial homotopy II, Bull. Amer. Math. Soc.55, 453-496,1949.

APA	Şahan T (2019). Further remarks on liftings of crossed modules. , 743 - 752. 10.15672/HJMS.2018.554
Chicago	Şahan Tunçar Further remarks on liftings of crossed modules. (2019): 743 - 752. 10.15672/HJMS.2018.554
MLA	Şahan Tunçar Further remarks on liftings of crossed modules. , 2019, ss.743 - 752. 10.15672/HJMS.2018.554
AMA	Şahan T Further remarks on liftings of crossed modules. . 2019; 743 - 752. 10.15672/HJMS.2018.554
Vancouver	Şahan T Further remarks on liftings of crossed modules. . 2019; 743 - 752. 10.15672/HJMS.2018.554
IEEE	Şahan T "Further remarks on liftings of crossed modules." , ss.743 - 752, 2019. 10.15672/HJMS.2018.554
ISNAD	Şahan, Tunçar. "Further remarks on liftings of crossed modules". (2019), 743-752. https://doi.org/10.15672/HJMS.2018.554

APA	Şahan T (2019). Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics, 48(3), 743 - 752. 10.15672/HJMS.2018.554
Chicago	Şahan Tunçar Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics 48, no.3 (2019): 743 - 752. 10.15672/HJMS.2018.554
MLA	Şahan Tunçar Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics, vol.48, no.3, 2019, ss.743 - 752. 10.15672/HJMS.2018.554
AMA	Şahan T Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics. 2019; 48(3): 743 - 752. 10.15672/HJMS.2018.554
Vancouver	Şahan T Further remarks on liftings of crossed modules. Hacettepe Journal of Mathematics and Statistics. 2019; 48(3): 743 - 752. 10.15672/HJMS.2018.554
IEEE	Şahan T "Further remarks on liftings of crossed modules." Hacettepe Journal of Mathematics and Statistics, 48, ss.743 - 752, 2019. 10.15672/HJMS.2018.554
ISNAD	Şahan, Tunçar. "Further remarks on liftings of crossed modules". Hacettepe Journal of Mathematics and Statistics 48/3 (2019), 743-752. https://doi.org/10.15672/HJMS.2018.554