Further remarks on liftings of crossed modules

Yıl 2019, Cilt: 48 Sayı: 3, 743 - 752, 15.06.2019

Tunçar Şahan

Öz

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

Anahtar Kelimeler

crossed module, lifting, groupoid action, derivation

Kaynakça

• 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.
• R. Brown, Higher dimensional group theory, In: Low Dimensional Topology, London Math. Soc. Lect. Notes 48, 215-238. Cambridge Univ. Press, 1982.
• R. Brown, Topology and Groupoids, BookSurge LLC, North Carolina, 2006.
• R. Brown, G. Danesh-Naruie and J.P.L. Hardy, Topological Groupoids: II. Covering Morphisms and G-Spaces, Math. Nachr. 74, 143-156, 1976.
• R. Brown and P.J. Higgins, On the connection between the second relative homotopy groups of some related space, Proc. London Math. Soc. 36, 193-212, 1978.
• R. Brown, P.J. Higgins and R. Sivera, Nonabelian Algebraic Topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts Math. 15, 2011.
• R. Brown and J. Huebschmann, Identities among relations. In: Low Dimentional Topology, London Math. Soc. Lect. Notes 48, 153-202. Cambridge Univ. Press, 1982.
• R. Brown and O. Mucuk, Covering groups of non-connected topological groups revis- ited, Math. Proc. Camb. Phil. Soc. 115, 97-110, 1994.
• 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.
• W.H. Cockcroft, On the homomorphisms of sequences, Proc. Cambridge. Phil. Soc. 45, 521-532, 1952.
• P.J. Higgins, Categories and groupoids, Van Nostrand, New York, 1971.
• J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helvetici 55, 302-314, 1980.
• J.-L. Loday, Cohomologie et groupes de Steinberg relatifs, J. Algebra 54, 178-202, 1978.
• J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24, 179-202, 1982.
• A.S.T. Lue, Cohomology of groups relative to a variety, J. Algebra 69, 155-174, 1981.
• J.F. Martins, The fundamental 2-crossed complex of a reduced CW-complex, Homol- ogy Homotopy Appl. 13 (2), 129-157, 2011.
• O. Mucuk, Covering groups of non-connected topological groups and the monodromy groupoid of a topological groupoid, PhD Thesis, University of Wales, Bangor, 1993.
• O. Mucuk and T. Şahan, Covering groupoids of categorical groups, Hacet. J. Math. Stat. 42 (4), 419-430, 2013.
• O. Mucuk and T. Şahan, Coverings and crossed modules of topological groups with operations, Turk. J. Math. 38 (5), 833-845, 2014.
• O. Mucuk and T. Şahan, Group-groupoid actions and liftings of crossed modules, Georgian Math. J., doi:10.1515/gmj-2018-0001, 2017.
• O. Mucuk, T. Şahan and N. Alemdar, Normality and Quotients in Crossed Modules and Group-groupoids, Appl. Categor. Struct. 23, 415-428, 2015.
• K. Norrie, Actions and Automorphisms of crossed modules, Bull. Soc. Math. France 118, 129-146, 1990.
• J.H.C. Whitehead, Note on a previous paper entitled “On adding relations to homo- topy group", Ann. Math. 47, 806-810, 1946.
• J.H.C. Whitehead, On operators in relative homotopy groups, Ann. Math. 49, 610- 640, 1948.
• J.H.C. Whitehead, Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, 1949.