Higher-dimensional self-acting groups: From crossed modules to internal categories

References

[1] I. Akça and Z. Arvasi, Simplicial and crossed Lie algebras, Homology Homotopy Appl. 4 (2002), no. 1, 43–57. 10.4310/HHA.2002.v4.n1.a4Search in Google Scholar

[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 (2013), no. 2, 223–238. 10.1515/gmj-2013-0021Search in Google Scholar

[3] N. Alemdar and O. Mucuk, The liftings of R-modules to covering groupoids, Hacet. J. Math. Stat. 41 (2012), no. 6, 813–822. Search in Google Scholar

[4] Z. Arvasi and U. Ege, Higher-dimensional free crossed modules of commutative algebras, Algebras Groups Geom. 19 (2002), no. 2, 129–145. Search in Google Scholar

[5] Z. Arvasi and I. Kocayusufoglu, 2-crossed complexes and crossed resolutions of Lie algebras, Algebras Groups Geom. 16 (1999), no. 4, 451–479. Search in Google Scholar

[6] A. Aytekin, J. M. Casas and E. Ö. Uslu, Semi-complete crossed modules of Lie algebras, J. Algebra Appl. 11 (2012), no. 5, Article ID 1250096. 10.1142/S021949881250096XSearch in Google Scholar

[7] J. C. Baez and J. Dolan, Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), no. 11, 6073–6105. 10.1063/1.531236Search in Google Scholar

[8] Y. Boyaci, T. S. Kuzpinari and E. Ö. Uslu, Split extension classifiers in the category of precrossed modules of commutative algebras, Turkish J. Math. 39 (2015), no. 5, 683–697. 10.3906/mat-1503-78Search in Google Scholar

[9] R. Brown, Groupoids and crossed objects in algebraic topology, Homology Homotopy Appl. 1 (1999), 1–78. 10.4310/HHA.1999.v1.n1.a1Search in Google Scholar

[10] R. Brown and J. Huebschmann, Identities among relations, Low-Dimensional Topology (Bangor 1979), London Math. Soc. Lecture Note Ser. 48, Cambridge University, Cambridge (1982), 153–202. 10.1017/CBO9780511758935.010Search in Google Scholar

[11] R. Brown and O. Mucuk, Covering groups of nonconnected topological groups revisited, Math. Proc. Cambridge Philos. Soc. 115 (1994), no. 1, 97–110. 10.1017/S0305004100071942Search in Google Scholar

[12] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cah. Topol. Géom. Différ. 17 (1976), no. 4, 343–362. Search in Google Scholar

[13] 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. 10.1016/1385-7258(76)90068-8Search in Google Scholar

[14] E. I. Çağlayan, n-isoclinic Lie crossed modules, Filomat 36 (2022), no. 14, 4935–4946. 10.2298/FIL2214935CSearch in Google Scholar

[15] J. M. Casas, T. Datuashvili and M. Ladra, Universal strict general actors and actors in categories of interest, Appl. Categ. Structures 18 (2010), no. 1, 85–114. 10.1007/s10485-008-9166-zSearch in Google Scholar

[16] T. Datuashvili, Central series for groups with action and Leibniz algebras, Georgian Math. J. 9 (2002), no. 4, 671–682. Search in Google Scholar

[17] T. Datuashvili, Witt’s theorem for groups with action and free Leibniz algebras, Georgian Math. J. 11 (2004), no. 4, 691–712. 10.1515/GMJ.2004.691Search in Google Scholar

[18] T. Datuashvili and T. Şahan, Actions and semi-direct products in categories of groups with action, preprint (2021), https://arxiv.org/pdf/2111.13542. Search in Google Scholar

[19] T. Datuashvili and T. Şahan, Actions and semi-direct products in categories of groups with action, Hacet. J. Math. Stat. 52 (2023), no. 1, 103–113. 10.15672/hujms.1028848Search in Google Scholar

[20] T. Datuashvili and T. Şahan, Pentactions and action representability in the category of reduced groups with action, Georgian Math. J. 30 (2023), no. 2, 195–209. 10.1515/gmj-2022-2205Search in Google Scholar

[21] A. Fröhlich, Non-Abelian homological algebra. I. Derived functors and satellites, Proc. Lond. Math. Soc. (3) 11 (1961), 239–275. 10.1112/plms/s3-11.1.239Search in Google Scholar

[22] J. Huebschmann, Crossed n-fold extensions of groups and cohomology, Comment. Math. Helv. 55 (1980), no. 2, 302–313. 10.1007/BF02566688Search in Google Scholar

[23] G. Janelidze, Internal crossed modules, Georgian Math. J. 10 (2003), no. 1, 99–114. 10.1515/GMJ.2003.99Search in Google Scholar

[24] J.-L. Loday, Cohomologie et groupe de Steinberg relatifs, J. Algebra 54 (1978), no. 1, 178–202. 10.1016/0021-8693(78)90025-XSearch in Google Scholar

[25] J.-L. Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982), no. 2, 179–202. 10.1016/0022-4049(82)90014-7Search in Google Scholar

[26] J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math. (2) 39 (1993), no. 3–4, 269–293. Search in Google Scholar

[27] J.-L. Loday and T. Pirashvili, Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann. 296 (1993), no. 1, 139–158. 10.1007/BF01445099Search in Google Scholar

[28] A. S.-T. Lue, Crossed homomorphisms of Lie algebras, Proc. Cambridge Philos. Soc. 62 (1966), 577–581. 10.1017/S030500410004024XSearch in Google Scholar

[29] A. S.-T. Lue, Cohomology of groups relative to a variety, J. Algebra 69 (1981), no. 1, 155–174. 10.1016/0021-8693(81)90136-8Search in Google Scholar

[30] O. Mucuk, B. Kılıçarslan, T. Şahan and N. Alemdar, Group-groupoids and monodromy groupoids, Topology Appl. 158 (2011), no. 15, 2034–2042. 10.1016/j.topol.2011.06.048Search in Google Scholar

[31] O. Mucuk and T. Şahan, Covering groupoids of categorical groups, Hacet. J. Math. Stat. 42 (2013), no. 4, 419–430. Search in Google Scholar

[32] O. Mucuk and T. Şahan, Coverings and crossed modules of topological groups with operations, Turkish J. Math. 38 (2014), no. 5, 833–845. 10.3906/mat-1307-53Search in Google Scholar

[33] O. Mucuk and T. Şahan, Group-groupoid actions and liftings of crossed modules, Georgian Math. J. 26 (2019), no. 3, 437–447. 10.1515/gmj-2018-0001Search in Google Scholar

[34] 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. 10.1007/s10485-013-9335-6Search in Google Scholar

[35] K. Norrie, Actions and automorphisms of crossed modules, Bull. Soc. Math. France 118 (1990), no. 2, 129–146. 10.24033/bsmf.2140Search in Google Scholar

[36] K. J. Norrie, Crossed modules and analogues of group theorems, Ph.D. thesis, King’s Collage, 1987. Search in Google Scholar

[37] A. Odabaş and E. S. Yılmaz, 2-dimensional groups with action: The category of crossed module of groups with action, preprint (2022), https://arxiv.org/abs/2201.05350. Search in Google Scholar

[38] G. Orzech, Obstruction theory in algebraic categories. I, II, J. Pure Appl. Algebra 2 (1972), 287–314, 315–340. 10.1016/0022-4049(72)90009-6Search in Google Scholar

[39] 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. 10.1017/S0013091500026766Search in Google Scholar

[40] S. Temel, T. Şahan and O. Mucuk, Crossed modules, double group-groupoids and crossed squares, Filomat 34 (2020), no. 6, 1755–1769. 10.2298/FIL2006755TSearch in Google Scholar

[41] E. Ö. Uslu, A. F. Aslan and A. Odabaş, On groups with action on itself, Georgian Math. J. 26 (2019), no. 3, 459–470. 10.1515/gmj-2017-0036Search in Google Scholar

[42] J. H. C. Whitehead, Combinatorial homotopy. I, Bull. Amer. Math. Soc. 55 (1949), 213–245. 10.1090/S0002-9904-1949-09175-9Search in Google Scholar

[43] J. H. C. Whitehead, Combinatorial homotopy. II, Bull. Amer. Math. Soc. 55 (1949), 453–496. 10.1090/S0002-9904-1949-09213-3Search in Google Scholar

[44] E. Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937), 152–160. 10.1515/crll.1937.177.152Search in Google Scholar