Normality and quotient in the category of crossed modules within the category of groups with operations
DOI:
https://doi.org/10.5269/bspm.v38i7.46499Abstract
In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and then give same properties of such crossed modules in groups with operations.
References
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8. Datuashvili, T., Categorical, homological and homotopical properties of algebraic objects, J. Math. Sci. 225(3), 383-533, (2017). https://doi.org/10.1007/s10958-017-3474-5
9. Datuashvili, T., Cohomologically trivial internal categories in categories of groups with operations, Appl. Categ. Structures 3, 221-237, (1995). https://doi.org/10.1007/BF00878442
10. Datuashvili, T., Cohomology of internal categories in categories of groups with operations, Categorical Topology and its Relation to Analysis, algebra and combinatorics, Ed. J. Adamek and S. Mac Lane (Prague, 1988), World Sci. Publishing, Teaneck, NJ, (1989).
11. Datuashvili, T., Kan extensions of internal functors: Nonconnected case, J.Pure Appl.Algebra 167, 195-202, (2002). https://doi.org/10.1016/S0022-4049(01)00035-4
12. Datuashvili, T.,Whitehead homotopy equivalence and internal category equivalence of crossed modules in categories of groups with operations, Proc. A. Razmadze Math.Inst. 113, 3-30, (1995). https://doi.org/10.1007/BF00878442
13. Gursoy, M.H., Aslan, H., Ë™ Icen, Ë™ I., Generalized crossed modules and group-groupoids, Turk. J. Math. 41(6), 1535-1551, (2017). https://doi.org/10.3906/mat-1602-63
14. Higgins, P.J., Groups with multiple operators, Proc. London Math. Soc. 3(6) 366-416, (1956).
https://doi.org/10.1112/plms/s3-6.3.366
15. Huebschmann, J., Crossed n-fold extensions of groups and cohomology, Comment. Math. Helvetici 55, 302-313, (1980). https://doi.org/10.1007/BF02566688
16. Loday, J.-L., Cohomologie et groupes de Steinberg relatifs, J. Algebra 54, 178-202, (1978).
https://doi.org/10.1016/0021-8693(78)90025-X
17. Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24, 179-202, (1982).
https://doi.org/10.1016/0022-4049(82)90014-7
18. Lue, A.S.T., Cohomology of groups relative to a variety, J. Algebra 69, 155-174, (1981).
https://doi.org/10.1016/0021-8693(81)90136-8
19. Mucuk, O., Akız, H. F., Monodromy groupoids of an internal groupoid in topological groups with operations, Filomat 29(10), 2355-2366, (2015). https://doi.org/10.2298/FIL1510355M
20. Mucuk, O., Demir, S., Normality and quotient in crossed modules over groupoids and double groupoids, Turk. J. Math. 42(5), 2336-2347, (2018). https://doi.org/10.3906/mat-1606-126
21. Mucuk, O., Kılı¸carslan, B., Sahan T., Alemdar, N., Group-groupoids and monodromy groupoids, Topology Appl. 158(15), 2034-2042, (2011). https://doi.org/10.1016/j.topol.2011.06.048
22. Mucuk, O., Sahan, T., Alemdar, N., Normality and quotients in crossed modules and group groupoids, Appl. Categor. Struct. 23, 415-428, (2015). https://doi.org/10.1007/s10485-013-9335-6
23. Mucuk, O., Sahan, T., Coverings and crossed modules of topological groups with operations, Turk. J. Math. 38(5), 833-845, (2014). https://doi.org/10.3906/mat-1307-53
24. Orzech, G., Obstruction theory in algebraic categories I, II, J. Pure Appl. Algebra 2, 287-314, 315-340, (1972).
https://doi.org/10.1016/0022-4049(72)90009-6
25. Norrie, K., Actions and Automorphisms of crossed modules, Bull. Soc. Math. France, 118, 129-146, (1990).
https://doi.org/10.24033/bsmf.2140
26. Porter, T., Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373-381, (1987). https://doi.org/10.1017/S0013091500026766
27. Sahan, T.: English title: Normality and quotients in some algebraic categories, PhD Thesis (in Turkish), University of Erciyes (2014).
28. Temel, S., Normality and quotient in crossed modules over groupoids and 2-Groupoids, Korean J. Math., 27(1), 151-163, (2019).
29. Whitehead, J.H.C., Note on a previous paper entitled "On adding relations to homotopy group", Ann. of Math. 47, 806-810, (1946). https://doi.org/10.2307/1969237
30. Whitehead, J.H.C., Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, (1949).
https://doi.org/10.1090/S0002-9904-1949-09213-3
2. Baez, J.C., Lauda, A.D., Higher-dimensional algebra V: 2-groups, Theory Appl. Categ. 12, 423-491, (2004).
3. Brown, R., Higher dimensional group theory. In: Low Dimensional Topology, London Math. Soc. Lect. Notes, 48, pp. 215-238. Cambridge Univ. Press, (1982). https://doi.org/10.1017/CBO9780511758935
4. Brown, R., Groupoids and crossed objects in algebraic topology, Homology Homotopy Appl. 1, 1-78, (1999).
https://doi.org/10.4310/HHA.1999.v1.n1.a1
5. Brown, R., Huebschmann, J., Identities among relations. In: Low Dimentional Topology, London Math. Soc. Lect. Notes, 48, pp. 153-202. Cambridge Univ. Press (1982). https://doi.org/10.1017/CBO9780511758935.010
6. Brown, R., Mucuk, O., Covering groups of non-connected topological groups revisited, Math. Proc. Camb. Phil. Soc. 115, 97-110, (1994). https://doi.org/10.1017/S0305004100071942
7. Brown, R., Spencer C. B., G-groupoids, crossed modules and the fundamental groupoid of a topological group, Proc. Konn. Ned. Akad. v. Wet. 79, 296-302, (1976). https://doi.org/10.1016/1385-7258(76)90068-8
8. Datuashvili, T., Categorical, homological and homotopical properties of algebraic objects, J. Math. Sci. 225(3), 383-533, (2017). https://doi.org/10.1007/s10958-017-3474-5
9. Datuashvili, T., Cohomologically trivial internal categories in categories of groups with operations, Appl. Categ. Structures 3, 221-237, (1995). https://doi.org/10.1007/BF00878442
10. Datuashvili, T., Cohomology of internal categories in categories of groups with operations, Categorical Topology and its Relation to Analysis, algebra and combinatorics, Ed. J. Adamek and S. Mac Lane (Prague, 1988), World Sci. Publishing, Teaneck, NJ, (1989).
11. Datuashvili, T., Kan extensions of internal functors: Nonconnected case, J.Pure Appl.Algebra 167, 195-202, (2002). https://doi.org/10.1016/S0022-4049(01)00035-4
12. Datuashvili, T.,Whitehead homotopy equivalence and internal category equivalence of crossed modules in categories of groups with operations, Proc. A. Razmadze Math.Inst. 113, 3-30, (1995). https://doi.org/10.1007/BF00878442
13. Gursoy, M.H., Aslan, H., Ë™ Icen, Ë™ I., Generalized crossed modules and group-groupoids, Turk. J. Math. 41(6), 1535-1551, (2017). https://doi.org/10.3906/mat-1602-63
14. Higgins, P.J., Groups with multiple operators, Proc. London Math. Soc. 3(6) 366-416, (1956).
https://doi.org/10.1112/plms/s3-6.3.366
15. Huebschmann, J., Crossed n-fold extensions of groups and cohomology, Comment. Math. Helvetici 55, 302-313, (1980). https://doi.org/10.1007/BF02566688
16. Loday, J.-L., Cohomologie et groupes de Steinberg relatifs, J. Algebra 54, 178-202, (1978).
https://doi.org/10.1016/0021-8693(78)90025-X
17. Loday, J.-L., Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24, 179-202, (1982).
https://doi.org/10.1016/0022-4049(82)90014-7
18. Lue, A.S.T., Cohomology of groups relative to a variety, J. Algebra 69, 155-174, (1981).
https://doi.org/10.1016/0021-8693(81)90136-8
19. Mucuk, O., Akız, H. F., Monodromy groupoids of an internal groupoid in topological groups with operations, Filomat 29(10), 2355-2366, (2015). https://doi.org/10.2298/FIL1510355M
20. Mucuk, O., Demir, S., Normality and quotient in crossed modules over groupoids and double groupoids, Turk. J. Math. 42(5), 2336-2347, (2018). https://doi.org/10.3906/mat-1606-126
21. Mucuk, O., Kılı¸carslan, B., Sahan T., Alemdar, N., Group-groupoids and monodromy groupoids, Topology Appl. 158(15), 2034-2042, (2011). https://doi.org/10.1016/j.topol.2011.06.048
22. Mucuk, O., Sahan, T., Alemdar, N., Normality and quotients in crossed modules and group groupoids, Appl. Categor. Struct. 23, 415-428, (2015). https://doi.org/10.1007/s10485-013-9335-6
23. Mucuk, O., Sahan, T., Coverings and crossed modules of topological groups with operations, Turk. J. Math. 38(5), 833-845, (2014). https://doi.org/10.3906/mat-1307-53
24. Orzech, G., Obstruction theory in algebraic categories I, II, J. Pure Appl. Algebra 2, 287-314, 315-340, (1972).
https://doi.org/10.1016/0022-4049(72)90009-6
25. Norrie, K., Actions and Automorphisms of crossed modules, Bull. Soc. Math. France, 118, 129-146, (1990).
https://doi.org/10.24033/bsmf.2140
26. Porter, T., Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc. 30, 373-381, (1987). https://doi.org/10.1017/S0013091500026766
27. Sahan, T.: English title: Normality and quotients in some algebraic categories, PhD Thesis (in Turkish), University of Erciyes (2014).
28. Temel, S., Normality and quotient in crossed modules over groupoids and 2-Groupoids, Korean J. Math., 27(1), 151-163, (2019).
29. Whitehead, J.H.C., Note on a previous paper entitled "On adding relations to homotopy group", Ann. of Math. 47, 806-810, (1946). https://doi.org/10.2307/1969237
30. Whitehead, J.H.C., Combinatorial homotopy II, Bull. Amer. Math. Soc. 55, 453-496, (1949).
https://doi.org/10.1090/S0002-9904-1949-09213-3
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2019-10-14
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