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Colimits of crossed modules in modified categories of interest
1. | Department of Mathematics, Pamukkale University, Denizli, Turkey |
2. | Department of Mathematics and Statistics, Masaryk University, Brno, Czech Republic |
In this paper, we give the constructions of the coequalizer and coproduct objects for the category of crossed modules, in a modified category of interest (MCI). In other words, we prove that the corresponding category is finitely cocomplete.
References:
[1] |
M. Alp,
Pullbacks of crossed modules and cat1-groups, Turkish J. Math., 22 (1998), 273-281.
|
[2] |
M. Alp,
Pullbacks of crossed modules and Cat1-commutative algebras, Turkish J. Math., 30 (2006), 237-246.
|
[3] |
M. Alp and B. Davvaz,
Pullback and pushout crossed polymodules, Proc. Indian Acad. Sci., Math. Sci., 125 (2015), 11-20.
doi: 10.1007/s12044-015-0212-0. |
[4] |
Y. Boyaci, J. M. Casas, T. Datuashvili and E. Ö. Uslu,
Actions in modified categories of interest with application to crossed modules, Theory Appl. Categ., 30 (2015), 882-908.
|
[5] |
R. Brown,
Coproducts of crossed $P$-modules: Applications to second homotopy groups and to the homology of groups, Topology, 23 (1984), 337-345.
doi: 10.1016/0040-9383(84)90016-8. |
[6] |
R. Brown,
From groups to groupoids: A brief survey, Bull. Lond. Math. Soc., 19 (1987), 113-134.
doi: 10.1112/blms/19.2.113. |
[7] |
R. Brown,
Modelling and computing homotopy types: I, Indag. Math. (N.S.), 29 (2018), 459-482.
doi: 10.1016/j.indag.2017.01.009. |
[8] |
R. Brown and C. D. Wensley,
On finite induced crossed modules and the homotopy $2$-type of mapping cones, Theory Appl. Categ., 1 (1995), 54-70.
|
[9] |
J. M. Casas, R. F. Casado, E. Khmaladze and M. Ladra, More on crossed modules in Lie, Leibniz, associative and diassociative algebras, J. Algebra Appl., 16 (2017), 1750107, 17 pp.
doi: 10.1142/S0219498817501079. |
[10] |
J. Casas, T. Datuashvili and M. Ladra, Actors in categories of interest, arXiv: math/0702574. Google Scholar |
[11] |
J. M. Casas, T. Datuashvili and M. Ladra,
Universal strict general actors and actors in categories of interest, Appl. Categ. Struct., 18 (2010), 85-114.
doi: 10.1007/s10485-008-9166-z. |
[12] |
J. M. Casas and M. Ladra,
Colimits in the crossed modules category in Lie algebras, Georgian Math. J., 7 (2000), 461-474.
doi: 10.1515/GMJ.2000.461. |
[13] |
K. Emir and S. Çetin,
Limits in modified categories of interest, Bull. Iran. Math. Soc., 43 (2017), 2617-2634.
|
[14] |
K. Emir and H. Gülsün Akay,
Pullback crossed modules in the category of racks, Hacet. J. Math. Stat., 48 (2019), 140-149.
|
[15] |
P. J. Higgins,
Groups with multiple operators, Proc. Lond. Math. Soc., 6 (1956), 366-416.
doi: 10.1112/plms/s3-6.3.366. |
[16] |
J.-L. Loday,
Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra, 24 (1982), 179-202.
doi: 10.1016/0022-4049(82)90014-7. |
[17] |
S. MacLane and J. H. C. Whitehead,
On the $3$-type of a complex, Proc. Natl. Acad. Sci. U.S.A., 36 (1950), 41-48.
doi: 10.1073/pnas.36.1.41. |
[18] |
G. Orzech,
Obstruction theory in algebraic categories. I, J. Pure Appl. Algebra, 2 (1972), 287-314.
doi: 10.1016/0022-4049(72)90008-4. |
[19] |
T. Porter,
Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc., 30 (1987), 373-381.
doi: 10.1017/S0013091500026766. |
[20] |
N. Shammu, Algebraic and Categorical Structure of Categories of Crossed Modules of Algebras, University College of North Wales, 1992. Google Scholar |
[21] |
J. H. C. Whitehead,
Combinatorial homotopy. II, Bull. Amer. Math. Soc., 55 (1949), 453-496.
doi: 10.1090/S0002-9904-1949-09213-3. |
show all references
References:
[1] |
M. Alp,
Pullbacks of crossed modules and cat1-groups, Turkish J. Math., 22 (1998), 273-281.
|
[2] |
M. Alp,
Pullbacks of crossed modules and Cat1-commutative algebras, Turkish J. Math., 30 (2006), 237-246.
|
[3] |
M. Alp and B. Davvaz,
Pullback and pushout crossed polymodules, Proc. Indian Acad. Sci., Math. Sci., 125 (2015), 11-20.
doi: 10.1007/s12044-015-0212-0. |
[4] |
Y. Boyaci, J. M. Casas, T. Datuashvili and E. Ö. Uslu,
Actions in modified categories of interest with application to crossed modules, Theory Appl. Categ., 30 (2015), 882-908.
|
[5] |
R. Brown,
Coproducts of crossed $P$-modules: Applications to second homotopy groups and to the homology of groups, Topology, 23 (1984), 337-345.
doi: 10.1016/0040-9383(84)90016-8. |
[6] |
R. Brown,
From groups to groupoids: A brief survey, Bull. Lond. Math. Soc., 19 (1987), 113-134.
doi: 10.1112/blms/19.2.113. |
[7] |
R. Brown,
Modelling and computing homotopy types: I, Indag. Math. (N.S.), 29 (2018), 459-482.
doi: 10.1016/j.indag.2017.01.009. |
[8] |
R. Brown and C. D. Wensley,
On finite induced crossed modules and the homotopy $2$-type of mapping cones, Theory Appl. Categ., 1 (1995), 54-70.
|
[9] |
J. M. Casas, R. F. Casado, E. Khmaladze and M. Ladra, More on crossed modules in Lie, Leibniz, associative and diassociative algebras, J. Algebra Appl., 16 (2017), 1750107, 17 pp.
doi: 10.1142/S0219498817501079. |
[10] |
J. Casas, T. Datuashvili and M. Ladra, Actors in categories of interest, arXiv: math/0702574. Google Scholar |
[11] |
J. M. Casas, T. Datuashvili and M. Ladra,
Universal strict general actors and actors in categories of interest, Appl. Categ. Struct., 18 (2010), 85-114.
doi: 10.1007/s10485-008-9166-z. |
[12] |
J. M. Casas and M. Ladra,
Colimits in the crossed modules category in Lie algebras, Georgian Math. J., 7 (2000), 461-474.
doi: 10.1515/GMJ.2000.461. |
[13] |
K. Emir and S. Çetin,
Limits in modified categories of interest, Bull. Iran. Math. Soc., 43 (2017), 2617-2634.
|
[14] |
K. Emir and H. Gülsün Akay,
Pullback crossed modules in the category of racks, Hacet. J. Math. Stat., 48 (2019), 140-149.
|
[15] |
P. J. Higgins,
Groups with multiple operators, Proc. Lond. Math. Soc., 6 (1956), 366-416.
doi: 10.1112/plms/s3-6.3.366. |
[16] |
J.-L. Loday,
Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra, 24 (1982), 179-202.
doi: 10.1016/0022-4049(82)90014-7. |
[17] |
S. MacLane and J. H. C. Whitehead,
On the $3$-type of a complex, Proc. Natl. Acad. Sci. U.S.A., 36 (1950), 41-48.
doi: 10.1073/pnas.36.1.41. |
[18] |
G. Orzech,
Obstruction theory in algebraic categories. I, J. Pure Appl. Algebra, 2 (1972), 287-314.
doi: 10.1016/0022-4049(72)90008-4. |
[19] |
T. Porter,
Extensions, crossed modules and internal categories in categories of groups with operations, Proc. Edinb. Math. Soc., 30 (1987), 373-381.
doi: 10.1017/S0013091500026766. |
[20] |
N. Shammu, Algebraic and Categorical Structure of Categories of Crossed Modules of Algebras, University College of North Wales, 1992. Google Scholar |
[21] |
J. H. C. Whitehead,
Combinatorial homotopy. II, Bull. Amer. Math. Soc., 55 (1949), 453-496.
doi: 10.1090/S0002-9904-1949-09213-3. |
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