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September  2020, 28(3): 1227-1238. doi: 10.3934/era.2020067

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

* Corresponding author: emir@math.muni.cz

Received  April 2020 Revised  May 2020 Published  July 2020

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.

Citation: Ali Aytekin, Kadir Emir. Colimits of crossed modules in modified categories of interest. Electronic Research Archive, 2020, 28 (3) : 1227-1238. doi: 10.3934/era.2020067
References:
[1]

M. Alp, Pullbacks of crossed modules and cat1-groups, Turkish J. Math., 22 (1998), 273-281.   Google Scholar

[2]

M. Alp, Pullbacks of crossed modules and Cat1-commutative algebras, Turkish J. Math., 30 (2006), 237-246.   Google Scholar

[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.  Google Scholar

[4]

Y. BoyaciJ. M. CasasT. Datuashvili and E. Ö. Uslu, Actions in modified categories of interest with application to crossed modules, Theory Appl. Categ., 30 (2015), 882-908.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

J. Casas, T. Datuashvili and M. Ladra, Actors in categories of interest, arXiv: math/0702574. Google Scholar

[11]

J. M. CasasT. 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.  Google Scholar

[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.  Google Scholar

[13]

K. Emir and S. Çetin, Limits in modified categories of interest, Bull. Iran. Math. Soc., 43 (2017), 2617-2634.   Google Scholar

[14]

K. Emir and H. Gülsün Akay, Pullback crossed modules in the category of racks, Hacet. J. Math. Stat., 48 (2019), 140-149.   Google Scholar

[15]

P. J. Higgins, Groups with multiple operators, Proc. Lond. Math. Soc., 6 (1956), 366-416.  doi: 10.1112/plms/s3-6.3.366.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

M. Alp, Pullbacks of crossed modules and cat1-groups, Turkish J. Math., 22 (1998), 273-281.   Google Scholar

[2]

M. Alp, Pullbacks of crossed modules and Cat1-commutative algebras, Turkish J. Math., 30 (2006), 237-246.   Google Scholar

[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.  Google Scholar

[4]

Y. BoyaciJ. M. CasasT. Datuashvili and E. Ö. Uslu, Actions in modified categories of interest with application to crossed modules, Theory Appl. Categ., 30 (2015), 882-908.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[10]

J. Casas, T. Datuashvili and M. Ladra, Actors in categories of interest, arXiv: math/0702574. Google Scholar

[11]

J. M. CasasT. 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.  Google Scholar

[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.  Google Scholar

[13]

K. Emir and S. Çetin, Limits in modified categories of interest, Bull. Iran. Math. Soc., 43 (2017), 2617-2634.   Google Scholar

[14]

K. Emir and H. Gülsün Akay, Pullback crossed modules in the category of racks, Hacet. J. Math. Stat., 48 (2019), 140-149.   Google Scholar

[15]

P. J. Higgins, Groups with multiple operators, Proc. Lond. Math. Soc., 6 (1956), 366-416.  doi: 10.1112/plms/s3-6.3.366.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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