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

Colimits of crossed modules in modified categories of interest

Ali Aytekin 1, and  Kadir Emir 2,,

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.

Keywords: Colimit, crossed module, modified category of interest.
Mathematics Subject Classification: Primary: 17A30; Secondary: 18G45, 18A30.
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. 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.   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. 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.  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. 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.   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. 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.  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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