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Year 2021, Volume: 42 Issue: 1, 99 - 114, 29.03.2021

Özgün Gürmen Alansal , Ummahan Ege Arslan

https://doi.org/10.17776/csj.727906

Abstract

References

  • [1] Porter T., Some Categorical Results in the Theory of Crossed Modules in Commutative Algebras., J. Algebra, 109 (1987) 415-429.
  • [2] Shammu N. M., Algebraic and Categorical Structure of Categories of Crossed Modules of Algebras., Ph.D. Thesis, U.C.N.W, (1992).
  • [3] Porter T., Homology of Commutative Algebras and an Invariant of Simis and Vasconceles., J. Algebra, 99(1986) 458-465.
  • [4] Brown R., Higgins P. J., On the Connection between the Second Relative Homotopy Groups of Some Related Spaces, Proc. London Math. Soc.,36(3) (1978) 193-212.
  • [5] Brown R., Higgins P. J., Sivera R., Nonabelian Algebraic Topology, European Mathematical Society Tracts in Mathematics, 15 (2011).
  • [6] Brown R., Sivera R., Algebraic Colimit Calculations in Homotopy Theory using Fibred and Cofibred Categories, Theory and Applications of Categories, 22 (2009) 221-251.
  • [7] Odabaş A, Ulualan E, Braided Regular Crossed Modeles Bifibered over Regular Groupoids, Turkish Journal of Mathematics, 41 (2017) 1385-1403.
  • [8] Arslan Ege U., Akça İ.İ, Irmak Onarlı G., Avcıoğlu O., Fibrations of 2-crossed modules, Mathematical Methods in the Applied Sciences, 42(16) (2019) 5293-5304.
  • [9] Whitehead J. H. C., Combinatorial Homotopy I and II., Bull. Amer. Math. Soc.,55 (1949) 231-245 and 453-496.
  • [10] Mac Lane S., Extension and Obstructions for Rings., Illinois Journal of Mathematics, 121 (1958), 316-345.
  • [11] Dedecker P., Lue, A. S. T., A Non-abelian Two dimensional Cohomology for Associative Algebras., Bull. Amer. Soc., 72 (1966) 1044-1050
  • [12] Lue A. S. T., Non-abelian Cohomology of Associative Algebras., Quart. J. Math. Oxford Ser., 2 (1968) 159-180.
  • [13] Arvasi Z., Ege U., Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures,11 (2003) 487-506.
  • [14] Arvasi Z., Porter T., Simplicial and Crossed Resolutions of Commutative Algebras, Journal of Algebras, 181 (1996) 426-448.
  • [15] Casas J. M., Ladra M., Colimits in the Crossed Modules Category in Lie Algebras. Georgian Mathematical Journal, 7(3) (2000) 461-474.

Crossed modules bifibred over k-Algebras

Year 2021, Volume: 42 Issue: 1, 99 - 114, 29.03.2021

Özgün Gürmen Alansal , Ummahan Ege Arslan

https://doi.org/10.17776/csj.727906

Abstract

In this paper we examine on a pair of adjoint functors (ϕ^* ,ϕ_*)for a subcategory of the category of crossed modules over commutative algebras where ϕ_*: XMod/P → XMod/Q, induced, and ϕ^*:XMod/Q → XMod/P, pullback (co-induced), which enables us to move from crossed Q-modules to crossed P-modules by an algebra morphism ϕ : P → Q. We show that this adjoint functor pair (ϕ^*,ϕ_*) makes p∶ XMod → k-Alg into a bi- fibred category over k-Alg, the category of commutative algebras, where p is given by p(C,R,∂) = R. Also, we give some examples and results on induced crossed modules in the case when ϕ is an epimorphism or the inclusion of an ideal. 

Keywords

induced crossed modules base change funktors fibred and co-fibred categories

References

  • [1] Porter T., Some Categorical Results in the Theory of Crossed Modules in Commutative Algebras., J. Algebra, 109 (1987) 415-429.
  • [2] Shammu N. M., Algebraic and Categorical Structure of Categories of Crossed Modules of Algebras., Ph.D. Thesis, U.C.N.W, (1992).
  • [3] Porter T., Homology of Commutative Algebras and an Invariant of Simis and Vasconceles., J. Algebra, 99(1986) 458-465.
  • [4] Brown R., Higgins P. J., On the Connection between the Second Relative Homotopy Groups of Some Related Spaces, Proc. London Math. Soc.,36(3) (1978) 193-212.
  • [5] Brown R., Higgins P. J., Sivera R., Nonabelian Algebraic Topology, European Mathematical Society Tracts in Mathematics, 15 (2011).
  • [6] Brown R., Sivera R., Algebraic Colimit Calculations in Homotopy Theory using Fibred and Cofibred Categories, Theory and Applications of Categories, 22 (2009) 221-251.
  • [7] Odabaş A, Ulualan E, Braided Regular Crossed Modeles Bifibered over Regular Groupoids, Turkish Journal of Mathematics, 41 (2017) 1385-1403.
  • [8] Arslan Ege U., Akça İ.İ, Irmak Onarlı G., Avcıoğlu O., Fibrations of 2-crossed modules, Mathematical Methods in the Applied Sciences, 42(16) (2019) 5293-5304.
  • [9] Whitehead J. H. C., Combinatorial Homotopy I and II., Bull. Amer. Math. Soc.,55 (1949) 231-245 and 453-496.
  • [10] Mac Lane S., Extension and Obstructions for Rings., Illinois Journal of Mathematics, 121 (1958), 316-345.
  • [11] Dedecker P., Lue, A. S. T., A Non-abelian Two dimensional Cohomology for Associative Algebras., Bull. Amer. Soc., 72 (1966) 1044-1050
  • [12] Lue A. S. T., Non-abelian Cohomology of Associative Algebras., Quart. J. Math. Oxford Ser., 2 (1968) 159-180.
  • [13] Arvasi Z., Ege U., Annihilators, Multipliers and Crossed Modules, Applied Categorical Structures,11 (2003) 487-506.
  • [14] Arvasi Z., Porter T., Simplicial and Crossed Resolutions of Commutative Algebras, Journal of Algebras, 181 (1996) 426-448.
  • [15] Casas J. M., Ladra M., Colimits in the Crossed Modules Category in Lie Algebras. Georgian Mathematical Journal, 7(3) (2000) 461-474.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Natural Sciences
Authors

Özgün Gürmen Alansal 0000-0003-2851-986X

Ummahan Ege Arslan 0000-0002-2995-0718

Publication Date March 29, 2021
Submission Date April 27, 2020
Acceptance Date January 13, 2021
Published in Issue Year 2021Volume: 42 Issue: 1

Cite

APA Gürmen Alansal, Ö., & Ege Arslan, U. (2021). Crossed modules bifibred over k-Algebras. Cumhuriyet Science Journal, 42(1), 99-114. https://doi.org/10.17776/csj.727906