EN
Crossed modules bifibred over k-Algebras
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
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.
Details
Primary Language
English
Subjects
Mathematical Sciences
Journal Section
Research Article
Authors
Publication Date
March 29, 2021
Submission Date
April 27, 2020
Acceptance Date
January 13, 2021
Published in Issue
Year 2021 Volume: 42 Number: 1
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