Özgün GÜRMEN ALANSAL, Ummahan EGE ARSLAN

Crossed modules bifibred over k-Algebras

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

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[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.