Değişmeli Cebirler için Çaprazlanmış Köşe ve Moore Bikompleks

Yıl 2024, Cilt: 2 Sayı: 1, 12 - 18, 30.06.2024

Hatice Binbir , Özgün Gürmen Alansal

Öz

Bir simplisel cebir, homotopi tiplerine karşılık gelen cebirsel yapıları modeller. Bu bağlamda, Moore kompleksinin boyutu ≤ 1 olan simplisel cebirler, çaprazlanmış modül yapısını vermektedir. Moore kompleksinin boyutu ≤ 2 olduğunda 2-çaprazlanmış modüller, çaprazlanmış kareler veya cat2-cebirlere denk yapılar elde edilmektedir. Simplisel cebirin ilk bileşeni birim alındığında indirgenmiş simplisel cebir yapısı oluşur ve bu yapı 1- bağlantılı homotopi tiplerine modelleme yapar. Bu çalışmada bisimplisel cebirlerin, çaprazlanmış köşe kavramını nasıl modellediği göstererildi. Değişmeli cebirler üzerinde çaprazlanmış kare kategorisi ile Moore kompleksinin boyutu 2 olan simplisel değişmeli cebirler kategorisinin denkliğinden yararlanarak bu yapıda NE_00={0} alırsak elde ettiğimiz yeni yapının bir çaprazlanmış köşe yapısı oluşturduğunu ispatladık.

Anahtar Kelimeler

Çaprazlanmış modül, Çaprazlanmış köşe, bisimplisel cebir, Moore kompleks

Crossed Corner of Commutative Algebra and Moore BiComplex

Öz

A simplicial algebra models algebraic structures that correspond to homotopy types. In this regard, simplicial algebras with a dimension of Moore complex ≤ 1 give the crossed modulus structure. When the length of the Moore complex is ≤ 2, 2-crossed modules, crossed squares, or structures equivalent to cat2 algebras are obtained. When the first component of simplicial algebra is taken as a unit, a reduced simplicial algebra structure is formed, and this structure models 1-connected homotopy types. In this study, we will show how bisimplicial algebras model the concept of crossed corners. In this structure, taking advantage of the equivalence of the category of crossed squares on commutative algebras and the category of simplicial commutative algebras of the Moore complex of length 2. If we take NE_00={0}, we have proved that the new structure obtained forms a crossed corner structure.

Anahtar Kelimeler

Crossed modules, Crossed corner, Bisimplicial algebra, Moore complex

Kaynakça

• Alp, M. (1999). Characterization of crossed corner, algebras, Groups and Geometries, 16(2). 173–182.
• Alp, M. (1999). Applications of crossed corner, Algebras, Groups and Geometries, 16(2), 337–344.
• Alp, M., Bekir, A., Ulualan, E. (2001). Relation between crossed square and crossed corner, Journal of Science and Technology of Dumlupınar University. (002), 89–96.
• Arvasi, Z. (1997). Crossed Squares and 2-Crossed Modules of Commutative Algebras, Theory and Applications of Categories 3(7). 160–181.
• Arvasi, Z., Porter, T. (1997). Higher dimensional Peiffer elements in simplicial commutative algebras, Theory and Applications of Categories 3(1), 1–23.
• Arvasi, Z., Ulualan, E. (2007). Quadratic and 2-crossed modules of algebras, Algebra Colloquium, 14.4, 669-686.
• Aytekin, A. (2019). Categorical structures of Lie-Rinehart crossed module, Turkish Journal of Mathematics, 43(1), 511–522.
• Conduchè D. (2003), Simplicial crossed modules and mapping cones, Georgian Math. Journal. 10 , 623–636. Binbir H. (2024), Crossed corners and related structures, Yüksek lisans tezi Kütahya Dumlupınar University, Kütahya.
• Ellis, G. J. (1988). Higher dimensional crossed modules of algebras, Journal of Pure and Applied Algebra. 52. 277–282.
• Guin-Walery, D., Loday, J. L. (1981). Obsruction á l’excision en K-theories Algébrique, in: E. M. Friedlander, M. R. Stein (Eds.), Algebraic K-Theory Evanston 1980, Vol. 854 of Lecture Notes Mathematics, Springer, Berlin, 1981, pp. 179–216.
• Gürmen Alansal, Ö. (2021). Peiffer Pairings in Multisimplicial Groups and Crossed n-Cubes and Applications for Bisimplicial Groups, Turkish Journal of Mathematics 45(1), 360-386.
• Gürmen Alansal, Ö. (2023). Crossed Corner and Reduced Simplicial Commutative Algebras. Journal of New Theory (45), 95-104.
• Gürmen Alansal, Ö., Ulualan, E. (2023). Bisimplicial commutative algebras and crossed squares, Fundamental Journal of Mathematics and Applications, 6, 177–187.
• Gürmen Alansal, Ö. (2024). Cubical simplicial algebras and related crossed structures, Filomat 38(9), 3121–3135.
• Odabas A., Ulualan, E. (2016)., On free quadratic modules of algebras, Bull. Malaysian Math. Sci. Soc., 39:3, 1059-–1074.
• Porter, T. (1986). Homology of Commutative Algebras and an Invariant of Simis and Vasconceles, Journal of Algebra, 99, 458–465.
• Shammu, N. M. (1992). Algebraic and Categorical Structure of categories of Crossed Modules of algebras, Doctoral Dissertation North Carolina Wilmington University Bangor.
• Whitehead, J. H. C. (1949). Combinatorial Homotopy II, Bulletin of the American Mathematical Society, 55, 453–496