Değişmeli Cebirler için Çaprazlanmış Köşe ve Moore Bikompleks

Year 2024, Volume: 2 Issue: 1, 12 - 18, 30.06.2024

Hatice Binbir , Özgün Gürmen Alansal

Abstract

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.

Keywords

Çaprazlanmış modül, Çaprazlanmış köşe, bisimplisel cebir, Moore kompleks

Crossed Corner of Commutative Algebra and Moore BiComplex

Abstract

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.

Keywords

Crossed modules, Crossed corner, Bisimplicial algebra, Moore complex

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