On relations among quadratic modules

YILMAZ, Elis SOYLU; Yilmaz, Koray

doi:10.1002/mma.8129

Cited by 3 publications

(3 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This structure is an algebraic model for homotopy connected 2-types of topological spaces. Recall that a crossed module is a group homomorphism ∂ : M → P together with an action of P on M , written p m, for p ∈ P and m ∈ M , satisfying the conditions ∂( p m) = p∂(m)p −1 and ∂m m ′ = mm ′ m −1 , for all m, m ′ ∈ M, p ∈ P. We denote the category of crossed modules of groups by X M. For further work about some categorical and algebraic properties of crossed modules in various settings and their examples, see to [20][21][22][23][24].…”

Section: Crossed Modules and Regular Groupoidsmentioning

confidence: 99%

Whiskered Groupoids and Crossed Modules with Diagrams

Güler,

Ulualan

2024

Journal of New Theory

View full text Add to dashboard Cite

In this study, we investigate the relationships between the category of crossed modules of groups and the category of whiskered groupoids. Our first aim is to construct a crossed module structure over groups from a whiskered groupoid with the objects set - a group (regular groupoid) - using the usual functor between the categories of crossed modules and cat groups. Conversely, the second aim is to construct a whiskered groupoid structure with the objects set, which is a group, from a crossed module of groups. While establishing this relationship, we frequently used arrow diagrams representing morphisms to make the axioms more comprehensible. We provide the conditions for the bimorphisms in a whiskered groupoid and give the relations between this structure and internal groupoids in the category of whiskered groupoids with the objects set as a group.

show abstract

Section: Crossed Modules and Regular Groupoidsmentioning

confidence: 99%

Whiskered Groupoids and Crossed Modules with Diagrams

Güler,

Ulualan

2024

Journal of New Theory

View full text Add to dashboard Cite

show abstract

“…The quasi quadratic modules over Lie algebras has been studied in [6]. For further work about the 2-dimensional crossed modules, see [7].…”

Section: Introductionmentioning

confidence: 99%

Crossed Corner and Reduced Simplicial Commutative Algebras

GÜRMEN ALANSAL

2023

Journal of New Theory

View full text Add to dashboard Cite

In this paper, we describe the crossed corner of commutative algebras and present the relation between the category of crossed corners of commutative algebras and the category of reduced simplicial commutative algebras with Moore complex of length 2. We provide a passage from crossed corners to bisimplicial algebras. In this construction, we utilize the Artin-Mazur codiagonal functor from reduced bisimplicial algebras to simplicial algebras and the hypercrossed complex pairings in the Moore complex of a simplicial algebra. Using the coskeleton functor from the category of $k$-truncated simplicial algebras to the category simplicial algebras with Moore complex of length $k$, we see that the length of Moore complex of the reduced simplicial algebra obtained from a crossed corner is 2.

show abstract

“…Ulualan and Uslu have adapted this algebraic 3-type model to the Lie algebras, [7] as well as Arvasi and Ulualan have worked on that of commutative algebra case, [8], [9]. Many studies have been conducted in these contexts for various algebraic cases, including the homotopy theory and some categorical results, [10][11][12][13][14]. Carrasco and Porter have initially introduced 2-quasi-crossed modules of groups in [15] as an auxiliary tool that they are in between 2-pre-crossed modules and 2-crossed modules.…”

Section: Introductionmentioning

confidence: 99%