Peiffer pairings in multisimplicial groups and crossed n-cubes and applications for bisimplicial groups

Alansal, Özgün Gürmen; Ulualan, Erdal

doi:10.3906/mat-1912-75

Cited by 3 publications

(3 citation statements)

References 12 publications

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“…For further details, see to [4,8,13,14]. The following sources [15][16][17][18] also cover various aspects of this area, including the simplicial objects within categories of some algebraic structures, and could be valuable for the reader's reference.…”

Section: Preliminariesmentioning

confidence: 99%

Whiskered Groupoids and Crossed Modules with Diagrams

Güler,

Ulualan

2024

Journal of New Theory

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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.

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Section: Preliminariesmentioning

confidence: 99%

Whiskered Groupoids and Crossed Modules with Diagrams

Güler,

Ulualan

2024

Journal of New Theory

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“…, then it is called a reduced bisimplicial algebra. If N E p,q = {0}, for p + q ≥ k + 1, then the Moore bicomplex is of length k. For 2-dimensional version of C α,β functions for bisimplicial algebras, see [18].…”

Section: Now Consider the Product Category ∆[N] × ∆[N] Whose Objects ...mentioning

confidence: 99%

Crossed Corner and Reduced Simplicial Commutative Algebras

GÜRMEN ALANSAL

2023

Journal of New Theory

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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.

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“…Porter in [12] has proven the equivalence between the category of n-types of simplicial groups and the category of crossed n-cubes. In [9], Gürmen-Alansal and Ulualan generalised these pairings for the Moore bicomplex in bisimplicial groups. It can be seen the role of these pairings for the relations among bisimplicial groups and crossed squares.…”

Section: Introductionmentioning

confidence: 99%

Bisimplicial Commutative Algebras and Crossed Squares

GÜRMEN ALANSAL,

ULUALAN

2023

Fundamental Journal of Mathematics and Applications

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A simplicial commutative algebra with Moore complex of length 1 gives a crossed module structure over commutative algebras. In this study, we will give 2-dimensional version of this result by giving hypercrossed complex pairings for a bisimplicial algebra and its Moore bicomplex. We give a detailed calculation in low dimensions for these pairings to see their role in the structures of crossed squares and bisimplicial algebras. In this context, we prove that if the Moore bicomplex of a bisimplicial commutative algebra is of length 1, then it gives a crossed square structure over commutative algebras.

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Peiffer pairings in multisimplicial groups and crossed n-cubes and applications for bisimplicial groups

Cited by 3 publications

References 12 publications

Whiskered Groupoids and Crossed Modules with Diagrams

Whiskered Groupoids and Crossed Modules with Diagrams

Crossed Corner and Reduced Simplicial Commutative Algebras

Bisimplicial Commutative Algebras and Crossed Squares

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