2-Dimensional Groups with Action : The Category of Crossed Module of Groups with Action (original) (raw)

Computing 3-dimensional groups: Crossed squares and cat2-groups

Journal of Symbolic Computation

The category XSq of crossed squares is equivalent to the category Cat2 of cat 2-groups. Functions for computing with these structures have been developed in the package XMod written using the GAP computational discrete algebra programming language. This paper includes details of the algorithms used. It contains tables listing the 1, 000 isomorphism classes of cat 2-groups on groups of order at most 30.

Crossed modules, double group-groupoids and crossed squares

Filomat, 2020

The purpose of this paper is to obtain the notion of crossed module over group-groupoids considering split extensions and prove a categorical equivalence between these types of crossed modules and double group-groupoids. This equivalence enables us to produce various examples of double groupoids. We also prove that crossed modules over group-groupoids are equivalent to crossed squares.

Generalized crossed modules and group-groupoids

TURKISH JOURNAL OF MATHEMATICS, 2017

In this present work, we present the concept of a crossed module over generalized groups and we call it a "generalized crossed module". We also define a generalized group-groupoid. Furthermore, we show that the category of generalized crossed modules is equivalent to that of generalized group-groupoids whose object sets are abelian generalized group.

A higher-dimensional categorical perspective on 2-crossed modules

A higher-dimensional categorical perspective on 2-crossed modules, 2024

In this study, we will express the 2-crossed module of groups from a higher-dimensional categorical perspective. According to simplicial homotopy theory, a 2-crossed module is the Moore complex of a 2-truncated simplicial group. Therefore, the 2-crossed module is an algebraic homotopy model for the homotopy 3-types. Tricategories are a three-dimensional generalization of the bicategory concept. Any tricategory is triequivalent to the Gray category, where Gray is a category enriched over the monoidal category 2Cat equipped with the Gray tensor product. Briefly, a Gray category is a semi-strict 3-category for homotopy 3-types. Naturally, the tricategory perspective is used in homotopy theory. The 2-crossed module is associated with the concept of the Gray category. The aim of this study is to obtain a single object tricategory from any 2-crossed module of groups.

Two-sided crossed products of groups

Filomat, 2016

In this paper, we first define a new version of the crossed product of groups under the name of two-sided crossed product. Then we present a generating and relator sets for this new product over cyclic groups. In a separate section, by using the monoid presentation of the two-sided crossed product of cyclic groups, we obtain the complete rewriting system and normal forms of elements of this new group construction.

Group-groupoid actions and liftings of crossed modules

Georgian Mathematical Journal

The aim of this paper is to define the notion of lifting via a group morphism for a crossed module and give some properties of this type of liftings. Further, we obtain a criterion for a crossed module to have a lifting crossed module. We also prove that the category of the lifting crossed modules of a certain crossed module is equivalent to the category of group-groupoid actions on groups, where the group-groupoid corresponds to the crossed module.

Adjunction between crossed modules of groups and algebras

Journal of Homotopy and Related Structures, 2014

We construct a pair of adjoint functors between the categories of crossed modules of groups and associative algebras and establish a one-to-one correspondence between module structures over a crossed module of groups and its respective crossed module of associative algebras.