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.

Liftings of crossed modules in the category of groups with operations

Boletim da Sociedade Paranaense de Matemática

In this paper we define the notion of lifting of a crossed module via the morphism in groups with operations and give some properties of this type of liftings. Further we prove that the lifting crossed modules of a certain crossed module are categorically equivalent to the internal groupoid actions on groups with operations, where the internal groupoid corresponds to the crossed module.

Groups up to congruence relation and from categorical groups to c-crossed modules

Journal of Homotopy and Related Structures

We introduce a notion of c-group, which is a group up to congruence relation and consider the corresponding category. Extensions, actions and crossed modules (c-crossed modules) are defined in this category and the semi-direct product is constructed. We prove that each categorical group gives rise to c-groups and to a c-crossed module, which is a connected, special and strict c-crossed module in the sense defined by us. The results obtained here will be applied in the proof of an equivalence of the categories of categorical groups and connected, special and strict c-crossed modules.

Group-2-gruopoids and 2g-crossed modules

Hacettepe Journal of Mathematics and Statistics, 2018

In this paper we define the group structure on a 2-groupoid name group-2-groupoid. Corresponding to a group-2-groupoid, a 2G-crossed module is obtained on the structure of crossed modules. Then we prove the categorical equivalence between group-2-groupoids and 2G-crossed modules.