Coverings and crossed modules of topological groups with operations

It is a well known result in the covering groups that a subgroup $G$ of the fundamental group at the identity of a semi-locally simply connected topological group determines a covering morphism of topological groups with characteristic group $G$. In this paper we generalize this result to a large class of algebraic objects called topological groups with operations, including topological groups. We also give the cover of crossed modules within topological groups with operations.Comment: 15 pages, research paper, LaTeX2e, xypic

Normality and quotient in crossed modules, cat1^1-groups and internal groupoids within groups with operations

In this paper we define the notions of normal subcrossed module and quotient crossed module within groups with operations; and using the equivalence of crossed modules over groups with operations and internal groupoids we prove how normality and quotient concepts are related in these two categories. Further we prove an equivalence of crossed modules over groups with operations and cat$^1$-groups with operations for a certain algebraic category; and then by this equivalence we determine normal and quotient objects in the category of cat$^{1}$-groups with operations. Finally we characterize the coverings of cat$^{1}$-groups with operations.Comment: 30 pages, research paper, LaTeX2e, xypi

Covering groupoids of categorical groups

If $X$ is a topological group, then its fundamental groupoid $\pi_1X$ is a group-groupoid which is a group object in the category of groupoids. Further if $X$ is a path connected topological group which has a simply connected cover, then the category of covering spaces of $X$ and the category of covering groupoids of $\pi_1X$ are equivalent. In this paper we prove that if $(X,x_0)$ is an $H$-group, then the fundamental groupoid $\pi_1X$ is a categorical group. This enable us to prove that the category of the covering spaces of an $H$-group $(X,x_0)$ is equivalent to the category of covering groupoid of the categorical group $\pi_1X$.Comment: 15 pages, research article, LaTeX2e, xypi

Some Results for the Local Subgroupoids

The notion of local subgroupoids as generalition of a local equivalence relations was defined by the first author and R.Brown. Here we investigate some relations between transitive components and coherence properties of the local subgroupoids.Comment: 9 pages, A

Extendibility, monodromy and local triviality for topological groupoids

A groupoid is a small category in which each morphism has an inverse. A topological groupoid is a groupoid in which both sets of objects and morphisms have topologies such that all groupoid structure maps are continuous. The notion of monodromy groupoid of a topological groupoid generalises those of fundamental groupoid and universal covering. It was earlier proved that the monodromy of a locally sectionable topological groupoid has a topological groupoid structure satisfying some properties. In this paper a similar problem is studied for compatible locally trivial topological groupoids.Comment: 9 pages, A

Covering morphisms of internal groupoids in the models of a semi-abelian theory

In this paper, for given an algebraic theory $T$ whose category $C$ of models is semi-abelian, we consider the topological models of $T$ called topological $T$-algebras and obtain some results related to the fundamental groups of topological $T$-algebras. We also deal with the internal groupoid structure in the category of models providing that the fundamental groupoid deduces a functor from topological $T$-algebras to the internal groupoids in $C$ and prove a criterion for the lifting of such an internal groupoid structure to the covering groupoids.Comment: 19 pages, research paper, LaTeX2e, xypi

Group-groupoid actions and liftings of crossed modules

The aim of this paper is to define the notion of lifting of a crossed module via a group morphism and give some properties of this type of the lifting. Further we obtain a criterion for a crossed module to have a lifting of crossed module. We also prove that the liftings of a certain crossed module constitute a category; and that this category is equivalent to the category of covers of that crossed module and hence to the category of group-groupoid actions of the corresponding groupoid to that crossed module.Comment: 18 pages, research paper, LaTeX2e, xypi

Crossed modules, double group-groupoids and crossed squares

In this paper using split extensions of group-groupoids we obtain the notion of crossed modules over group-grouoids which are also called 2-groups and we prove a categorical equivalence of these types of crossed modules and double group-groupoids which are internal to the category of group-groupoids. This equivalence enables us to produce more examples of double groupoids.Comment: 15 pages, research paper, LaTeX2e, xypi

Holonomy and monodromy groupoids

We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.Comment: 12 pages, LaTeX2E, xypic 3-rd Conference Geometry and Topology of Manifolds, Krynica 29.04.2001-5.05.2001 (Poland

Coverings of internal groupoids and crossed modules in the category of groups with operations

In this paper we prove some results on the covering morphisms of internal groupoids. We also give a result on the coverings of the crossed modules of groups with operations.Comment: 16 pages, research paper, LaTeX2e, xypic. arXiv admin note: text overlap with arXiv:1601.0709