119 research outputs found
Coverings and crossed modules of topological groups with operations
It is a well known result in the covering groups that a subgroup 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 . 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, cat-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-groups with operations for a certain algebraic category; and then by
this equivalence we determine normal and quotient objects in the category of
cat-groups with operations. Finally we characterize the coverings of
cat-groups with operations.Comment: 30 pages, research paper, LaTeX2e, xypi
Covering groupoids of categorical groups
If is a topological group, then its fundamental groupoid is a
group-groupoid which is a group object in the category of groupoids. Further if
is a path connected topological group which has a simply connected cover,
then the category of covering spaces of and the category of covering
groupoids of are equivalent. In this paper we prove that if
is an -group, then the fundamental groupoid is a categorical group.
This enable us to prove that the category of the covering spaces of an
-group is equivalent to the category of covering groupoid of the
categorical group .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 whose category of models
is semi-abelian, we consider the topological models of called topological
-algebras and obtain some results related to the fundamental groups of
topological -algebras. We also deal with the internal groupoid structure in
the category of models providing that the fundamental groupoid deduces a
functor from topological -algebras to the internal groupoids in 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
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