Local Group-Groupoids (original) (raw)
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Topological Group-Groupoids and Equivalent Categories
Yüzüncü Yıl Üniversitesi Fen Bilimleri Enstitüsü Dergisi
The concept of groupoid was offered by Brandt (1926). The structure of the topological groupoid was given by Ehresmann (1958). A groupoid action is a significant appliance in algebraic topology offered by Ehresmann. Another algebraic notion is a covering given by Brown (1988). The topological group-groupoids (Γ-groupoid) were first put forward by Icen & Ozcan (2001). The definition of coverings of topological Γ groupoid and actions of topological Γ-groupoid were also presented by Icen et al. (2005). In this paper, we are going to create a category TΓGpdCov(Γ) of covering morphisms of TΓ-groupoid and a category TΓGpdOp(Γ) of actions of TΓ-groupoid. We will then prove that these categories are equivalent.
Extendibility, monodromy, and local triviality for topological groupoids
International Journal of Mathematics and Mathematical Sciences, 2001
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 maps of groupoid structure are continuous. The notion of monodromy groupoid of a topological groupoid generalizes those of fundamental groupoid and universal cover. It was earlier proved that the monodromy groupoid of a locally sectionable topological groupoid has the structure of a topological groupoid satisfying some properties. In this paper a similar problem is studied for compatible locally trivial topological groupoids.
On topological groupoids and multiple identities
2009
This paper studies some properties of (n, m)-homogeneous isotopies of medial topological groupoids. It also examines the relationship between paramediality and associativity. We extended some affirmations of the theory of topological groups on the class of topological (n, m)-homogeneous primitive goupoids with divisions. Mathematics subject classification: 20N15.
2023
The primary goal of this paper is to provide a comprehensive presentation of the fundamental groupoid, along with the necessary category theory required to define it and prove its fundamental properties. This will lead to the proof of the Seifert Van Kampen theorem for fundamental groupoids. From there, we will discuss basic notions of 2-category theory: this will allow us to explore the categorical properties of the fundamental groupoid when viewed as a costack over the category of 2-groupoids. We will show that, for a “nice" class of topological space, the fundamental groupoid is a terminal object in this category: this provides a purely categorical description of the fundamental groupoid. The focal point of this project is the discussion of this result, originally formulated and proved by Ilia Pirashvili in 2015. It serves as the main subject of investigation, offering valuable insights that will inform our concluding remarks and motivate questions for further research.
On weaker forms for concepts in theory of topological groupoids
Journal of the Egyptian Mathematical Society, 2013
In this paper, we investigate the topologically weak concepts of topological groupoids by giving the concepts of a-topological groupoid and a-topological subgroupoid. Furthermore, we show the role of the density condition to allow a-topological subgroupoid inherited properties from a-topological groupoid and the irresoluteness property for the structure maps in a-topological groupoid is studied. We also give some results about the fibers of a-topological groupoids.
Covering groupoids of categorical rings
Filomat, 2015
A categorical group is a kind of categorization of group and similarly a categorical ring is a categorization of ring. For a topological group X, the fundamental groupoid ?X is a group object in the category of groupoids, which is also called in literature group-groupoid or 2-group. If X is a path connected topological group which has a simply connected cover, then the category of covering groups of X and the category of covering groupoids of ?X are equivalent. Recently it was proved that if (X, x0) is an H-group, then the fundamental groupoid ?X is a categorical group and the category of the covering spaces of (X, x0) is equivalent to the category of covering groupoids of the categorical group ?X. The purpose of this paper is to present similar results for rings and categorical rings.
A note on topological semigroup-groupoid
2013
In this paper we prove that the set of homotopy classes of paths in topological semigroup is a semigroup-groupoid. Further, we define the category TSGCov/X of topological semigroup coverings of X and prove that its equivalent to the category SGpGpdCov/ of covering groupoids of the semigroup-groupoid . We also prove that the topological semigroup structure of a topological semigroup-groupoid lifts to a universal topological covering groupoid.
Actions of Double Group-groupoids and Covering Morphism
GAZI UNIVERSITY JOURNAL OF SCIENCE, 2020
• Covering morphism of double groupoids derived by action double groupoid was considered. • Action double group-groupoid on a group-groupoid was characterized. • Covering morphism of double group-groupoids was obtained. • A categorical equivalence was proved.