scholarly journals A topological fibrewise fundamental groupoid

2015 ◽  
Vol 17 (2) ◽  
pp. 37-51 ◽  
Author(s):  
David Michael Roberts
Keyword(s):  
Fundamental Groupoid

scholarly journals The fundamental groupoid as a terminal costack

2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Ilia Pirashvili
Keyword(s):  
Topological Spaces ◽  
Fundamental Groupoid

AbstractIn this paper we prove that for good topological spaces the assignment

scholarly journals Extendibility, monodromy, and local triviality for topological groupoids

2001 ◽  
Vol 27 (3) ◽  
pp. 131-140 ◽  
Author(s):  
Osman Mucuk ◽  
İlhan İçen
Keyword(s):  
Universal Cover ◽  
Small Category ◽  
Topological Groupoid ◽  
Fundamental Groupoid ◽  
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 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.

Monoid Presentations and Associated Groupoids

1998 ◽  
Vol 08 (02) ◽  
pp. 141-152 ◽  
Author(s):  
N. D. Gilbert
Keyword(s):  
Fundamental Group ◽  
Group Completion ◽  
Monoid Presentation ◽  
Fundamental Groupoid ◽  
Monoid Structure

We consider properties of a 2-complex associated by Squier to a monoid presentation. We show that the fundamental groupoid admits a monoid structure, and we establish a relationship between its group completion and the fundamental group of the 2-complex. We also treat a modified complex, due to Pride, for monoid presentations of groups, and compute the structure of the fundamental groupoid in this setting.

On the Algebra of Semigroup Diagrams

1997 ◽  
Vol 07 (03) ◽  
pp. 313-338 ◽  
Author(s):  
Vesna Kilibarda
Keyword(s):  
Homotopy Theory ◽  
Structural Information ◽  
Geometric Method ◽  
Fundamental Groups ◽  
Free Products ◽  
Semigroup Presentation ◽  
Fundamental Groupoid ◽  
Low Dimensional ◽  
Semigroup Diagrams ◽  
Natural Way

In this work we enrich the geometric method of semigroup diagrams to study semigroup presentations. We introduce a process of reduction on semigroup diagrams which leads to a natural way of multiplying semigroup diagrams associated with a given semigroup presentation. With respect to this multiplication the set of reduced semigroup diagrams is a groupoid. The main result is that the groupoid [Formula: see text] of reduced semigroup diagrams over the presentation S = <X:R> may be identified with the fundamental groupoid γ (KS) of a certain 2-dimensional complex KS. Consequently, the vertex groups of the groupoid [Formula: see text] are isomorphic to the fundamental groups of the complex KS. The complex we discovered was first considered in the paper of Craig Squier, published only recently. Steven Pride has also independently defined a 2-dimensional complex isomorphic to KS in relation to his work on low-dimensional homotopy theory for monoids. Some structural information about the fundamental groups of the complex KS are presented. The class of these groups contains all finitely generated free groups and is closed under finite direct and finite free products. Many additional results on the structure of these groups may be found in the paper of Victor Guba and Mark Sapir.

scholarly journals Computational Paths and the Fundamental Groupoid of a Type

2021 ◽  
Author(s):  
Arthur F. Ramos ◽  
Ruy J. G. B. de Queiroz ◽  
Anjolina G. de Oliveira
Keyword(s):  
Category Theory ◽  
Fundamental Concept ◽  
Fundamental Groupoid

Using computational paths as the fundamental concept, we show that we can leverage Category Theory to propose the concept of fundamental groupoid of a type.

scholarly journals COSIMPLICIAL SPACES AND COCYCLES

2014 ◽  
Vol 15 (3) ◽  
pp. 445-470
Author(s):  
J. F. Jardine
Keyword(s):  
Cohomology Theory ◽  
Postnikov Tower ◽  
Fundamental Groupoid ◽  
Cosimplicial Space

Standard results from non-abelian cohomology theory specialize to a theory of torsors and stacks for cosimplicial groupoids. The space of global sections of the stack completion of a cosimplicial groupoid $G$ is weakly equivalent to the Bousfield–Kan total complex of $BG$ for all cosimplicial groupoids $G$. The $k$-invariants for the Postnikov tower of a cosimplicial space $X$ are naturally elements of stack cohomology for the stack associated to the fundamental groupoid ${\it\pi}(X)$ of $X$. Cocycle-theoretic ideas and techniques are used throughout the paper.

scholarly journals Groupoids and the algebra of rewriting in group presentations

2020 ◽  
Author(s):  
N. D. Gilbert ◽  
E. A. McDougall
Keyword(s):  
Reduced Form ◽  
Rewriting Systems ◽  
Rewriting System ◽  
Group Presentations ◽  
Rewrite Rules ◽  
Fundamental Groupoid

Abstract Presentations of groups by rewriting systems (that is, by monoid presentations), have been fruitfully studied by encoding the rewriting system in a 2-complex—the Squier complex—whose fundamental groupoid then describes the derivation of consequences of the rewrite rules. We describe a reduced form of the Squier complex, investigate the structure of its fundamental groupoid, and show that key properties of the presentation are still encoded in the reduced form.

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