groupoids

groupoids

0.1

Introduction

Several classes of

groupoids

and large groupoids shall be considered in this topic with pertinent examples that illustrate the construction of groupoids through several

extensions

of the much simpler (and global) group

symmetry

to both higher order symmetries and

dimensions

, as well as internal (or local, partial) plus external symmetry. Considered as powerful tools for investigating both

Abelian

and

non-Abelian structures

, groupoids are now essential for understanding

topology

, and are one of the important–

if not the most important

–

concepts

in

algebraic topology

(

[

1

]

)

0.2

Groupoids and topology

Groupoids are

generalizations

or extensions of the concept of group, supergroup, ‘

virtual group

’, and paragroup, in several ways; one may simply extend the notion of a group viewed as an one-object

category

to a

many-object category with group-like

elements

and all

invertible

morphisms

. Another enrichment of the notion of a group–as in the case of

topological groups

– is the concept of

topological groupoid

𝖦

. One can also think of a groupoid as a class of linked groups, and further extend the latter groupoid

definition

to higher dimensions through ‘geometric’-algebraic constructions, for example, to

double groupoids

, cubic groupoids, …, groupoid categories, groupoid

supercategories

, and so on. Crossed modules of groups and crossed complexes also correspond to such extended groupoids.

For precise definitions of specific classes of groupoids, see also groupoid and topological groupoid definitions,
as well as those entries listed next as examples.

0.3

Additional examples

of major classes of groupoids defining the several extensions and enrichment possibilities of the notions of group and group symmetry introduced in the above definition are the subject of several other entries:

1.

2-groupoids

(please see

groupoid categories

)

2.

Double groupoids;

homotopy double groupoid

of a Hausdorff space

3.

Higher homotopy groupoids

and the higher dimensional, generalized

van Kampen theorems

4.

Groupoid category

5.

Crossed complexes

6.

Higher dimensional algebra

(HDA)

7.

Groupoid

super-categories

(

n

-categories, etc.)

8.

Groupoid supercategories

References

1

R. Brown. 2006.

Topology and Groupoids

. Booksurge PLC.

2

R. Brown. 2008.

Nonabelian Algebraic Topology

.

preprint

, (two volumes).

Title

groupoids

Canonical

name

Groupoids

Date of creation

2013-03-22 18:15:32

Last modified on

2013-03-22 18:15:32

Owner

bci1 (20947)

Last modified by

bci1 (20947)

Numerical id

43

Author

bci1 (20947)

Entry type

Topic

Classification

msc 55U05

Classification

msc 55U35

Classification

msc 55U40

Classification

msc 18G55

Classification

msc 18B40

Synonym

groupoid categories

Synonym

topological groupoids

Synonym

supergroups

Related topic

Groupoid

Related topic

GroupoidCategory

Related topic

GroupoidHomomorphisms

Related topic

HomotopyDoubleGroupoidOfAHausdorffSpace

Related topic

TopologicalGroupoid

Related topic

QuantumGroups

Related topic

GeneralizedVanKampenTheoremsHigherDimensional

Related topic

EquivalentRepresentationsOfGroupoids

Related topic

C_cG

Related topic

GroupoidAndGroupRepresentationsRelate

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