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

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