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 nonAbelian 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 oneobject category^{} to a manyobject category with grouplike 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^{} $\U0001d5a6$. 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.
2groupoids (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.

7.
Groupoid supercategories ($n$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  20130322 18:15:32 
Last modified on  20130322 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 