0.1 Introduction

Several classes of groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and large groupoids shall be considered in this topic with pertinent examples that illustrate the construction of groupoids through several extensionsPlanetmathPlanetmathPlanetmath of the much simpler (and global) group symmetryPlanetmathPlanetmath to both higher order symmetries and dimensions, as well as internal (or local, partial) plus external symmetry. Considered as powerful tools for investigating both AbelianMathworldPlanetmath and non-Abelian structuresPlanetmathPlanetmath, groupoids are now essential for understanding topologyPlanetmathPlanetmath, and are one of the important–if not the most importantconcepts in algebraic topology ([1])

0.2 Groupoids and topology

Groupoids are generalizationsPlanetmathPlanetmath 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 categoryMathworldPlanetmath to a many-object category with group-like elements and all invertible morphismsMathworldPlanetmath. Another enrichment of the notion of a group–as in the case of topological groupsMathworldPlanetmath– is the concept of topological groupoidPlanetmathPlanetmathPlanetmathPlanetmath 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 groupoidsPlanetmathPlanetmathPlanetmath, cubic groupoids, …, groupoid categories, groupoid supercategoriesPlanetmathPlanetmath, 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. 1.

    2-groupoids (please see groupoid categories)

  2. 2.

    Double groupoids; homotopy double groupoidPlanetmathPlanetmath of a Hausdorff space

  3. 3.

    Higher homotopy groupoids and the higher dimensional, generalized van Kampen theoremsMathworldPlanetmath

  4. 4.

    Groupoid category

  5. 5.

    Crossed complexes

  6. 6.
  7. 7.

    Groupoid super-categories (n-categories, etc.)

  8. 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