PlanetPhysics/Groupoids

This is a contributed topic on groupoids and their applications.

Groupoids are a key concept in modern topology, and especially in algebraic topology; they may be considered as one of the simplest, special types of categories.

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 ()

Groupoids

Def.

1. a "magma: a set with a total binary operation" or
2. a "set with a partial binary operation that is associative and has inverses and identities"

is called a groupoid.

"A groupoid is a category in which every morphism is an isomorphism."

Topology

Def.

1. a "branch of mathematics studying those properties of a geometric figure or solid that are not changed by stretching, bending and similar homeomorphisms", or
2. a "collection τ of subsets of a set X such that the empty set and X are both members of τ and τ is closed under arbitrary unions and finite intersections"

is called a topology.

Def. a "branch of topology that associates objects from abstract algebra to topological spaces" is called an algebraic topology.

Brief Description

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 ${\mathsf {G}}$. 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.

Additional examples

These are 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 #higher homotopy groupoids and the higher dimensional, generalized Van Kampen theorems #Groupoid category
3. Crossed complexes
4. higher dimensional algebra (HDA)
5. Groupoid super-categories ($n$-categories, etc.)
6. Groupoid supercategories