Definition 0.1.

A 2-groupoid is a 2-category whose morphismsMathworldPlanetmathPlanetmath are all invertible, that is, ones such that, each 1-arrow (morphism) is invertible with respect to the morphism composition.

Remark 0.1.

Note, however that ω-groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath has a distinct meaning from that of ω-category).

An important reason for studying 2–categories, and especially 2-groupoids, is to use them as coefficient objects for non-Abelian Cohomology theories. Thus, some double groupoidsPlanetmathPlanetmathPlanetmath defined over Hausdorff spaces that are non-AbelianMathworldPlanetmathPlanetmath (or non-commutative) are relevant to non-Abelian Algebraic Topology (NAAT) and http://planetphysics.org/?op=getobj&from=lec&id=61NAQAT (or NA-QAT).

One needs to distinguish between a 2-groupoid and a double-groupoid as the two concepts are very different. Interestingly, some double groupoids defined over Hausdorff spaces that are non-Abelian (or non-commutative) have true two-dimensional geometric representations with special properties that allow generalizationsPlanetmathPlanetmath of important theorems in algebraic topology and higher dimensional algebra, such as the generalized van Kampen theoremMathworldPlanetmath with significant consequences that cannot be obtained through AbelianMathworldPlanetmath means.

Furthermore, whereas the definition of an n-groupoid is a straightforward generalization of a 2-groupoid, the notion of a multiple groupoid is not at all an obvious generalization or extensionPlanetmathPlanetmathPlanetmath of the concept of double groupoid.

Title 2-groupoid
Canonical name 2groupoid
Date of creation 2013-03-22 19:21:09
Last modified on 2013-03-22 19:21:09
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 17
Author bci1 (20947)
Entry type Definition
Classification msc 55Q35
Classification msc 55Q05
Classification msc 20L05
Classification msc 18D05
Classification msc 18-00
Synonym 2-category with invertible morphisms
Defines 2-groupoid
Defines HDA
Defines higher dimensional algebra
Defines (m-1) arrows