# 2-groupoid

###### Definition 0.1.

A 2-groupoid is a 2-category whose morphisms 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 $\omega$-groupoid has a distinct meaning from that of $\omega$-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 groupoids defined over Hausdorff spaces that are non-Abelian (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 generalizations of important theorems in algebraic topology and higher dimensional algebra, such as the generalized van Kampen theorem with significant consequences that cannot be obtained through Abelian 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 extension 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 Defines