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.