globular $\omega $groupoid
Definition 0.1.
An $\omega $groupoid^{} has a distinct meaning from that of $\omega $category^{}, although certain authors restrict its definition to the latter by adding the restriction^{} of invertible morphisms^{}, and thus also assimilate the $\omega $groupoid with the $\mathrm{\infty}$groupoid. Ronald Brown and Higgins showed in 1981 that $\mathrm{\infty}$groupoids and crossed complexes are equivalent^{}. Subsequently,in 1987, these authors introduced the tensor products^{} and homotopies^{} for $\omega $groupoids and crossed complexes. “It is because the geometry of convex sets is so much more complicated in dimensions^{} $\mathrm{>}\mathrm{1}$ than in dimension $\mathrm{1}$ that new complications emerge for the theories of higher order group theory and of higher homotopy groupoids.”
However, in order to introduce a precise and useful definition of globular $\omega $groupoids one needs to define first the $n$globe ${G}^{n}$ which is the subspace^{} of an Euclidean^{} $n$space ${R}^{n}$ of points $x$ such that that their norm $x\le 1$, but with the cell structure^{} for $n\ge 1$ specified in Section^{} 1 of R. Brown (2007). Also, one needs to consider a filtered space that is defined as a compactly generated space ${X}_{\mathrm{\infty}}$ and a sequence^{} of subspaces ${X}_{*}$. Then, the $n$globe ${G}^{n}$ has a skeletal filtration^{} giving a filtered space $G^{n}{}_{*}$.
Thus, a fundamental globular $\omega $groupoid of a filtered (topological) space is defined by using an $n$globe with its skeletal filtration (R. Brown, 2007 available from: arXiv:math/0702677v1 [math.AT]). This is analogous to the fundamental cubical omega–groupoid of Ronald Brown and Philip Higgins (1981ac) that relates the construction to the fundamental crossed complex of a filtered space. Thus, as shown in R. Brown (2007: http://arxiv.org/abs/math/0702677), the crossed complex associated to the free globular omegagroupoid on one element of dimension $n$ is the fundamental crossed complex of the $n$globe.
more to come… entry in progress
Remark 0.1.
An important reason for studying $n$–categories, and especially $n$groupoids, is to use them as coefficient objects for nonAbelian Cohomology theories. Thus, some double groupoids^{} defined over Hausdorff spaces that are nonAbelian^{} (or noncommutative) are relevant to nonAbelian Algebraic Topology (NAAT) and http://planetphysics.org/?op=getobj&from=lec&id=61NAQAT (or NAQAT).
Furthermore, whereas the definition of an $n$groupoid is a straightforward generalization^{} of a 2groupoid, the notion of a multiple groupoid is not at all an obvious generalization or extension^{} of the concept of double groupoid.
References
 1 Brown, R. and Higgins, P.J. (1981). The algebra^{} of cubes. J. Pure Appl. Alg. 21 : 233–260.
 2 Brown, R. and Higgins, P. J. Colimit^{} theorems for relative homotopy groups. J.Pure Appl. Algebra 22 (1) (1981) 11–41.
 3 Brown, R. and Higgins, P. J. The equivalence of $\mathrm{\infty}$groupoids and crossed complexes. Cahiers Topologie G$\mathrm{\xe9}$om. Diff$\mathrm{\xe9}$rentielle 22 (4) (1981) 371–386.

4
Brown, R., Higgins, P. J. and R. Sivera,: 2011. “NonAbelian Algebraic Topology”, EMS Publication.
http://www.bangor.ac.uk/ mas010/nonabat.html ;
http://www.bangor.ac.uk/ mas010/nonabt/partI010604.pdf  5 Brown, R. and G. Janelidze: 2004. Galois theory and a new homotopy double groupoid^{} of a map of spaces, Applied Categorical Structures 12: 6380.
Title  globular $\omega $groupoid 

Canonical name  Globularomegagroupoid 
Date of creation  20130322 19:21:02 
Last modified on  20130322 19:21:02 
Owner  bci1 (20947) 
Last modified by  bci1 (20947) 
Numerical id  42 
Author  bci1 (20947) 
Entry type  Definition 
Classification  msc 55Q35 
Classification  msc 55Q05 
Classification  msc 20L05 
Classification  msc 18D05 
Classification  msc 1800 
Defines  filtered space 
Defines  ${G}^{n}$ 
Defines  $n$globe 
Defines  fundamental globular $\omega $groupoid of a filtered topological space 