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 (1981a-c) 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 omega-groupoid 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 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).

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.