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.

Title	globular $\omega$-groupoid
Canonical name	Globularomegagroupoid
Date of creation	2013-03-22 19:21:02
Last modified on	2013-03-22 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 18-00
Defines	filtered space
Defines	$G^n$
Defines	$n$-globe
Defines	fundamental globular $\omega$-groupoid of a filtered topological space