globular ω-groupoid

Definition 0.1.

An ω-groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath has a distinct meaning from that of ω-categoryMathworldPlanetmath, although certain authors restrict its definition to the latter by adding the restrictionPlanetmathPlanetmath of invertible morphismsMathworldPlanetmath, and thus also assimilate the ω-groupoid with the -groupoid. Ronald Brown and Higgins showed in 1981 that -groupoids and crossed complexes are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. Subsequently,in 1987, these authors introduced the tensor productsPlanetmathPlanetmath and homotopiesMathworldPlanetmathPlanetmath for ω-groupoids and crossed complexes. “It is because the geometry of convex sets is so much more complicated in dimensionsMathworldPlanetmathPlanetmath >1 than in dimension 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 ω-groupoids one needs to define first the n-globe Gn which is the subspaceMathworldPlanetmath of an EuclideanPlanetmathPlanetmath n-space Rn of points x such that that their norm ||x||1, but with the cell structureMathworldPlanetmath for n1 specified in SectionPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath 1 of R. Brown (2007). Also, one needs to consider a filtered space that is defined as a compactly generated space X and a sequencePlanetmathPlanetmath of subspaces X*. Then, the n-globe Gn has a skeletal filtrationPlanetmathPlanetmath giving a filtered space Gn*.

Thus, a fundamental globular ω-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:, 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 groupoidsPlanetmathPlanetmathPlanetmath defined over Hausdorff spaces that are non-AbelianMathworldPlanetmathPlanetmath (or non-commutative) are relevant to non-Abelian Algebraic Topology (NAAT) and (or NA-QAT).

Furthermore, whereas the definition of an n-groupoid is a straightforward generalizationPlanetmathPlanetmath of a 2-groupoid, the notion of a multiple groupoid is not at all an obvious generalization or extensionPlanetmathPlanetmathPlanetmath of the concept of double groupoid.


  • 1 Brown, R. and Higgins, P.J. (1981). The algebraPlanetmathPlanetmath of cubes. J. Pure Appl. Alg. 21 : 233–260.
  • 2 Brown, R. and Higgins, P. J. ColimitMathworldPlanetmath theorems for relative homotopy groups. J.Pure Appl. Algebra 22 (1) (1981) 11–41.
  • 3 Brown, R. and Higgins, P. J. The equivalence of -groupoids and crossed complexes. Cahiers Topologie Géom. Différentielle 22 (4) (1981) 371–386.
  • 4 Brown, R., Higgins, P. J. and R. Sivera,: 2011. “Non-Abelian Algebraic Topology”, EMS Publication. mas010/nonab-a-t.html ; mas010/nonab-t/partI010604.pdf
  • 5 Brown, R. and G. Janelidze: 2004. Galois theory and a new homotopy double groupoidPlanetmathPlanetmath of a map of spaces, Applied Categorical Structures 12: 63-80.
Title globular ω-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 Gn
Defines n-globe
Defines fundamental globular ω-groupoid of a filtered topological space