An -groupoid has a distinct meaning from that of -category, although certain authors restrict its definition to the latter by adding the restriction of invertible morphisms, and thus also assimilate the -groupoid with the -groupoid. Ronald Brown and Higgins showed in 1981 that -groupoids and crossed complexes are equivalent. Subsequently,in 1987, these authors introduced the tensor products and homotopies for -groupoids and crossed complexes. “It is because the geometry of convex sets is so much more complicated in dimensions than in dimension 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 -groupoids one needs to define first the -globe which is the subspace of an Euclidean -space of points such that that their norm , but with the cell structure for specified in Section 1 of R. Brown (2007). Also, one needs to consider a filtered space that is defined as a compactly generated space and a sequence of subspaces . Then, the -globe has a skeletal filtration giving a filtered space .
Thus, a fundamental globular -groupoid of a filtered (topological) space is defined by using an -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 is the fundamental crossed complex of the -globe.
more to come… entry in progress
An important reason for studying –categories, and especially -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).
- 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 -groupoids and crossed complexes. Cahiers Topologie Gom. Diffrentielle 22 (4) (1981) 371–386.
Brown, R., Higgins, P. J. and R. Sivera,: 2011. “Non-Abelian Algebraic Topology”, EMS Publication.
http://www.bangor.ac.uk/ mas010/nonab-a-t.html ;
- 5 Brown, R. and G. Janelidze: 2004. Galois theory and a new homotopy double groupoid of a map of spaces, Applied Categorical Structures 12: 63-80.
|Date of creation||2013-03-22 19:21:02|
|Last modified on||2013-03-22 19:21:02|
|Last modified by||bci1 (20947)|
|Defines||fundamental globular -groupoid of a filtered topological space|