double groupoid with connection
1 Double Groupoid with Connection
1.1 Introduction: Geometrically defined double groupoid with connection
In the setting of a geometrically defined double groupoid with connection, as in , (resp. ), there is an appropriate notion of geometrically thin square. It was proven in , (Theorem 5.2 (resp. , Proposition 4)), that in the cases there specified geometrically and algebraically thin squares coincide.
1.2 Basic definitions
1.2.1 Double Groupoids
where is a set of ‘points’, are ‘horizontal’ and ‘vertical’ groupoids, and is a set of ‘squares’ with two compositions.
A map where and are (finite) simplicial complexes is PWL (piecewise linear) if there exist subdivisions of and relative to which is simplicial.
We briefly recall here the related concepts involved:
A square in a topological space is thin if there is a factorisation of ,
where is a tree and is piecewise linear (PWL, as defined next) on the boundary of .
- 1 Ronald Brown: Topology and Groupoids, BookSurge LLC (2006).
- 2 Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, Math. Nachr., 71: 273–286.
- 3 Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., Theory and pplications of Categories 10, 71–93.
- 4 Ronald Brown R, P.J. Higgins, and R. Sivera.: Non-Abelian algebraic topology,(in preparation),(2008). http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf(available here as PDF) , http://www.bangor.ac.uk/ mas010/publicfull.htmsee also other available, relevant papers at this website.
- 5 R. Brown and J.–L. Loday: Homotopical excision, and Hurewicz theorems, for –cubes of spaces, Proc. London Math. Soc., 54:(3), 176–192,(1987).
- 6 R. Brown and J.–L. Loday: Van Kampen Theorems for diagrams of spaces, Topology, 26: 311–337 (1987).
- 7 R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths (Preprint), 1986.
- 8 R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom. Diff., 17 (1976), 343–362.
|Title||double groupoid with connection|
|Date of creation||2013-03-22 19:19:40|
|Last modified on||2013-03-22 19:19:40|
|Last modified by||bci1 (20947)|