Introduction

Geometrically defined double groupoid with connection

Basic definitions

Definition 0.1   A map where and are (finite) simplicial complexes is PWL (piecewise linear) if there exist subdivisions of and relative to which is simplicial.

Remarks

We briefly recall here the related concepts involved:

Definition 0.2   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 .

Definition 0.3   A tree, is defined here as the underlying space of a finite -connected -dimensional simplicial complex boundary of .

Bibliography

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). (available here as PDF) , see 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.