0.1 Introduction
In the setting of a geometrically defined double groupoid with connection, as in [2], (resp. [3]), there
is an appropriate notion of geometrically thin square. It was proven in [2], (theorem 5.2 (resp. [3],
proposition 4)), that in the cases there specified geometrically and algebraically thin squares
coincide.
0.2 Geometrically defined double groupoid with connection
0.2.1 Basic definitions
Definition 0.1. A map Φ : |K|→|L| where K and L are (finite) simplicial complexes is PWL
(piecewise linear) if there exist subdivisions of K and L relative to which Φ is simplicial.
0.2.2 Remarks
We briefly recall here the related concepts involved:
Definition 0.2. A square u : I2→X in a topological space X is thin if there is a factorisation
of u,
where Ju is a tree and Φu is piecewise linear (PWL, as defined next) on the boundary ∂I2
of I2.
Definition 0.3. A tree, is defined here as the underlying space |K| of a finite 1-connected
1-dimensional simplicial complex K boundary ∂I2 of I2.
References
[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 n-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.
