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double groupoid with connection (Definition)

Introduction: Geometrically defined double groupoid with connection

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.

Basic definitions

Definition 0.1   A map $\Phi : \vert K\vert \longrightarrow \vert L\vert $ where $K $ and $L $ are (finite) simplicial complexes is PWL (piecewise linear) if there exist subdivisions of $K $ and $L $ relative to which $\Phi$ is simplicial.

Remarks

We briefly recall here the related concepts involved:

Definition 0.2   A square $u:I^{2} \longrightarrow X $ in a topological space $X $ is thin if there is a factorisation of $u $,

$\displaystyle u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow} J_{u} \stackrel{p_{u}}{\longrightarrow} X, $
where $J_{u}$ is a tree and $\Phi_{u} $ is piecewise linear (PWL, as defined next) on the boundary $\partial{I}^{2} $ of $I^{2} $.
Definition 0.3   A tree, is defined here as the underlying space $\vert K\vert $ of a finite $1 $-connected $1 $-dimensional simplicial complex $K $ boundary $\partial{I}^{2} $ of $I^{2} $.

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 $n$-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.



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Cross-references: boundary, tree, topological, concepts, simplicial, piecewise linear, simplicial complexes, thin squares, proposition, theorem, square, geometrically defined double groupoid with connection

This is version 1 of double groupoid with connection, born on 2009-01-31.
Object id is 451, canonical name is DoubleGroupoidWithConnection.
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Physics Classification: 02. (Mathematical methods in physics)
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