double groupoid with connection (original) (raw)
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 [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_.
1.2 Basic definitions
1.2.1 Double Groupoids
Definition 1.2.
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.
1.3 Remarks
We briefly recall here the related concepts involved:
Definition 1.3.
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 theboundary ∂I2 of I2.
Definition 1.4.
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
- 4 Ronald Brown R, P.J. Higgins, and R. Sivera.: Non-Abelian
- 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
- 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.