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 [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 1.1.

Generally, the geometry of squares and their compositions lead to a common representation, or definition of a double groupoidPlanetmathPlanetmathPlanetmath in the following form:

𝒟= \xymatrix @=3pc S \ar@¡1ex¿ [r] ^s^1 \ar@¡-1ex¿ [r] _t^1 \ar@¡1ex¿ [d]^  t_2 \ar@¡-1ex¿ [d]_s_2 & H \ar[l] \ar@¡1ex¿ [d]^ t \ar@¡-1ex¿ [d]_s
V \ar[u] \ar@¡1ex¿ [r] ^s \ar@¡-1ex¿ [r] _t & M \ar[l] \ar[u]

where M is a set of ‘points’, H,V are ‘horizontal’ and ‘vertical’ groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and S is a set of ‘squares’ with two compositions.

The laws for a double groupoid are also defined, more generally, for any topological spaceMathworldPlanetmath 𝕋, and make it also describable as a groupoid internal to the category of groupoidsPlanetmathPlanetmath.

Definition 1.2.

A map Φ:|K||L| where K and L are (finite) simplicial complexesMathworldPlanetmath 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:I2X 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 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.


  • 1 Ronald Brown: Topology and Groupoids, BookSurge LLC (2006).
  • 2 Brown, R., and Hardy, J.P.L.:1976, Topological groupoidsPlanetmathPlanetmathPlanetmathPlanetmath I: universal constructions, Math. Nachr., 71: 273–286.
  • 3 Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoidPlanetmathPlanetmath of a Hausdorff space., Theory and pplications of CategoriesMathworldPlanetmath 10, 71–93.
  • 4 Ronald Brown R, P.J. Higgins, and R. Sivera.: Non-AbelianMathworldPlanetmathPlanetmath algebraic topology,(in preparation),(2008). mas010/nonab-t/partI010604.pdf(available here as PDF) , mas010/publicfull.htmsee 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 TheoremsMathworldPlanetmath 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
Canonical name DoubleGroupoidWithConnection
Date of creation 2013-03-22 19:19:40
Last modified on 2013-03-22 19:19:40
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 11
Author bci1 (20947)
Entry type Topic
Classification msc 55U40
Classification msc 18E05
Classification msc 18D05
Defines connection
Defines double groupoid