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 groupoid^{} in the following form:
$$ | (1.1) |
where $M$ is a set of ‘points’, $H,V$ are ‘horizontal’ and ‘vertical’ groupoids^{}, and $S$ is a set of ‘squares’ with two compositions.
The laws for a double groupoid are also defined, more generally, for any topological space^{} $\mathbb{T}$, and make it also describable as a groupoid internal to the category of groupoids^{}.
Definition 1.2.
A map $\mathrm{\Phi}:|K|\u27f6|L|$ where $K$ and $L$ are (finite) simplicial complexes^{} is PWL (piecewise linear) if there exist subdivisions of $K$ and $L$ relative to which $\mathrm{\Phi}$ is simplicial.
1.3 Remarks
We briefly recall here the related concepts involved:
Definition 1.3.
A square $u:{I}^{2}\u27f6X$ in a topological space $X$ is thin if there is a factorisation of $u$,
$$u:{I}^{2}\stackrel{{\mathrm{\Phi}}_{u}}{\u27f6}{J}_{u}\stackrel{{p}_{u}}{\u27f6}X,$$ |
where ${J}_{u}$ is a tree and ${\mathrm{\Phi}}_{u}$ is piecewise linear (PWL, as defined next) on the boundary $\partial {I}^{2}$ of ${I}^{2}$.
Definition 1.4.
A tree, is defined here as the underlying space $|K|$ of a finite $1$-connected $1$-dimensional simplicial complex $K$ boundary $\partial {I}^{2}$ of ${I}^{2}$.
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). http://www.bangor.ac.uk/ mas010/nonab-t/partI010604.pdf(available here as PDF) , http://www.bangor.ac.uk/ 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 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.
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 |