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:

\mathcal{D}=\vbox{\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]}},

(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 $\Phi:|K|\longrightarrow|L|$ 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.

1.3 Remarks

We briefly recall here the related concepts involved:

Definition 1.3.

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

u:I^{2}\lx@stackrel{{\scriptstyle\Phi_{u}}}{{\longrightarrow}}J_{u}% \lx@stackrel{{\scriptstyle 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 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