# 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.: ,(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 DoubleGroupoidWithConnection 2013-03-22 19:19:40 2013-03-22 19:19:40 bci1 (20947) bci1 (20947) 11 bci1 (20947) Topic msc 55U40 msc 18E05 msc 18D05 connection double groupoid