# 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 , (resp. ), there is an appropriate notion of geometrically thin square. It was proven in , (Theorem 5.2 (resp. , 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.
 $\mathcal{D}=\vbox{\xymatrix@=3pc {S \ar@<1ex> [r] ^{s^1} \ar@<-1ex> [r] _{t^1} \ar@<1ex> [d]^{\, t_2} \ar@<-1ex> [d]_{s_2} & H \ar[l] \ar@<1ex> [d]^{\,t} \ar@<-1ex> [d]_s \\ V \ar[u] \ar@<1ex> [r] ^s \ar@<-1ex> [r] _t & M \ar[l] \ar[u]}},$ (1.1)
###### 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 $u$,

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

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

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