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 specifiedgeometrically 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) ifthere 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 thereis 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 atree and $\\Phi_{u}$ is piecewise linear (PWL, as defined next) on theboundary $\\partial{I}^{2}$ of $I^{2}$ .

Definition 1.4.

A tree, is defined here as the underlying space $|K|$ of afinite $1$ -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 homotopydouble 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.