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Homedouble groupoid with connection

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# 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 $u$,

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

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

- 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). (available here as PDF) , see 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.

## Mathematics Subject Classification

55U40*no label found*18E05

*no label found*18D05

*no label found*

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