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.
Definition 0.1
A map

where

and

are
(finite) simplicial complexes is
PWL (
piecewise linear) if
there exist subdivisions of

and

relative to which

is simplicial.
We briefly recall here the related concepts
involved:
Definition 0.2
A
square

in a
topological
space

is
thin if there
is a factorisation of

,
where

is a
tree and

is piecewise linear (PWL, as defined next) on the
boundary

of

.
Definition 0.3
A
tree, is defined here as the underlying space

of a
finite

-connected

-dimensional simplicial complex

boundary

of

.
- 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
-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.
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