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geometrically defined double groupoid with connection
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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.
We briefly recall here the related concepts involved:
Definition 0.2 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 0.3 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} $ .
- 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.
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Cross-references: boundary, tree, topological space, subdivisions, piecewise, simplicial complexes, finite, map, thin squares, proposition, theorem, square, thin
There is 1 reference to this entry.
This is version 30 of geometrically defined double groupoid with connection, born on 2008-07-20, modified 2009-02-01.
Object id is 10839, canonical name is GeometricallyAndorAlgebraicallyThinSquares.
Accessed 1125 times total.
Classification:
| AMS MSC: | 55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory) | | | 55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories) | | | 55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology) | | | 18D05 (Category theory; homological algebra :: Categories with structure :: Double categories, $2$-categories, bicategories and generalizations) |
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Pending Errata and Addenda
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