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Let us consider first the concept of a tree that enters in the definition of a thin square. Thus, a simplified notion of thin square is that of ``a continuous map from the unit square of the real plane into a Hausdorff space $X_H$ which factors through a tree'' ([1]).
Definition 0.2 A square map $ 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) on the boundary $ \partial{I}^{2} $ of $ I^{2} $ .
- 1
- R. Brown, K.A. Hardie, K.H. Kamps and T. Porter., A homotopy double groupoid of a Hausdorff space , Theory and Applications of Categories 10,(2002): 71-93.
- 2
- R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom.Diff., 17 (1976), 343-362.
- 3
- R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths Preprint, 1986.
- 4
- K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff Applied Categorical Structures, 8 (2000): 209-234.
- 5
- Al-Agl, F.A., Brown, R. and R. Steiner: 2002, Multiple categories: the equivalence of a globular and cubical approach, Adv. in Math, 170: 711-118.
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"thin square" is owned by bci1.
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See Also: copula
| Other names: |
AlgebraicallyThinSquares |
| Also defines: |
tree, square |
| Keywords: |
thin square, tree, square interval |
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Cross-references: thin, topological space, map, real numbers, unit, Cartesian product, interval, boundary, simplicial complex, finite
There are 60 references to this entry.
This is version 18 of thin square, born on 2008-09-04, modified 2009-02-02.
Object id is 10988, canonical name is ThinSquare.
Accessed 1878 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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