thin square

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 0.1   A tree, is defined here as the underlying space $ \vert K\vert $ of a finite $ 1 $ -connected $ 1 $ -dimensional simplicial complex $ K $ and boundary $ \partial{I}^{2} $ of $ I^{2} = I \times I $ (that is, a square (interval) defined here as the Cartesian product of the unit interval $ I :=[0,1]$ of real numbers).

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 $ ,

$\displaystyle 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} $ .

Bibliography

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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