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 which factors through a tree” ([1]).

Definition 0.1   A tree, is defined here as the underlying space of a finite -connected -dimensional simplicial complex and boundary of (that is, a square (interval) defined here as the Cartesian product of the unit interval of real numbers).

Definition 0.2   A square map in a topological space is thin if there is a factorisation of ,

where is a tree and is piecewise linear (PWL) on the boundary of .

Bibliography

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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.

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R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. Géom.Diff., 17 (1976), 343–362.

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R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales–Bangor, Maths Preprint, 1986.

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K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff Applied Categorical Structures, 8 (2000): 209-234.

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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.