PlanetPhysics/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 {\em a continuous map from the unit square of the real plane into a Hausdorff space which factors through a tree} ([1]).

A {\it 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).

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 .

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[1] [2] [3] [4] [5]

References

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  1. 1.0 1.1 R. Brown, K.A. Hardie, K.H. Kamps and T. Porter., A homotopy double groupoid of a Hausdorff space , {\it Theory and Applications of Categories} 10 ,(2002): 71-93.
  2. R. Brown and C.B. Spencer: Double groupoids and crossed modules, Cahiers Top. G\'eom.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.