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 XH which factors through a tree” ([1]).
Definition 0.1. A tree, is defined here as the underlying space |K| of a finite 1-connected
1-dimensional simplicial complex K and boundary ∂I2 of I2 = I × 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 : I2→X in a topological space X is thin if there is a
factorisation of u,
where Ju is a tree and Φu is piecewise linear (PWL) on the boundary ∂I2 of I2.
References
[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.
