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
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
![$ I :=[0,1]$](img8.png)
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

.
-
- 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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