%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: thin square
%%% Primary Category Code: 00.
%%% Filename: ThinSquare.tex
%%% Version: 3
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
Let us consider first the \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of a {\em 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 $X_H$ which factors through a tree}'' (\cite{BHKP}).
\begin{definition}
A {\it tree}, is defined here as the underlying space $ |K| $ of a
finite $ 1 $-connected $ 1 $-dimensional \htmladdnormallink{simplicial complex}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} $ K $ and
\htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} $ \partial{I}^{2} $ of $ I^{2} = I \times I $ (that is, a \emph{\htmladdnormallink{square}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html}} (interval) defined here as the Cartesian product of the unit interval $I :=[0,1]$ of real numbers).
\end{definition}
\begin{definition}
A \emph{square map} $ u:I^{2} \longrightarrow X $ in a \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} space $ X $ is \emph{thin} if there
is a factorisation of $ u $, $$ u : I^{2} \stackrel{\Phi_{u}}{\longrightarrow}
J_{u} \stackrel{p_{u}}{\longrightarrow} X, $$ where $J_{u}$ is a
\emph{tree} and $ \Phi_{u} $ is piecewise linear (PWL) on the
boundary $ \partial{I}^{2} $ of $ I^{2} $.
\end{definition}
\begin{thebibliography}{9}
\bibitem{BHKP}
R. Brown, K.A. Hardie, K.H. Kamps and T. Porter.,
\htmladdnormallink{A homotopy double groupoid of a Hausdorff space}{http://www.tac.mta.ca/tac/volumes/10/2/10-02.pdf} ,
{\it Theory and Applications of Categories} \textbf{10},(2002): 71-93.
\bibitem{BS1}
R. Brown and C.B. Spencer: Double groupoids and crossed modules, \emph{Cahiers Top. G\'eom.Diff.},
\textbf{17} (1976), 343--362.
\bibitem{BMos}
R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales--Bangor, Maths Preprint, 1986.
\bibitem{HKK}
K.A. Hardie, K.H. Kamps and R.W. Kieboom., A homotopy 2-groupoid of a Hausdorff
\emph{Applied Categorical Structures}, \textbf{8} (2000): 209-234.
\bibitem{Agl-Br-St2k2}
Al-Agl, F.A., Brown, R. and R. Steiner: 2002, Multiple categories: the equivalence of a globular and cubical approach, \emph{Adv. in Math}, \textbf{170}: 711-118.
\end{thebibliography}
\end{document}
