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%%% Primary Title: cubically thin homotopy
%%% Primary Category Code: 00.
%%% Filename: CubicallyThinHomotopy2.tex
%%% Version: 1
%%% Owner: bci1
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\begin{document}
\subsection{Cubically thin homotopy}
Let $u,u'$ be \htmladdnormallink{squares}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} in $X$ with common vertices.
\begin{enumerate}
\item A {\it cubically thin homotopy} $U:u\equiv^{\square}_T u'$
between $u$ and $u'$ is a cube $U\in R^{\square}_3(X)$ such that
\begin{itemize}
\item $U$ is a \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} between $u$ and $u',$
\begin{center}
i.e. $\partial^{-}_1 (U)=u,\enskip \partial^{+}_1 (U)=u',$\end{center}
\item $U$ is rel. vertices of $I^2,$
\begin{center}
i.e. $\partial^{-}_2\partial^{-}_2 (U),\enskip\partial^{-}_2
\partial^{+}_2 (U),\enskip
\partial^{+}_2\partial^{-}_2 (U),\enskip\partial^{+}_2
\partial^{+}_2 (U)$ are
constant,\end{center}
\item the faces $ \partial^{\alpha}_{i} (U) $ are thin for $ \alpha =
\pm 1, \ i = 1,2 $.
\end{itemize}
\item The square $u$ is {\it cubically} $T$-{\it equivalent} to
$u',$ denoted $u\equiv^{\square}_T u'$ if there is a cubically
thin homotopy between $u$ and $u'.$
\end{enumerate}
This definition enables one to construct $\boldsymbol{\rho}^{\square}_2 (X)$ , by defining a
\htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} of cubically thin homotopy on the set $R^{\square}_2(X)$ of squares.
\begin{thebibliography}{9}
\bibitem{HKK}
K.A. Hardie, K.H. Kamps and R.W. Kieboom, A homotopy 2-groupoid of a Hausdorff space,
\emph{Applied Cat. Structures}, \textbf{8} (2000): 209-234.
\bibitem{BHKP}
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} \textbf{10},(2002): 71-93.
\end{thebibliography}
\end{document}
