%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: homotopy addition lemma and corollary
%%% Primary Category Code: 00.
%%% Filename: HamiltonianAlgebroid.tex
%%% Version: 16
%%% Owner: bci1
%%% Author(s): bci1
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\begin{document}
\subsection{Homotopy addition lemma}
\emph{Let $f: \boldsymbol{\rho}^\square(X) \to \mathsf D$ be a \htmladdnormallink{morphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of
double groupoids
with connection. If $\alpha \in {\boldsymbol{\rho}^\square_2}(X)$ is thin, then $f(\alpha)$
is thin.}
\subsubsection{Remarks}
The \htmladdnormallink{groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} ${\boldsymbol{\rho}^\square_2}(X)$ employed here is as defined by the \htmladdnormallink{cubically thin homotopy}{http://planetphysics.us/encyclopedia/CubicallyThinHomotopy2.html} on the set
$R^{\square}_2(X)$ of squares. Additional explanations of the data, including \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} such as path groupoid and \htmladdnormallink{homotopy}{http://planetphysics.us/encyclopedia/ThinEquivalence.html} \htmladdnormallink{double groupoid}{http://planetphysics.us/encyclopedia/WeakHomotopy.html} are provided in an
attachment.
\subsection{Corollary}
\emph{Let $u : I^3\to X$ be a singular cube in a Hausdorff space $X$.
Then by restricting $u$ to the faces of $I^3$ and taking the
corresponding elements in $\boldsymbol{\rho}^{\square}_2 (X)$, we obtain a
cube in $\boldsymbol{\rho}^{\square} (X)$ which is commutative by the Homotopy
addition lemma for $\boldsymbol{\rho}^{\square} (X)$ (\cite{BHKP}, \htmladdnormallink{proposition}{http://planetphysics.us/encyclopedia/Predicate.html} 5.5). Consequently, if $f : \boldsymbol{\rho}^{\square} (X)\to \mathsf{D}$ is
a morphism of
double groupoids with connections, any singular cube
in $X$ determines a
\htmladdnormallink{commutative {3-shell}}{http://www.math.purdue.edu/research/atopology/BrownR-Kamps-Porter/vkt7.txt} in $\mathsf{D}$.}
\begin{thebibliography}{9}
\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}
