GeometricallyDefinedDoubleGroupoidWithConnection 2009-05-01 00:56:23 2009-05-01 00:56:23 Definition double groupoid with connection double groupoid with connection % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % define commands here \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym} \usepackage{xypic} \usepackage[mathscr]{eucal} \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote%%@ }}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\GL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} \newcommand{\G}{\mathcal G} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathbb G}} \newcommand{\dgrp}{{\mathbb D}} \newcommand{\desp}{{\mathbb D^{\rm{es}}}} \newcommand{\Geod}{{\rm Geod}} \newcommand{\geod}{{\rm geod}} \newcommand{\hgr}{{\mathbb H}} \newcommand{\mgr}{{\mathbb M}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathbb G)}} \newcommand{\obgp}{{\rm Ob(\mathbb G')}} \newcommand{\obh}{{\rm Ob(\mathbb H)}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\ghomotop}{{\rho_2^{\square}}} \newcommand{\gcalp}{{\mathbb G(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\glob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\oset}[1]{\overset {#1}{\ra}} \newcommand{\osetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \subsection{Introduction} In the setting of a geometrically defined double groupoid with connection, as in \cite{BH2}, (resp. \cite{BHKP}), there is an appropriate notion of \emph{geometrically thin} square. It was proven in \cite{BH2}, (Theorem 5.2 (resp. \cite{BHKP}, Proposition 4)), that in the cases there specified \emph{geometrically and algebraically thin squares coincide}. \subsection{Geometrically defined double groupoid with connection} \subsubsection{Basic definitions} \begin{definition} A map $ \Phi : |K| \longrightarrow |L| $ where $ K $ and $ L $ are (finite) simplicial complexes is \emph{PWL} ({\it piecewise linear}) if there exist subdivisions of $ K $ and $ L $ relative to which $ \Phi$ is simplicial. \end{definition} \subsubsection{Remarks} We briefly recall here the related concepts involved: \begin{definition} A \emph{square} $ u:I^{2} \longrightarrow X $ in a topological 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, as defined next) on the boundary $ \partial{I}^{2} $ of $ I^{2} $. \end{definition} \begin{definition} A {\it tree}, is defined here as the underlying space $ |K| $ of a finite $ 1 $-connected $ 1 $-dimensional simplicial complex $ K $ boundary $ \partial{I}^{2} $ of $ I^{2} $. \end{definition} \begin{thebibliography}{9} \bibitem{BR2k6} Ronald Brown: Topology and Groupoids, BookSurge LLC (2006). \bibitem{BH2} Brown, R., and Hardy, J.P.L.:1976, Topological groupoids I: universal constructions, \emph{Math. Nachr.}, 71: 273-286. \bibitem{BHKP} Brown, R., Hardie, K., Kamps, H. and T. Porter: 2002, The homotopy double groupoid of a Hausdorff space., \emph{Theory and pplications of Categories} \textbf{10}, 71-93. \bibitem{BRHS} Ronald Brown R, P.J. Higgins, and R. Sivera.: {\em Non-Abelian algebraic topology},({\em in preparation}),(2008). \PMlinkexternal{(available here as PDF)}{http://www.bangor.ac.uk/~mas010/nonab-t/partI010604.pdf} , \PMlinkexternal{see also other available, relevant papers at this website}{http://www.bangor.ac.uk/~mas010/publicfull.htm}. \bibitem{BL87a} R. Brown and J.-L. Loday: Homotopical excision, and Hurewicz theorems, for $n$-cubes of spaces, \emph{Proc. London Math. Soc.}, 54:(3), 176-192,(1987). \bibitem{BL87b} R. Brown and J.-L. Loday: Van Kampen Theorems for diagrams of spaces, \emph{Topology}, 26: 311-337 (1987). \bibitem{BM86} R. Brown and G. H. Mosa: Double algebroids and crossed modules of algebroids, University of Wales-Bangor, Maths ({\em Preprint}), 1986. \bibitem{BS76} R. Brown and C.B. Spencer: Double groupoids and crossed modules, {\em Cahiers Top. G\'eom. Diff.}, 17 (1976), 343-362. \end{thebibliography}