Talk:PlanetPhysics/Category of Groupoids 2

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: category of groupoids
%%% Primary Category Code: 00.
%%% Filename: CategoryOfGroupoids2.tex
%%% Version: 6
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}



\begin{document}

 \section{Category of Groupoids}

\subsection{Properties}
The category of groupoids, $G_{pd}$, has several important properties not available for \htmladdnormallink{groups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, although it does contain the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of groups as a full subcategory. One such important property is that $Gpd$ is \emph{cartesian closed}. Thus, if $J$ and $K$ are two \htmladdnormallink{groupoids}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}, one can form a groupoid $GPD(J,K)$ such that if $G$ also is a groupoid then there exists a \htmladdnormallink{natural equivalence}{http://planetphysics.us/encyclopedia/IsomorphismClass.html} $$Gpd(G \times J, K) \rightarrow Gpd(G, GPD(J,K))$$.


Other important properties of $G_{pd}$ are:

\begin{enumerate}

\item The category $G_{pd}$ also has a unit interval \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $I$, which is the groupoid with two objects $0,1$ and exactly one arrow $0 \rightarrow 1$;

\item The groupoid $I$ has allowed the development of a useful
\htmladdnormallink{Homotopy Theory}{http://planetmath.org/encyclopedia/HomotopyCategory2.html} for groupoids that leads to analogies between groupoids and spaces or \htmladdnormallink{manifolds}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html}; effectively, groupoids may be viewed as ``adding the spatial notion of a `place' or location'' to that of a group. In this context, the \htmladdnormallink{homotopy category}{http://planetphysics.us/encyclopedia/HomotopyCategory.html} plays an important role;

\item Groupoids extend the notion of invertible \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} by comparison with that available for groups; such invertible operations also occur in the theory of inverse \htmladdnormallink{semigroups}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. Moreover, there are interesting \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} beteen inverse semigroups and ordered groupoids. Such \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} are thus applicable to \htmladdnormallink{sequential machines}{http://planetphysics.us/encyclopedia/AAT.html} and automata whose \htmladdnormallink{state spaces}{http://planetphysics.us/encyclopedia/StableAutomaton.html} are semigroups. Interestingly, the category of finite automata, just like $G_{pd}$ is also \emph{cartesian closed};

\item The category $G_{pd}$ has a variety of \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, such as: quotient morphisms, \htmladdnormallink{retractions}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html}, \htmladdnormallink{covering}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} morphisms, fibrations, universal morphisms, (in contrast to only the \htmladdnormallink{epimorphisms}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} and \htmladdnormallink{monomorphisms}{http://planetphysics.us/encyclopedia/InjectiveMap.html} of group theory);

\item A \htmladdnormallink{monoid}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} object, $END(J)= GPD(J,J)$, also exists in the category of groupoids, that contains a maximal subgroup object denoted here as $AUT(J)$. Regarded as a group object in the category groupoids, $AUT(J)$ is equivalent to a \htmladdnormallink{crossed module}{http://planetphysics.us/encyclopedia/CubicalHigherHomotopyGroupoid.html} $C_M$, which in the case when $J$ is a group is the traditional crossed module $J\rightarrow Aut(J)$, defined by the inner automorphisms.

\end{enumerate}

\begin{thebibliography} {9}

\bibitem{MJP1999}
May, J.P. 1999, \emph{A Concise Course in Algebraic Topology.}, The University of Chicago Press: Chicago

\bibitem{BR-JG2k4}
R. Brown and G. Janelidze.(2004). Galois theory and a new homotopy double groupoid of a map of spaces.(2004).
{\em Applied Categorical Structures},\textbf{12}: 63-80. Pdf file in arxiv: math.AT/0208211

\bibitem{PJH71}
P. J. Higgins. 1971. \emph{Categories and Groupoids.}, Originally published by: Van Nostrand Reinhold, 1971. Republished in: \emph{Reprints in Theory and Applications of Categories}, No. 7 (2005) pp 1-195:
http://www.tac.mta.ca/tac/reprints/articles/7/tr7.pdf


\end{thebibliography} 

\end{document}