locally compact groupoidsLocallyCompactGroupoids2008-08-04 03:56:422009-02-01 21:46:48Topiclocally compact groupoidgroupoid as a small categorytopological groupoidanalytic groupoidlocally compact topological groupsecond countable locally compact groupoidlocally compact groupoidssecond countable topological spacelocally compact Hausdorff spacetopological groupoidscontinuous inversion maps% this is the default PlanetMath preamble. as your knowledge
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\def\D{\mathsf{D}}\section{Locally compact groupoids}
This is a specific topic entry defining the basics of locally compact groupoids and related concepts.
Let us first recall the related concepts of groupoid and \emph{topological groupoid},
together with the appropriate notations needed to define a \emph{locally compact groupoid}.
\subsubsection{Groupoids and topological groupoids: categorical definitions}
Recall that a groupoid $\grp$ is a small category with inverses
over its set of objects $X = Ob(\grp)$~. One writes $\grp^y_x$ for
the set of morphisms in $\grp$ from $x$ to $y$~.
\emph{A topological groupoid} consists of a space $\grp$, a distinguished subspace
$\grp^{(0)} = \obg \subset \grp$, called {\it the space of objects} of $\grp$,
together with maps
\begin{equation}
r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} }
\end{equation}
called the {\it range} and {\it source maps} respectively,
together with a law of composition
\begin{equation}
\circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{
~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) =
r(\gamma_2)~ \}~ \lra ~\grp~,
\end{equation}
such that the following hold~:~
\begin{enumerate}
\item[(1)]
$s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ
\gamma_2) = r(\gamma_1)$~, for all $(\gamma_1, \gamma_2) \in
\grp^{(2)}$~.
\item[(2)]
$s(x) = r(x) = x$~, for all $x \in \grp^{(0)}$~.
\item[(3)]
$\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma =
\gamma$~, for all $\gamma \in \grp$~.
\item[(4)]
$(\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ
(\gamma_2 \circ \gamma_3)$~.
\item[(5)]
Each $\gamma$ has a two--sided inverse $\gamma^{-1}$ with $\gamma
\gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)$~.
Furthermore, only for topological groupoids the inverse map needs be continuous.
It is usual to call $\grp^{(0)} = Ob(\grp)$ {\it the set of objects}
of $\grp$~. For $u \in Ob(\grp)$, the set of arrows $u \lra u$ forms a
group $\grp_u$, called the \emph{isotropy group of $\grp$ at $u$}.
\end{enumerate}
Thus, as is well kown, a topological groupoid is just a groupoid internal to the
\PMlinkname{category of topological spaces and continuous maps}{CategoryOfPointedTopologicalSpaces}. The notion of internal groupoid has proved significant in a number of fields, since groupoids generalize bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to ref. \cite{Brown2006}.
\subsection{Locally compact and analytic groupoids}
\begin{definition}
A \emph{locally compact groupoid} $\grp_{lc}$ is defined as a groupoid that has also the topological structure of a second countable, \PMlinkname{locally compact Hausdorff space}{LocallyCompactHausdorffSpace}, and if the product and also inversion maps are continuous. Moreover, each $\grp_{lc}^u$ as well as the unit space $\grp_{lc}^0$ is closed in $\grp_{lc}$.
\end{definition}
\begin{remark}
The locally compact Hausdorff second countable spaces are {\em analytic}.
One can therefore say also that $\grp_{lc}$ is analytic.
When the groupoid $\grp_{lc}$ has only one object in its object space, that is, when it becomes a group, the above
definition is restricted to that of a \emph{locally compact topological group}; it is then a special case of a one-object category with all of its morphisms being invertible, that is also endowed with a locally compact, topological structure.
\end{remark}
\begin{thebibliography}{9}
\bibitem{Brown2006}
R. Brown. (2006). \emph{Topology and Groupoids}. BookSurgeLLC
\end{thebibliography}