Talk:PlanetPhysics/Locally Compact Groupoid

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\begin{document}

 \begin{definition}
A \emph{locally compact groupoid} $\grp_{lc}$ is defined as a groupoid that has also the \htmladdnormallink{topological structure}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of a second countable, \htmladdnormallink{locally compact Hausdorff space}{http://planetphysics.us/encyclopedia/LocallyCompactHausdorffSpaces.html}, 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}

\textbf{Remarks:}
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 \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in its object space, that is, when it becomes a \htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}, the above definition is restricted to that of a
\emph{locally compact \htmladdnormallink{topological group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}}; it is then a special case of a one-object \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} with all of its \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} being invertible, that is also endowed with a locally compact, topological structure.

Let us also recall the related \htmladdnormallink{concepts}{http://planetphysics.us/encyclopedia/PreciseIdea.html} of groupoid and \emph{\htmladdnormallink{topological groupoid}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html}}, together with the appropriate notations needed to define a
\emph{locally compact groupoid}.

\textbf{Groupoids and \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} Groupoids}

Recall that a groupoid $\grp$ is a \htmladdnormallink{small category}{http://planetphysics.us/encyclopedia/Cod.html} 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 \htmladdnormallink{source maps}{http://planetphysics.us/encyclopedia/SmallCategory.html}} respectively,
together with a law of \htmladdnormallink{composition}{http://planetphysics.us/encyclopedia/Cod.html} \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
\emph{category of topological spaces and continuous maps}. The notion of internal groupoid has proved significant in a number of \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, since groupoids generalize bundles of groups, group actions, and \htmladdnormallink{equivalence relations}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}. For a further study of groupoids we refer the reader to ref. \cite{Brown2006}.

\begin{thebibliography}{9}

\bibitem{Brown2006}
R. Brown. (2006). \emph{Topology and Groupoids}. BookSurgeLLC
\end{thebibliography} 

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