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\begin{document}
\section{Groupoid definitions}
\begin{definition}
A \emph{groupoid} $\grp$ is simply defined as a \htmladdnormallink{small category}{http://planetphysics.us/encyclopedia/Cod.html} with inverses over its set of \htmladdnormallink{objects}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $X = Ob(\grp)$. One often writes $\grp^y_x$ for the set of \htmladdnormallink{morphisms}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} in $\grp$ from $x$ to $y$.
\end{definition}
\begin{definition}
\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
\htmladdnormallink{group}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} $\grp_u$, called the \emph{isotropy group of $\grp$ at $u$}.
\end{enumerate}
\end{definition}
Thus, as it is well kown, a topological groupoid is just a groupoid internal to the \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} of \htmladdnormallink{topological}{http://planetphysics.us/encyclopedia/CoIntersections.html} spaces and continuous maps. The notion of internal groupoid has proved significant in a number of \htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}, since groupoids generalise bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to Brown (2006).
Several examples of groupoids are:
\begin{itemize}
\item (a) locally compact groups, transformation groups, and any group in general:
\item (b) equivalence relations
\item (c) tangent bundles
\item (d) the \htmladdnormallink{tangent groupoid}{http://planetphysics.us/encyclopedia/MoyalDeformation.html} \item (e) holonomy groupoids for foliations
\item (f) Poisson groupoids
\item (g) \htmladdnormallink{graph}{http://planetphysics.us/encyclopedia/Cod.html} groupoids.
\end{itemize}
As a simple, helpful example of a groupoid, consider (b) above. Thus, let \textit{R} be an \emph{equivalence relation} on a set X. Then \textit{R} is a groupoid under the following \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html}:
$(x, y)(y, z) = (x, z), (x, y)^{-1} = (y, x)$. Here, $\grp^0 = X $, (the diagonal of $X \times X$ ) and $r((x, y)) = x, s((x, y)) = y$.
Therefore, $ R^2$ = $\left\{((x, y), (y, z)) : (x, y), (y, z) \in R \right\} $.
When $R = X \times X $, \textit{R} is called a \textit{trivial} groupoid. A special case of a \htmladdnormallink{trivial groupoid}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} is
$R = R_n = \left\{ 1, 2, . . . , n \right\}$ $\times $ $\left\{ 1, 2, . . . , n \right\} $. (So every \textit{i} is equivalent to every \textit{j}). Identify $(i,j) \in R_n$ with the \htmladdnormallink{matrix}{http://planetphysics.us/encyclopedia/Matrix.html} unit $e_{ij}$. Then the groupoid $R_n$ is just \htmladdnormallink{matrix multiplication}{http://planetphysics.us/encyclopedia/Matrix.html} except that we only multiply $e_{ij}, e_{kl}$ when $k = j$, and $(e_{ij} )^{-1} = e_{ji}$. We do not really lose anything by restricting the multiplication, since the pairs $e_{ij}, {e_{kl}}$ excluded from groupoid multiplication just give the 0 product in normal algebra anyway.
For a groupoid $\grp_{lc}$ to be a \htmladdnormallink{locally compact groupoid}{http://planetphysics.us/encyclopedia/LocallyCompactGroupoid.html} means that $\grp_{lc}$ is required to be a (second countable) \htmladdnormallink{locally compact Hausdorff space}{http://planetphysics.us/encyclopedia/LocallyCompactHausdorffSpaces.html}, and the product and also inversion maps are required to be continuous. Each $\grp_{lc}^u$ as well as the unit space $\grp_{lc}^0$ is closed in $\grp_{lc}$. What replaces the left \htmladdnormallink{Haar measure}{http://planetphysics.us/encyclopedia/HigherDimensionalQuantumAlgebroid.html} on $\grp_{lc}$ is a \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of measures $\lambda^u$ ($u \in \grp_{lc}^0$), where $\lambda^u$ is a positive \htmladdnormallink{regular}{http://planetphysics.us/encyclopedia/CoIntersections.html} Borel measure on $\grp_{lc}^u$ with dense support. In addition, the $\lambda^u~$ 's are required to vary continuously (when integrated against $f \in C_c(\grp_{lc}))$ and to form an invariant family in the sense that for each x, the map $y \mapsto xy$ is a measure preserving \htmladdnormallink{homeomorphism}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} from $\grp_{lc}^s(x)$ onto $\grp_{lc}^r(x)$. Such a system $\left\{ \lambda^u \right\}$ is called a \emph{left \htmladdnormallink{Haar system}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html}} for the locally compact groupoid $\grp_{lc}$.
\end{document}
