GroupoidGroupoid52009-02-26 12:26:082009-02-26 12:36:56Definitiontopological groupoidspace of groupoid objectsequivalence relationgroupoid homomorphismgroupoidgroupoid representationsHaar systems with measure associated with locally compact groupoids% this is the default PlanetPhysics preamble.
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\def\D{\mathsf{D}}\section{Groupoid definitions}
\begin{definition}
A \emph{groupoid} $\grp$ is simply defined as a small category with inverses over its set of objects $X = Ob(\grp)$. One often writes $\grp^y_x$ for the set of morphisms 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 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}
\end{definition}
Thus, as it is well kown, a topological groupoid is just a groupoid internal to the category of topological spaces and continuous maps. The notion of internal groupoid has proved significant in a number of fields, 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 tangent groupoid
\item (e) holonomy groupoids for foliations
\item (f) Poisson groupoids
\item (g) graph 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 operations:
$(x, y)(y, z) = (x, z), (x, y)^{-1} = (y, x)$. 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 trivial groupoid 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 matrix unit $e_{ij}$. Then the groupoid $R_n$ is just matrix multiplication 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 locally compact groupoid means that $\grp_{lc}$ is required to be a (second countable) locally compact Hausdorff space, 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 Haar measure on $\grp_{lc}$ is a system of measures $\lambda^u$ ($u \in \grp_{lc}^0$), where $\lambda^u$ is a positive regular 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 homeomorphism from $\grp_{lc}^s(x)$ onto $\grp_{lc}^r(x)$. Such a system $\left\{ \lambda^u \right\}$ is called a \emph{left Haar system} for the locally compact groupoid $\grp_{lc}$.