# PlanetPhysics/Groupoid5

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## Groupoid definitions

A groupoid $\displaystyle \grp$ is simply defined as a small category with inverses over its set of objects $\displaystyle X = Ob(\grp)$ . One often writes $\displaystyle \grp^y_x$ for the set of morphisms in $\displaystyle \grp$ from $x$  to $y$ .

A topological groupoid consists of a space $\displaystyle \grp$ , a distinguished subspace $\displaystyle \grp^{(0)} = \obg \subset \grp$ , called {\it the space of objects} of $\displaystyle \grp$ , together with maps

$\displaystyle r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} }$

called the {\it range} and {\it source maps} respectively,

together with a law of composition

$\displaystyle \circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{ ~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) = r(\gamma_2)~ \}~ \lra ~\grp~,$

such that the following hold~:~

\item[(1)] [/itex]s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ \gamma_2) = r(\gamma_1)$~,forall$ (\gamma_1, \gamma_2) \in \grp^{(2)}$\displaystyle ~. \item[(2)] $s(x) = r(x) = x$ ~, for all $\displaystyle x \in \grp^{(0)}$ ~. \item[(3)]$\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma$~,forall$ \gamma \in \grp$\displaystyle ~. \item[(4)]$ (\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ (\gamma_2 \circ \gamma_3)$\displaystyle ~. \item[(5)] Each $\gamma$ has a two--sided inverse $\gamma ^{-1}$ with$\gamma \gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)$\displaystyle ~. Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call [itex]\grp^{(0)} = Ob(\grp)$ {\it the set of objects} of $\displaystyle \grp$ ~. For $\displaystyle u \in Ob(\grp)$ , the set of arrows $\displaystyle u \lra u$ forms a group $\displaystyle \grp_u$ , called the isotropy group of $\displaystyle \grp$ at $u$  .

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:

• (a) locally compact groups, transformation groups, and any group in general:
• (b) equivalence relations
• (c) tangent bundles
• (d) the tangent groupoid
• (e) holonomy groupoids for foliations
• (f) Poisson groupoids
• (g) graph groupoids.

As a simple, helpful example of a groupoid, consider (b) above. Thus, let R be an equivalence relation on a set X. Then R is a groupoid under the following operations: $(x,y)(y,z)=(x,z),(x,y)^{-1}=(y,x)$ . Here, $\displaystyle \grp^0 = X$ , (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 $R=X\times X$ , R is called a 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 i is equivalent to every 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 $\displaystyle \grp_{lc}$ to be a locally compact groupoid means that $\displaystyle \grp_{lc}$ is required to be a (second countable) locally compact Hausdorff space, and the product and also inversion maps are required to be continuous. Each $\displaystyle \grp_{lc}^u$ as well as the unit space $\displaystyle \grp_{lc}^0$ is closed in $\displaystyle \grp_{lc}$ . What replaces the left Haar measure on $\displaystyle \grp_{lc}$ is a system of measures $\lambda ^{u}$  ($\displaystyle u \in \grp_{lc}^0$ ), where $\lambda ^{u}$  is a positive regular Borel measure on $\displaystyle \grp_{lc}^u$ with dense support. In addition, the $\lambda ^{u}~$  's are required to vary continuously (when integrated against $\displaystyle f \in C_c(\grp_{lc}))$ and to form an invariant family in the sense that for each x, the map $y\mapsto xy$  is a measure preserving homeomorphism from $\displaystyle \grp_{lc}^s(x)$ onto $\displaystyle \grp_{lc}^r(x)$ . Such a system $\left\{\lambda ^{u}\right\}$  is called a left Haar system for the locally compact groupoid $\displaystyle \grp_{lc}$ .