# 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 ${\displaystyle x}$  to ${\displaystyle y}$ .

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)${\displaystyle ~,forall}$ (\gamma_1, \gamma_2) \in \grp^{(2)}$\displaystyle ~. \item[(2)] $s(x) = r(x) = x$ ~, for all $\displaystyle x \in \grp^{(0)}$ ~. \item[(3)]$\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma${\displaystyle ~,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 ${\displaystyle \gamma ^{-1}}$ 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 ${\displaystyle u}$  .

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: ${\displaystyle (x,y)(y,z)=(x,z),(x,y)^{-1}=(y,x)}$ . Here, $\displaystyle \grp^0 = X$ , (the diagonal of ${\displaystyle X\times X}$  ) and ${\displaystyle r((x,y))=x,s((x,y))=y}$ .

Therefore, ${\displaystyle R^{2}}$  = ${\displaystyle \left\{((x,y),(y,z)):(x,y),(y,z)\in R\right\}}$ . When ${\displaystyle R=X\times X}$ , R is called a trivial groupoid. A special case of a trivial groupoid is ${\displaystyle R=R_{n}=\left\{1,2,...,n\right\}}$  ${\displaystyle \times }$  ${\displaystyle \left\{1,2,...,n\right\}}$ . (So every i is equivalent to every j ). Identify ${\displaystyle (i,j)\in R_{n}}$  with the matrix unit ${\displaystyle e_{ij}}$ . Then the groupoid ${\displaystyle R_{n}}$  is just matrix multiplication except that we only multiply ${\displaystyle e_{ij},e_{kl}}$  when ${\displaystyle k=j}$ , and ${\displaystyle (e_{ij})^{-1}=e_{ji}}$ . We do not really lose anything by restricting the multiplication, since the pairs ${\displaystyle e_{ij},{e_{kl}}}$  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 ${\displaystyle \lambda ^{u}}$  ($\displaystyle u \in \grp_{lc}^0$ ), where ${\displaystyle \lambda ^{u}}$  is a positive regular Borel measure on $\displaystyle \grp_{lc}^u$ with dense support. In addition, the ${\displaystyle \lambda ^{u}~}$  '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 ${\displaystyle y\mapsto xy}$  is a measure preserving homeomorphism from $\displaystyle \grp_{lc}^s(x)$ onto $\displaystyle \grp_{lc}^r(x)$ . Such a system ${\displaystyle \left\{\lambda ^{u}\right\}}$  is called a left Haar system for the locally compact groupoid $\displaystyle \grp_{lc}$ .