# PlanetPhysics/Locally Compact Groupoid

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A locally compact groupoid  $\displaystyle \grp_{lc}$
is defined as a groupoid that has also the topological structure of a second countable, locally compact Hausdorff space, and if the product and also inversion maps are continuous. Moreover, each $\displaystyle \grp_{lc}^u$
as well as the unit space $\displaystyle \grp_{lc}^0$
is closed in $\displaystyle \grp_{lc}$
.


Remarks: The locally compact Hausdorff second countable spaces are analytic . One can therefore say also that $\displaystyle \grp_{lc}$ is analytic. When the groupoid $\displaystyle \grp_{lc}$ has only one object in its object space, that is, when it becomes a group, the above definition is restricted to that of a locally compact topological group; it is then a special case of a one-object category with all of its morphisms being invertible, that is also endowed with a locally compact, topological structure.

Let us also recall the related concepts of groupoid and topological groupoid, together with the appropriate notations needed to define a locally compact groupoid .

Recall that a groupoid $\displaystyle \grp$ is a small category with inverses over its set of objects $\displaystyle X = Ob(\grp)$ ~. One 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 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 generalize bundles of groups, group actions, and equivalence relations. For a further study of groupoids we refer the reader to ref. .