PlanetPhysics/Locally Compact Groupoid

A locally compact groupoid ${\mathcal {G}}_{lc}$ 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 ${\mathcal {G}}_{lc}^{u}$ as well as the unit space ${\mathcal {G}}_{lc}^{0}$ is closed in ${\mathcal {G}}_{lc}$.

Remarks: The locally compact Hausdorff second countable spaces are analytic . One can therefore say also that ${\mathcal {G}}_{lc}$ is analytic. When the groupoid ${\mathcal {G}}_{lc}$ 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 .

Groupoids

Recall that a groupoid ${\mathcal {G}}$ is a small category with inverses over its set of objects $X=Ob({\mathcal {G}})$~. One writes ${\mathcal {G}}_{x}^{y}$ for the set of morphisms in ${\mathcal {G}}$ from $x$ to $y$~. A topological groupoid consists of a space ${\mathcal {G}}$, a distinguished subspace ${\mathcal {G}}^{(0)}={\rm {Ob({\mathsf {G)}}\subset {\mathcal {G}}}}$, called the space of objects of ${\mathcal {G}}$, together with maps

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called the range and source maps respectively,

together with a law of composition

$\circ ~:~{\mathcal {G}}^{(2)}:={\mathcal {G}}\times _{{\mathcal {G}}^{(0)}}{\mathcal {G}}=\{~(\gamma _{1},\gamma _{2})\in {\mathcal {G}}\times {\mathcal {G}}~:~s(\gamma _{1})=r(\gamma _{2})~\}~\longrightarrow ~{\mathcal {G}}~,$

such that the following hold~:~

(1) $s(\gamma _{1}\circ \gamma _{2})=r(\gamma _{2})~,~r(\gamma _{1}\circ \gamma _{2})=r(\gamma _{1})$~, for all $(\gamma _{1},\gamma _{2})\in {\mathcal {G}}^{(2)}$.

(2) $s(x)=r(x)=x$~, for all $x\in {\mathcal {G}}^{(0)}$.

(3) $\gamma \circ s(\gamma )=\gamma ~,~r(\gamma )\circ \gamma =\gamma$~, for all $\gamma \in {\mathcal {G}}$~.

(4) $(\gamma _{1}\circ \gamma _{2})\circ \gamma _{3}=\gamma _{1}\circ (\gamma _{2}\circ \gamma _{3})$.

(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 ${\mathcal {G}}^{(0)}=Ob({\mathcal {G}})$ the set of objects of ${\mathcal {G}}$~. For $u\in Ob({\mathcal {G}})$, the set of arrows $u\longrightarrow u$ forms a group ${\mathcal {G}}_{u}$, called the isotropy group of ${\mathcal {G}}$ 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. .