PlanetPhysics/Locally Compact Groupoid

A locally compact groupoid ${𝒢}_{lc}$ is defined as a groupoid that has also the [[../TrivialGroupoid/|topological structure]] of a second countable, [[../LocallyCompactHausdorffSpaces/|locally compact Hausdorff space]], and if the product and also inversion maps are continuous. Moreover, each ${𝒢}_{lc}^u$ as well as the unit space ${𝒢}_{lc}^0$ is closed in ${𝒢}_{lc}$.

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

Let us also recall the related [[../PreciseIdea/|concepts]] of groupoid and [[../GroupoidHomomorphism2/|topological groupoid]], together with the appropriate notations needed to define a locally compact groupoid .

Groupoids

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

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called the range and [[../SmallCategory/|source maps]] respectively,

together with a law of [[../Cod/|composition]]

$\circ :{𝒢}^{(2)}:=𝒢{\times }_{{𝒢}^{(0)}}𝒢=\{({\gamma }_1,{\gamma }_2)\in 𝒢\times 𝒢:s({\gamma }_1)=r({\gamma }_2)\}⟶𝒢,$

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 $({\gamma }_1,{\gamma }_2)\in {𝒢}^{(2)}$.

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

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

(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 ${𝒢}^{(0)}=Ob(𝒢)$ the set of objects of $𝒢$~. For $u\in Ob(𝒢)$, the set of arrows $u⟶u$ forms a group ${𝒢}_u$, called the isotropy group of $𝒢$ 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 [[../CosmologicalConstant/|fields]], since groupoids generalize bundles of groups, group actions, and [[../TrivialGroupoid/|equivalence relations]]. For a further study of groupoids we refer the reader to ref. .