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 $$\mathcal G^y_x$$ 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 $$ r,s~:~ \mathcal G @ [r]^r [r]_s & \mathcal G^{(0)} $$

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. .