PlanetPhysics/Locally Compact Groupoid

A locally compact groupoid 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 as well as the unit space is closed in .

Remarks: The locally compact Hausdorff second countable spaces are analytic . One can therefore say also that is analytic. When the groupoid 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 is a small category with inverses over its set of objects ~. One writes for the set of morphisms in from to ~. A topological groupoid consists of a space , a distinguished subspace , called the space of objects of , together with maps

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called the range and source maps respectively,

together with a law of composition

such that the following hold~:~

(1) ~, for all .

(2) ~, for all .

(3) ~, for all ~.

(4) .

(5) Each has a two--sided inverse with .

Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call the set of objects of ~. For , the set of arrows forms a group , called the isotropy group of at .

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. [1].

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References

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  1. 1.0 1.1 R. Brown. (2006). Topology and Groupoids . BookSurgeLLC