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].