# 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]}.

## All Sources

[edit | edit source]^{[1]}

## References

[edit | edit source]- ↑
^{1.0}^{1.1}R. Brown. (2006).*Topology and Groupoids*. BookSurgeLLC