# PlanetPhysics/Locally Compact Groupoid

\newcommand{\sqdiagram}[9]{**Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}}**
}

Alocally compact groupoidFailed to parse (unknown function "\grp"): {\displaystyle \grp_{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, eachFailed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u}as well as the unit spaceFailed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^0}is closed inFailed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}.

**Remarks:**
The locally compact Hausdorff second countable spaces are *analytic* .
One can therefore say also that **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}}**
is analytic.
When the groupoid **Failed to parse (unknown function "\grp"): {\displaystyle \grp_{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 and \htmladdnormallink{topological** {http://planetphysics.us/encyclopedia/CoIntersections.html} Groupoids}

Recall that a groupoid **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
is a small category with inverses
over its set of objects **Failed to parse (unknown function "\grp"): {\displaystyle X = Ob(\grp)}**
~. One writes **Failed to parse (unknown function "\grp"): {\displaystyle \grp^y_x}**
for
the set of morphisms in **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
from to ~.
*A topological groupoid* consists of a space **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
, a distinguished subspace
**Failed to parse (unknown function "\grp"): {\displaystyle \grp^{(0)} = \obg \subset \grp}**
, called {\it the space of objects} of **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
,
together with maps

Failed to parse (unknown function "\xymatrix"): {\displaystyle r,s~:~ \xymatrix{ \grp \ar@<1ex>[r]^r \ar[r]_s & \grp^{(0)} } }

called the {\it range} and {\it source maps} respectively,

together with a law of composition

Failed to parse (unknown function "\grp"): {\displaystyle \circ~:~ \grp^{(2)}: = \grp \times_{\grp^{(0)}} \grp = \{ ~(\gamma_1, \gamma_2) \in \grp \times \grp ~:~ s(\gamma_1) = r(\gamma_2)~ \}~ \lra ~\grp~, }

such that the following hold~:~

\item[(1)]
</math>s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ
\gamma_2) = r(\gamma_1)(\gamma_1, \gamma_2) \in
\grp^{(2)}**Failed to parse (unknown function "\item"): {\displaystyle ~. \item[(2)] <math>s(x) = r(x) = x}**
~, for all **Failed to parse (unknown function "\grp"): {\displaystyle x \in \grp^{(0)}}**
~.

\item[(3)]
</math>\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma =
\gamma\gamma \in \grp**Failed to parse (unknown function "\item"): {\displaystyle ~. \item[(4)] }**
(\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ
(\gamma_2 \circ \gamma_3)**Failed to parse (unknown function "\item"): {\displaystyle ~. \item[(5)] Each <math>\gamma}**
has a two--sided inverse with </math>\gamma
\gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)**Failed to parse (unknown function "\grp"): {\displaystyle ~. Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call <math>\grp^{(0)} = Ob(\grp)}**
{\it the set of objects}
of **Failed to parse (unknown function "\grp"): {\displaystyle \grp}**
~. For **Failed to parse (unknown function "\grp"): {\displaystyle u \in Ob(\grp)}**
, the set of arrows **Failed to parse (unknown function "\lra"): {\displaystyle u \lra u}**
forms a
group **Failed to parse (unknown function "\grp"): {\displaystyle \grp_u}**
, called the *isotropy group of Failed to parse (unknown function "\grp"): {\displaystyle \grp}
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 SourcesEdit

^{[1]}

## ReferencesEdit

- ↑
^{1.0}^{1.1}R. Brown. (2006).*Topology and Groupoids*. BookSurgeLLC