PlanetPhysics/Locally Compact Groupoid

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A locally compact groupoid  Failed 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, each Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^u}
 as well as the unit space Failed to parse (unknown function "\grp"): {\displaystyle \grp_{lc}^0}
 is closed in Failed 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 \grpFailed 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. 1.0 1.1 R. Brown. (2006). Topology and Groupoids . BookSurgeLLC