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locally compact groupoids

(Topic)

Locally compact groupoids

This is a specific topic entry defining the basics of locally compact groupoids and related concepts.

Let us first recall the related concepts of groupoid and topological groupoid, together with the appropriate notations needed to define a locally compact groupoid.

Groupoids and topological groupoids: categorical definitions

Recall that a groupoid ${\mathsf{G}}$ is a small category with inverses over its set of objects $X = Ob({\mathsf{G}})$ . One writes ${\mathsf{G}}^y_x$ for the set of morphisms in ${\mathsf{G}}$ from $x$ to $y$ .

A topological groupoid consists of a space ${\mathsf{G}}$ , a distinguished subspace ${\mathsf{G}}^{(0)} = {\rm Ob(\mathsf{G)}}\subset {\mathsf{G}}$ , called the space of objects of ${\mathsf{G}}$ , together with maps

$\displaystyle r,s~:~ \xymatrix{ {\mathsf{G}}\ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)} }$

(1.1)

called the range and source maps respectively, together with a law of composition

$\displaystyle \circ~:~ {\mathsf{G}}^{(2)}: = {\mathsf{G}}\times_{{\mathsf{G}}^{... ...{\mathsf{G}}~:~ s(\gamma_1) = r(\gamma_2)~ \}~ {\longrightarrow}~{\mathsf{G}}~,$

(1.2)

such that the following hold :

(1): $s(\gamma_1 \circ \gamma_2) = r(\gamma_2)~,~ r(\gamma_1 \circ \gamma_2) = r(\gamma_1)$ , for all $(\gamma_1, \gamma_2) \in {\mathsf{G}}^{(2)}$ .
(2): $s(x) = r(x) = x$ , for all $x \in {\mathsf{G}}^{(0)}$ .
(3): $\gamma \circ s(\gamma) = \gamma~,~ r(\gamma) \circ \gamma = \gamma$ , for all $\gamma \in {\mathsf{G}}$ .
(4): $(\gamma_1 \circ \gamma_2) \circ \gamma_3 = \gamma_1 \circ (\gamma_2 \circ \gamma_3)$ .
(5): Each $\gamma$ has a two-sided inverse $\gamma^{-1}$ with $\gamma \gamma^{-1} = r(\gamma)~,~ \gamma^{-1} \gamma = s (\gamma)$ .
Furthermore, only for topological groupoids the inverse map needs be continuous. It is usual to call ${\mathsf{G}}^{(0)} = Ob({\mathsf{G}})$ the set of objects of ${\mathsf{G}}$ . For $u \in Ob({\mathsf{G}})$ , the set of arrows $u \lra u$ forms a group ${\mathsf{G}}_u$ , called the isotropy group of ${\mathsf{G}}$ at $u$ .

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].

Locally compact and analytic groupoids

Definition 1.1 A locally compact groupoid ${\mathsf{G}}_{lc}$ 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 ${\mathsf{G}}_{lc}^u$ as well as the unit space ${\mathsf{G}}_{lc}^0$ is closed in ${\mathsf{G}}_{lc}$ .

Remark 1.1 The locally compact Hausdorff second countable spaces are analytic.

One can therefore say also that ${\mathsf{G}}_{lc}$ is analytic.

When the groupoid ${\mathsf{G}}_{lc}$ 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.

Bibliography

1: R. Brown. (2006). Topology and Groupoids. BookSurgeLLC

"locally compact groupoids" is owned by bci1.

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Other names:

locally compact topological groupoids

Also defines:

locally compact groupoid, groupoid as a small category, topological groupoid, analytic groupoid, locally compact topological group, second countable locally compact groupoid

Keywords:

locally compact groupoids, second countable topological space, locally compact Hausdorff space, topological groupoids, continuous inversion maps

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Cross-references: invertible, category, analytic, Hausdorff, locally compact, closed, unit space, inversion, product, second countable, structure, equivalence relations, group actions, fields, number, isotropy group, group, continuous, topological groupoids, composition, source, range, maps, subspace, morphisms, objects, inverses, small category, groupoid
There are 26 references to this entry.

This is version 41 of locally compact groupoids, born on 2008-08-04, modified 2009-02-01.
Object id is 10915, canonical name is LocallyCompactGroupoids.
Accessed 2243 times total.

Classification:

AMS MSC:	22A22 (Topological groups, Lie groups :: Topological and differentiable algebraic systems :: Topological groupoids )
	55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
	18B40 (Category theory; homological algebra :: Special categories :: Groupoids, semigroupoids, semigroups, groups )
	81R50 (Quantum theory :: Groups and algebras in quantum theory :: Quantum groups and related algebraic methods)
	46L05 (Functional analysis :: Selfadjoint operator algebras :: General theory of $C^*$-algebras)
	81R15 (Quantum theory :: Groups and algebras in quantum theory :: Operator algebra methods)
	46M20 (Functional analysis :: Methods of category theory in functional analysis :: Methods of algebraic topology )

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