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locally compact groupoid (Definition)
Definition 0.1   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 ${\mathsf{G}}_{lc}$.

Remarks: 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.

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 topological Groupoids

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

$\displaystyle r,s~:~ \xymatrix{ {\mathsf{G}}\ar@<1ex>[r]^r \ar[r]_s & {\mathsf{G}}^{(0)} }$ (0.1)

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

$\displaystyle \circ~:~ {\mathsf{G}}^{(2)}: = {\mathsf{G}}\times_{{\mathsf{G}}^{... ...{\mathsf{G}}~:~ s(\gamma_1) = r(\gamma_2)~ \}~ {\longrightarrow}~{\mathsf{G}}~,$ (0.2)

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 $u {\longrightarrow}u$ 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].

Bibliography

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



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See Also: quantum transformation groupoid, quantum nano-automata, uniform continuity over locally compact quantum groupoids

Also defines:  groupoid
Keywords:  locally compact Hausdorff, gorupoid

Cross-references: equivalence relations, fields, composition, source maps, small category, topological, topological groupoid, concepts, morphisms, category, topological group, group, object, locally compact Hausdorff space, topological structure
There are 25 references to this object.

This is version 1 of locally compact groupoid, born on 2008-12-18.
Object id is 345, canonical name is LocallyCompactGroupoid.
Accessed 921 times total.

Classification:
Physics Classification: 03. (Quantum mechanics, field theories, and special relativity )
 03.65.Fd (Algebraic methods )
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