PlanetMath: equivalent representations of groupoids

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equivalent representations of groupoids

(Definition)

Definition 0.1 Two representations of groupoids $(\mu_i, U_{{\mathsf{G}}} * \H , L_i)$ , for $i=1,2$ are called equivalent if $\mu_1 \sim \mu_2$ , and if there also exists a fiber-preserving isomorphism of analytical Hilbert space bundles $v: (U_{{\mathsf{G}}}* \H _1)\vert _U \longrightarrow (U_{{\mathsf{G}}}* \H _2)\vert _U$ , where $U$ is a measurable subset of $U_{{\mathsf{G}}}$ of null complementarity; the isomorphism $v$ also has the following property: $\hat{v}[r(x)]\hat{L}_1(x) = \hat{L}_2 \hat{v}[d(x)]$ for $x \in {\mathsf{G}}\vert _U$ .

"equivalent representations of groupoids" is owned by bci1.

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Other names:

equivalence class of groupoid representations

Also defines:

equivalent representations of groupoids

Keywords:

representations, groupoids, equivalence class of groupoid representations, fiber-preserving

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Cross-references: property, null, subset, measurable, Hilbert space, isomorphism, equivalent, groupoids, representations

This is version 22 of equivalent representations of groupoids, born on 2008-07-27, modified 2009-06-04.
Object id is 10880, canonical name is EquivalentRepresentationsOfGroupoids.
Accessed 937 times total.

Classification:

AMS MSC:	55U99 (Algebraic topology :: Applied homological algebra and category theory :: Miscellaneous)
	55U40 (Algebraic topology :: Applied homological algebra and category theory :: Topological categories, foundations of homotopy theory)
	55P10 (Algebraic topology :: Homotopy theory :: Homotopy equivalences)
	55N20 (Algebraic topology :: Homology and cohomology theories :: Generalized homology and cohomology theories)
	55N33 (Algebraic topology :: Homology and cohomology theories :: Intersection homology and cohomology)
	18D05 (Category theory; homological algebra :: Categories with structure :: Double categories, $2$-categories, bicategories and generalizations)

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