quantum fundamental groupoid

Definition 0.1.

A quantum fundamental groupoid $F_{\\mathcal{Q}}$ is defined as a functor $F_{\\mathcal{Q}}:\\mathbf{H}_{B}\\to{\\mathcal{Q}}_{G}$ , where $\\mathbf{H}_{B}$ is the category of Hilbert space bundles, and ${\\mathcal{Q}}_{G}$ is the category of locally compact quantum groupoids and their homomorphisms.

0.1 Fundamental groupoid functors and functor categories

The natural setting for the definition of a quantum fundamental groupoid $F_{\\mathcal{Q}}$ is in one of the functor categories– that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor), $F_{\\mathcal{G}}$ , and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categories of quantum spaces ${\\mathcal{Q}}$ represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces $\\mathbf{H}_{B}$ .

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{\\mathcal{G}}:\\textbf{Top}\\to\\mathcal{G}_{2}$ , where Top is thecategory of topological spaces and $\\mathcal{G}_{2}$ is the groupoid category (http://planetmath.org/GroupoidCategory).

Example 0.1.

A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specializedone-dimensional CW-complexes whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on rigged Hilbert spaces (also called Frechét nuclear spaces).

单词	QuantumFundamentalGroupoid
释义	quantum fundamental groupoid Definition 0.1. A quantum fundamental groupoid $F_{\\mathcal{Q}}$ is defined as a functor $F_{\\mathcal{Q}}:\\mathbf{H}_{B}\\to{\\mathcal{Q}}_{G}$ , where $\\mathbf{H}_{B}$ is the category of Hilbert space bundles, and ${\\mathcal{Q}}_{G}$ is the category of locally compact quantum groupoids and their homomorphisms. 0.1 Fundamental groupoid functors and functor categories The natural setting for the definition of a quantum fundamental groupoid $F_{\\mathcal{Q}}$ is in one of the functor categories– that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor), $F_{\\mathcal{G}}$ , and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categories of quantum spaces ${\\mathcal{Q}}$ represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces $\\mathbf{H}_{B}$ . Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{\\mathcal{G}}:\\textbf{Top}\\to\\mathcal{G}_{2}$ , where Top is thecategory of topological spaces and $\\mathcal{G}_{2}$ is the groupoid category (http://planetmath.org/GroupoidCategory). Example 0.1. A specific example of a quantum fundamental groupoid can be given for spin foams of spin networks, with a spin foam defined as a functor between spin network categories. Thus, because spin networks or graphs are specializedone-dimensional CW-complexes whose cells are linked quantum spin states, their quantum fundamental groupoid is defined as a functor representation of CW-complexes on rigged Hilbert spaces (also called Frechét nuclear spaces).
随便看	completeness of semimartingale convergence CompletenessPrinciple completeness principle CompletenessTheoremForPropositionalLogic completeness theorem for propositional logic completeness theorem for propositional logic CompletenessTheoremForPropositionalLogic1 CompletenessUnderUcpConvergence completeness under ucp convergence CompletePartialOrdersDoNotAddSmallSubsets complete partial orders do not add small subsets CompleteQuadrilateral complete quadrilateral CompleteRingOfQuotients complete ring of quotients CompleteRingOfQuotientsOfReducedCommutativeRings complete ring of quotients of reduced commutative rings CompleteSemilattice complete semilattice CompleteSetOfPrimitiveOrthogonalIdempotents complete set of primitive orthogonal idempotents CompleteUltrafilterAndPartitions complete ultrafilter and partitions CompleteUltrametricField complete ultrametric field