quantum fundamental groupoid
Definition 0.1.
A quantum fundamental groupoid is defined as a functor
, where is the category
of Hilbert space
bundles, and 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
is in one of the functor categoriesβ that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor),
, and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categories
of quantum spaces represented by Hilbert space bundles or rigged Hilbert (also called FrechΓ©t) spaces .
Other related functor categories are those specified with the general definition of the fundamental groupoid functor, , where Top is the
category of topological spaces and 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 specialized
one-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 FrechΓ©t nuclear spaces).
Title | quantum fundamental groupoid |
Canonical name | QuantumFundamentalGroupoid |
Date of creation | 2013-03-22 18:15:53 |
Last modified on | 2013-03-22 18:15:53 |
Owner | bci1 (20947) |
Last modified by | bci1 (20947) |
Numerical id | 62 |
Author | bci1 (20947) |
Entry type | Definition |
Classification | msc 55Q05 |
Classification | msc 55U40 |
Classification | msc 20L05 |
Synonym | quantum spacetimes homotopy theory |
Related topic | FundamentalGroupoidFunctor |
Related topic | SpinNetworksAndSpinFoams |
Related topic | QuantumGroupoids2 |
Related topic | GroupoidCategory |
Related topic | SpinNetworksAndSpinFoams |
Related topic | FunctorCategories |
Related topic | FundamentalGroupoid |
Related topic | HomotopyGroups |
Related topic | HigherDimensionalAlgebraHDA |
Related topic | GeneralizedVanKampenTheoremsHigherDimensional |
Related topic | Quantu |
Defines | category of quantum groupoids |
Defines | FrechΓ©t spaces |