quantum fundamental groupoid

Definition 0.1.

A quantum fundamental groupoid $F_{𝒬}$ is defined as a functor$F_{𝒬}:{𝐇}_B\to {𝒬}_G$, where ${𝐇}_B$ is the category of Hilbert space bundles, and ${𝒬}_G$ 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 $F_{𝒬}$is in one of the functor categories– that of fundamental groupoid functors (http://planetmath.org/FundamentalGroupoidFunctor),$F_{𝒢}$, and their natural transformations (http://planetmath.org/NaturalTransformation) defined in the context of quantum categories of quantum spaces $𝒬$ represented by Hilbert space bundles or rigged Hilbert (also called Frechét) spaces ${𝐇}_B$.

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, $F_{𝒢}:\text{𝐓𝐨𝐩}\to {𝒢}_2$, where Top is thecategory of topological spaces and ${𝒢}_2$ 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).