PlanetPhysics/Fundamental Quantum Groupoid

\newcommand{\sqdiagram}[9]{Failed to parse (unknown function "\diagram"): {\displaystyle \diagram #1 \rto^{#2} \dto_{#4}& \eqno{\mbox{#9}}} }

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

Fundamental Groupoid Functors and Functor Categories

The natural setting for the definition of a quantum fundamental groupoid $F_{\mathbb {Q} }$ is in one of the functor categories-- that of fundamental groupoid functors, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}} , and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathbb {Q} }$ represented by Hilbert space bundles or `rigged' Hilbert (or Frech\'et) spaces ${\mathbb {H} }_{B}$.

Other related functor categories are those specified with the general definition of the fundamental groupoid functor, Failed to parse (unknown function "\grp"): {\displaystyle F_{\grp}: '''Top''' \to \grp_2} , where Top is the category of topological spaces and Failed to parse (unknown function "\grp"): {\displaystyle \grp_2} is the groupoid category.

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 Frech\'et nuclear spaces).