fundamental groupoid functors in quantum theories

Fundamental Groupoid Functors in Quantum Theories

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, $ F_{\mathcal G}$ , and their natural transformations defined in the context of quantum categories of quantum spaces $ {\mathcal Q}$ represented by Hilbert space bundles or `rigged' Hilbert (or Frechét) spaces $ {\H }_B$ .

Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category, $ {\mathcal Q}_F$ :

Definition 0.1   The quantum fundamental groupoid, QFG is defined by a functor $ F_{\mathcal Q}: \H _B \to {\mathcal Q}_G$ , where $ {\mathcal Q}_G$ is the category of quantum groupoids and their homomorphisms.

Fundamental Groupoid Functors

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 the category of topological spaces and $ \mathcal G_2$ is the groupoid category.

Specific Example of QFG

One can provide a physically relevant example of QFG as spin foams, or functors of spin networks; more precise the spin foams were defined as functors between spin network categories that realize dynamic transformations on the spin space. 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 representation of CW-complexes on `rigged' Hilbert spaces, that are called Frechét nuclear spaces.



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