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# 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, $F_{{\mathcal{G}}}$, and their natural transformations defined in the context of quantum categories of quantum spaces ${\mathcal{Q}}$ represented by Hilbert space bundles or rigged Hilbert (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 the
category of topological spaces and $\mathcal{G}_{2}$ is the groupoid category.

###### 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 Frechét nuclear spaces).

## Mathematics Subject Classification

55Q05*no label found*55U40

*no label found*20L05

*no label found*

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