1 Quantum Fundamental Groupoid
Definition 1.1. A quantum fundamental groupoid F𝒬 is defined as a functor
where ℋB is the category of Hilbert space bundles, and 𝒬G is the category of quantum
groupoids and their homomorphisms.
1.1 Fundamental groupoid functors and functor categories
The natural setting for the definition of a quantum fundamental groupoid F𝒬 is in one of the
functor categories—that of fundamental groupoid functors, FG, and their natural transformations,
defined in the context of quantum categories of quantum spaces 𝒬 represented by Hilbert space
bundles or “rigged” Hilbert (or Fréchet) spaces ℋB.
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 G2 is the groupoid category.
Example 1.1. 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 Fréchet
nuclear spaces).
