0.0.1 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 Frechét) spaces
˙B.
Other related functor categories are those specified with the general definition of the fundamental
groupoid functor, F𝒢 : Top →𝒢2, where Top is the category of topological spaces and 𝒢2 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).
