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 (also called 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).
