0.1 Fundamental Groupoid Functors in Quantum Theories
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.
Let us briefly recall the description of quantum fundamental groupoids in a quantum functor
category, 𝒬F :
Definition 0.1. The quantum fundamental groupoid, QFG is defined by a functor F𝒬 :
¡/span¿B →𝒬G, where 𝒬G is the category of quantum groupoids and their homomorphisms.
0.1.1 Fundamental Groupoid Functors
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.
0.1.2 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.
