fundamental groupoid functors in quantum theories
The natural setting for the definition of a quantum fundamental groupoid
is in one of the functor
categories- that of fundamental groupoid functors,
, and their natural transformations defined in the context of quantum categories
of quantum spaces
represented by Hilbert space bundles
or `rigged' Hilbert (or Frechét) spaces
.
Let us briefly recall the description of quantum fundamental groupoids in a quantum functor category,
:
Definition 0.1
The
quantum fundamental groupoid, QFG is defined by a functor

, where

is the
category
of
quantum groupoids and their
homomorphisms.
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
is the groupoid category.
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.
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