  Instead of formulating the state space of a quantum field theory over one big
Hilbert space, it has been proposed by Kijowski to describe quantum states as
projective families of density matrices over a collection of smaller, simpler
Hilbert spaces. Beside the physical motivations for this approach, it could
help designing a quantum state space holding the states we need. In
[Oko{\l}\'ow 2013, arXiv:1304.6330] the description of a theory of Abelian
connections within this framework was developed, an important insight being to
use building blocks labeled by combinations of edges and surfaces. The present
work generalizes this construction to an arbitrary gauge group G (in
particular, G is neither assumed to be Abelian nor compact). This involves
refining the definition of the label set, as well as deriving explicit formulas
to relate the Hilbert spaces attached to different labels.
  If the gauge group happens to be compact, we also have at our disposal the
well-established Ashtekar-Lewandowski Hilbert space, which is defined as an
inductive limit using building blocks labeled by edges only. We then show that
the quantum state space presented here can be thought as a natural extension of
the space of density matrices over this Hilbert space. In addition, it is
manifest from the classical counterparts of both formalisms that the projective
approach allows for a more balanced treatment of the holonomy and flux
variables, so it might pave the way for the development of more satisfactory
coherent states.
