So that we do not disturb the Freidel et al thread, I will start another based on Careful's remarks:

Re gauge groups: in the commutative case, at the heart of topos theory is a categorical equivalence known as Stone duality. One form of this is familiar to you as the ordinary Fourier transform. The equivalence singles out [itex]U(1)[/itex] as a special object amongst commutative spaces because it looks like both a space and something dual to this. The fact that Barrett's ideas appear prominently in Baratin-Freidel is no coincidence, because Barrett specialises in studying q-deformed Fourier transforms.

What one needs then is a noncommutative analogue...and the claim is that the whole of Connes' program of NCG is

Something like q-deformed [itex]SU(2) \times SU(2)[/itex] is a natural candidate for a noncommutative (quaternionic) self-dual object, in the sense above. I know of no proof of this...that would involve a proper understanding of higher Stone duality. Suffice it to say that these questions are equivalent to answering questions like the Riemann hypothesis...a generalisation of which would follow from an analysis of this duality. Take a look at the papers on Feynman diagrams and MZVs by Kreimer, Broadhurst and others. They are very numerical and easy to read.

I hope that is a start.

P.S. I

First comment: if you had made any effort whatsoever to look at category theory over the last few months you would realise that the assumption that it "isn't deep enough" might be a little premature. The reason for our anger is that it should be perfectly obvious that these technical results we are discussing fit into a larger framework.The reason why I am asking these similar questions over and over again when I hear these cries is because I believe a deeper failure in our theories to be responsible for the problems of QG while such lines of thought merely reflect a technical issue. Therefore, I would welcome any insight which shows me otherwise....

Re gauge groups: in the commutative case, at the heart of topos theory is a categorical equivalence known as Stone duality. One form of this is familiar to you as the ordinary Fourier transform. The equivalence singles out [itex]U(1)[/itex] as a special object amongst commutative spaces because it looks like both a space and something dual to this. The fact that Barrett's ideas appear prominently in Baratin-Freidel is no coincidence, because Barrett specialises in studying q-deformed Fourier transforms.

What one needs then is a noncommutative analogue...and the claim is that the whole of Connes' program of NCG is

**not deep enough**...but pure category theory is. The secret is cohomology. Some time in the 60s or 70s there was a split of Grothendieck's ideas (arising out of Weil conjectures and lots of fancy maths) into the Algebraic Geometry camp and the pure category theory camp. It is now clear to mathematicians that the latter is more powerful. There are also thousands upon thousands of expository articles on this history, some of them quite readable.Something like q-deformed [itex]SU(2) \times SU(2)[/itex] is a natural candidate for a noncommutative (quaternionic) self-dual object, in the sense above. I know of no proof of this...that would involve a proper understanding of higher Stone duality. Suffice it to say that these questions are equivalent to answering questions like the Riemann hypothesis...a generalisation of which would follow from an analysis of this duality. Take a look at the papers on Feynman diagrams and MZVs by Kreimer, Broadhurst and others. They are very numerical and easy to read.

I hope that is a start.

P.S. I

*am*the barmaid.
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