Quantum quandaries
For long time it has been clear that category theory might provide a fundamental formulation of quantum TGD. The problem has been that category theory seems to postulate quite too many objects. The reading of Quantum Quandaries by John Baez helped to see the situation in all its simplicity.
 Topological Quantum Field theories have extremely simple formulation as a functor from the category of cobordisms (topological evolutions between n1manifolds by connecting nmanifold) to the category of Hilbert spaces assignable to n1manifolds.
 Since lightlike partonic 3surfaces correspond to almost topological QFT, with the overall important "almost" coming just from the lightlikeness in the induced metric, the theory is nontrivial physically and nothing of the beauty of TQFT as a functor is lost. Cobordisms are however replaced by what I have christened Feynmann cobordisms generalizing the Feynman diagrams to 3D context: the ends of lightlike 3manifolds meet at the vertices which correspond to 2dimensional partonic surfaces.
 Also the counterparts of ordinary string diagrams having interpretation as ordinary cobordisms are possible but have nothing to do with particle reactions: the particle simply decomposes into several pieces and spinor fields propagate along different routes. This is the spacetime correlate for what happens in double slit experiment when photon travels along two different paths simultaneously.
 The intriguing results is that for n<4dimensional cobordisms unitary Smatrix exists only for trivial cobordisms. I wonder whether string theorists have considered the possible catastrophic consequences concerning the nonperturbative dream about the unique stringy Smatrix. In the zero energy ontology of TGD Smatrix appears as timelike entanglement coefficients and need not be unitary. I have already proposed that padic thermodynamics and thermodynamics in general could be regarded as an exact part of quantum theory in this framework and the basic mathematics of hyperfinite factors provides strong technical support for this idea. It could be that one cannot require unitarity in the case of Feynman cobordisms and that only the condition that Smatrix for a product of Feynman cobordisms is a product of Smatrices for composites. Hence the time parameter in Smatrix can be replaced with complex time parameter with imaginary part in the role of temperature without losing the product structure. pAdic thermodynamics and particle massication might be topological necessities in this framework.
For more details see the chapter Category Theory, Quantum TGD, and TGD Inspired Theory of Consciousness.
