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> In fact the identity functor gives a trivial example that does detect all smooth structures, but obviously we want something less tautological That's an intriguing comment. The identity functor is in a sense the strongest TQFT, while the constant functor is the weakest. But what additional constraints and properties would we expect for a typical TQFT? Should the target category have some kind of sums?
> It would also be nice if there were versions of the Pachner move theorem for cobordisms. I'm a bit surprised by this. The Pachner theorem comes from looking at specific triangulations of cylinders, so restricting bordisms to those that induce PL homeomorphisms on the boundaries. If you want to remove that restriction, you don't really have Pachner's theory anymore, but simply triangulated bordisms. I don't know what else you'd expect there.
> According to Manuel Bärenz's edit on nLab, it can be realized as a generalized DW theory, based on quantum groups instead of finite groups. I should really have said "with a ribbon (= braided spherical) fusion category" rather than "quantum group" back then.
