I have been reading Bakalov and Kirillov's Lectures on Tensor Categories and Modular Functors in which the authors state that a direct construction of a $C$-extended 3D TQFT from a $C$-extended 2D topological modular functor, where $C$ is a rigid semisimple abelian category, is 'at present unknown'. They refer to some partial results by Crane and Kohno involving Heegard splitting. They do show that a $C$-extended 2D topological modular functor gives $C$ the structure of a modular tensor category, and of course a modular tensor category gives a $C$-extended 3DTQFT, so such a construction must be possible. I was wondering whether any progress has been made in this area since the notes were written in 2000.
Many thanks in advance.
This post imported from StackExchange MathOverflow at 2014-09-21 08:45 (UCT), posted by SE-user dv1