TITLE: Reshetikhin-Turaev as a 3-2-1-theory
QUESTION [15 upvotes]: I keep reading that the Reshetikhin-Turaev construction actually yields a 3-2-1 tqft. I know the construction that associates to a suitably decorated surface a vector space built up from a hom-space in a modular tensor category and to a decorated 3-manifold a linear map between these vector spaces as described in the book by Bakalov and Kirillov or in Turaev's blue book. 
But to obtain a 3-2-1 theory, I would need a 2-category first. I suppose in this case this would be the category of all small linear categories. What I don't see is, how the Reshetikhin-Turaev construction associates a functor to surfaces and a natural transformation to 3-cobordisms. Or is this the wrong way to think about this?
I have the feeling that this was probably already asked and answered here, but I could not find it. If so, sorry.

REPLY [4 votes]: I really like Hiro Lee Tanaka's answer.  This is just a long comment. 
The Witten-Reshitkhin-Turaev TQFT needs more data than a three-manifold, or a surface.  You can decorate the three-manifolds with a framing, or a $p_1$ structure, and then decorate surfaces with a Lagrangian subspace of their first homology and an integer, and put colors on one manifolds, and the whole thing works... in theory.  Actual computations are a bear.  
The most successful modes of computation descend from coloring surgery diagrams to define morphisms between reduced skein modules as can be found in the seminal paper of Habegger, Masbaum and Vogel on TQFT derived from the Kauffman bracket.
Blanchet, C.; Habegger, N.; Masbaum, G.; Vogel, P. Topological quantum field theories derived from the Kauffman bracket. Topology 34 (1995), no. 4, 883–927. 
In practice all the extra data can be accounted for quite elegantly by accepting that the TQFT underlying WRT is four dimensional. I think this idea is due to Kevin Walker.
The four dimensional theory is so trivial, that you can account for it with homological data, and hence decategorify the theory to get a less elegant $3$-dimensional TQFT. The ambiguities you are eliminating are in fact just scalar multiplication by a power of the central charge, which is a root of unity.
Here is what I think the extension looks like from top down.
Choose a level $r$
The Crane-Yetter invariant is assigned to four manifolds.
The reduced Kauffman bracket skein module at level $r$ is assigned to three-manifolds.
The surface $F$ is assigned the category of left modules over the reduced Kauffman bracket skein algebra of the surface.  The morphisms in the category correspond to four dimensional cobordisms.
A one manifold is assigned a $2$-category. The objects are relative reduced skein modules of three manifolds, so that there is a disk in the boundary of the three-manifold for each circle in the one manifold, inside of which is an oriented arc that is the boundary of relative skeins.  The morphisms come from gluing three manifolds along the complement of the disk, and are induced by including relative skein modules.  The two morphisms are given by morphisms between the reduced skein modules induced by surgery diagrams corresponding to the four manifolds.
Pretty sketchy.
Practically speaking, current interest in quantum invariants is focused on their asymptotic behavior as the level goes to infinity. Generally people are focused on the magnitude of the invariants not the phase, so all the fancy book keeping to keep track of phase to get a three-dimensional theory is wasted effort.