TITLE: What's the right object to categorify a braided tensor category?
QUESTION [8 upvotes]: The yoga of categorification has gained a lot of popularity in recent years, and some techniques for it have made a lot of progress.  It's well-understood that, for example, a ring is probably categorified by a monoidal abelian (or triangulated) category.  
But I get a little more confused another step up.  If I have a braided tensor category, what sort of 2-category should I expect to categorify it?
Edit: I realized this wasn't the right question to ask; what I really wanted to know is What structure on a monoidal category would make its 2-category of module categories monoidal and braided?

REPLY [4 votes]: The free braided monoidal 2-category with duals on one self-dual object generator is the 2-category of 2-tangles. The original proof of Fisher had a gap, in part, because he relied on the original movie-move theorem. In our abstract, there was a slight mis-statement about the meaning of "movie" in that context. The error was fixed in CRS, and then Baez Langford gave a precise characterization of a free braided monoidal 2-category with duals. 
Ben's question might be along the lines of using Lauda-Khovanov construction to construct a representation theory of the categorification  of $u^+(sl(2))$ for example. Arron spoke about this in Riverside on Saturday morning. According to Arron's slides, there was hope that such representations can give a BM2Catw/D. 
Is anyone working on giving an explicit "other" braided monoidal 2-cat w/ duals?