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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?

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When you say "a ring is probably categorified by a monoidal category" I assume you mean by a monoidal additive category? Plain monoidal categories just categorify monoids. – Mike Shulman Nov 9 '09 at 2:01
Ring ~ monoidal category is close to the kind of categorification I am most interested in, but unfortunately I have no idea how to continue to the next step. – Reid Barton Nov 9 '09 at 2:26
Perhaps the more interesting question is, given a braided 2-category and a braided category, how does one justify the claim that one is a categorification of the other? – S. Carnahan Nov 9 '09 at 16:52

I'd guess you want braided monoidal (i.e. 2-monoidal) 2-category based on the guess that Grothendieck group moves you one step left on the Baez-Dolan periodic table. That is you want a 4-category with only one object and only one morphism.

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As Noah mentioned, you want a braided 2-category, or roughly equivalently, a 4-category with one object (i.e., one 0-morphism) and one 1-morphism. Under this translation, objects in the braided category become 2-endomorphisms of the 1-morphism, and the braided monoidal structure is given by the composition law for 2-endomorphisms, which should admit a canonical action of the E[2]-operad (assuming you chose a good model for n-categories).

I think the usual decategorification process involves removing all non-invertible morphisms above level n, and (if you don't believe in infinity-categories) collapsing isomorphism classes of n-morphisms to points. I may be missing some subtleties in the abelian/enriched situation, as it's been a while since I've read Baez-Dolan. In your case, n=3, so you can repeat the first sentence in this paragraph with the appropriate substitution. In the braided language (shifting down by 2), this means you remove the non-invertible 2-morphisms and contract the isomorphism classes of maps to be single maps. If your braided category was weak, this strictifies composition of 1-morphisms.

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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?

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That's exactly what I'm interested in, though I think in retrospect this was the wrong question to ask. – Ben Webster Nov 9 '09 at 23:52

I have a lot I would like to say about this, but it's not really in any coherent form in my head, so I'll content myself to one comment without justification: as the first step, I believe you will need to repeat the process that takes us from the natural numbers to the integers and from the category of R-modules to its derived category, applying it to the category of [whatever you think of as categorifying an R-module]s.

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