A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!
In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.
First recall the Baez-Dolan periodic table.
Here all the functors are the generalized “center” construction defined in HDA-1. In the first column, the center means exactly what you think it means. In the second column the first arrow, which we’ll call Z_1, is the Drinfel’d center, while the second arrow is the Mueger center.
A result of Mueger’s says that under some niceness conditions Z_2(Z_1(C)) is automatically just vector spaces! Hence the second column is a complex. The natural thing to do is to take homology. If you restrict your attention to fusion categories (which have duals, are semisimple, and have finitely many simple objects) of non-zero dimension, then the kernel of Z_2 is exactly modular categories. So what you’re doing is taking all modular categories (which form a monoid under external tensor product) modulo those modular tensor categories which are Drinfel’d centers. A priori this only gives you a monoid, but Davydov, Mueger, Nikshych, and Ostrik prove that it is a group.
What about the other columns? I think Kitaev mentioned in passing in his talk that the other columns also form complexes. But I’m a little confused about why. In particular there’s something fishy going on somewhere. You can continue the periodic table downward past the diagonal, but it stablizes. In particular, if you look at Z^2 in that region you get the identity instead of the functor sending everything to vector spaces. Anyone know what’s going on? Mueger’s paper mentions an analogue (due to Baez) in the 1st column where you extend the table upwards by one row. In the first column this gives you sets where the center becomes “the monoid of all endomorphisms.” What’s the homology there? We could also extend the other columns upwards. Then the second column would start with categories, and Z_0 would be the monoidal category of all endofunctors. Is Z_1(Z_0(C)) always vector spaces? If so, why? Can you extend this picture leftwards by looking at (-1)-categories?
Enough questions that I can’t answer, let’s go to one I can answer: “Why is this called the Witt group?” Well, one nice class of tractable examples of tensor categories are the pointed tensor categories (this means that for every V). Any braided pointed fusion category comes from an abelian group with a bilinear form. The abelian group is the Grothendieck group, and the bilinear form tells you by which scalar the braiding acts on the object . (You should think of this as a kind of strictification.) Modularity becomes the condition that the bilinear form is nondegenerate, while Drinfel’d doubles consist of precisely the hyperbolic spaces. Hence the “Witt group” restricted to the pointed case is nothing other than the classical Witt group.