Abstract

I will explain how to describe the space of all fusion 2-categories, and monoidal equivalences. The starting point is the observation that every fusion 2-category is Morita connected. In particular, an important part of our proof consists in understanding the Witt groups of braided fusion 1-categories. More precisely, we prove that the functor sending a symmetric fusion 1-category to the associated Witt space preserves limits. This can be used to show that fusion 2-categories are classified by a single non-degenerate braided fusion 1-category together with group-theoretic data. As consequences of our classification, we obtain Ocneanu rigidity and rank finiteness for fusion 2-categories, as well as strong constraints on the associated hypergroups. This is joint work in progress with Huston, Johnson-Freyd, Penneys, Plavnik, Nikshych, Reutter, and Yu.