Description:

Title: Fully dualizable linear categories

Abstract: Finite dimensional vector spaces over a field k admit a purely categorical description as the dualizable objects inside the symmetric monoidal category Vect of all k-vector spaces. Via the cobordism hypothesis, this provides a complete classification of one dimensional topological field theories with target Vect. The goal of this talk is to present an analogous picture in the context of categorified linear algebra and two dimensional topological field theory. More precisely, I will give a complete classification of the fully dualizable objects in a certain symmetric monoidal 2-category of categories linear over k. We will also discuss variants of this story in which k is allowed to be a commutative ring or even a connective commutative ring spectrum, with (derived) Brauer groups playing a central role.