Tannaka-Krein reconstruction for fusion 2-categories

Abstract

We reprove the classical Tannaka-Krein reconstruction theorem by finding a monoidal equivalence of categories between a 1-truncated sub-2-category of the slice 2-category $\mathsf{Mod}(\mathsf{Vec})/\mathsf{Vec}$ and the category of algebras. We then immediately generalize this approach to find a monoidal equivalence of 2-categories between a 2-truncated sub-3-category of the slice 3-category $\mathsf{Mod}(\mathsf{TwoVec})/\mathsf{TwoVec}$ and the category of algebras. As an immediate consequence, a finite semisimple 2-Hopf algebra $C$ can be recovered from its fusion 2-category of modules together with the monoidal fiber 2-functor to $\mathsf{TwoVec}$. Moreover, every fusion 2-category equipped with a monoidal functor to $\mathsf{TwoVec}$ is of this form.

35 pages, many figures. Comments welcome!