Tannaka-Krein reconstruction for fusion 2-categories

TL;DR

This paper extends the classical Tannaka-Krein reconstruction theorem to fusion 2-categories, establishing a monoidal equivalence with categories of algebras and enabling the recovery of 2-Hopf algebras from their module categories.

Contribution

It generalizes the Tannaka-Krein reconstruction to fusion 2-categories and provides a method to recover finite semisimple 2-Hopf algebras from their module 2-categories.

Findings

Reproved classical Tannaka-Krein theorem using monoidal equivalences.

Generalized the approach to 2-categories and fusion 2-categories.

Established a method to recover 2-Hopf algebras from 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 ${\sf Mod}({\sf Vec})/{\sf Vec}$ 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 ${\sf Mod}({\sf TwoVec})/{\sf TwoVec}$ 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 ${\sf TwoVec}$. Moreover, every fusion 2-category equipped with a monoidal functor to ${\sf TwoVec}$ is of this form.