nLab fusion 2-category

Idea

A categorification of the notion of fusion categories to 2-categories.

Properties

Tannaka-Krein reconstruction theorem

The Tannaka duality establishes a correspondence between (multi-)fusion categories and weak Hopf algebras. It is expected that an analogous statement holds for the higher categorification of this setting. The theorem corresponding to the specialization of this setting to fusion 2-categories and semisimple Hopf categories (as opposed to multi- fusion 2-categories and weak Hopf categories) appears in (Green (2023)).

Theorem

(Tannaka-Krein reconstruction theorem for fusion 2-categories)

There is a symmetric monoidal equivalence, contravariant at the level of 1-morphisms, between the full subcategory of $3Vec/2Vec$ consisting of locally faithful functors and the 2-category of 2-Hopf Algebras. The natural transformations associated to this equivalence reconstruct a semisimple Hopf category from its fusion 2-category of representations and fiber functor, and a fusion 2-category with fiber functor $F$ from the Hopf category $End(F)$.

References

• Christopher L. Douglas, David J. Reutter, Fusion 2-categories and a state-sum invariant for 4-manifolds [arXiv:1812.11933]

• Thibault Decoppet, Matthew Yu. Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories. (2023) [arXiv:2306.08117]

• David Green, Tannaka-Krein reconstruction for fusion 2-categories (2023) [arXiv:2309.05591]

• Thibault Decoppet, Peter Huston, Theo Johnson-Freyd, Dmitri Nikshych, David Penneys, Julia Plavnik, David Reutter, Matthew Yu. The Classification of Fusion 2-Categories (2024). (arXiv:2411.05907).

On the classification of fermionic (i.e. super-) fusion 2-categories:

• Thibault Decoppet. Extension Theory and Fermionic Strongly Fusion 2-Categories (2024). (arXiv:2403.03211).

In the context of “generalized global symmetries”:

• Wenjie Xi, Tian Lan, Longye Wang, Chenjie Wang, Wei-Qiang Chen, On a class of fusion 2-category symmetry: condensation completion of braided fusion category [arXiv:2312.15947]