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Link to original content: http://ncatlab.org/nlab/show/fusion 2-categories
Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
With braiding
With duals for objects
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
With duals for morphisms
monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
A categorification of the notion of fusion categories to 2-categories.
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)).
(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 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 from the Hopf category .
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:
In the context of “generalized global symmetries”:
Last revised on November 12, 2024 at 08:48:47. See the history of this page for a list of all contributions to it.