On fusion 2-categories

Thesis

The theory of fusion 1-categories has received much attention, and is therefore well established. Fusion 2-categories were recently introduced as a categorification of the notion of a fusion 1-category. The present thesis is devoted to the study of the general properties of fusion 2-categories.

At the heart of the definition of a fusion 2-category is the concept of finite semisimple 2-category over an algebraically closed field of characteristic zero. We generalize this concept by presenting the notion of a compact semisimple 2-category, which is sensible over an arbitrary field. Further, over an algebraically closed field or a real closed field, we prove that compact semisimple 2-categories are always finite.

Algebras in fusion 1-categories play a key role in the theory of fusion 1-categories. Accordingly, it is natural to study algebras in fusion 2-categories. We focus our attention on rigid algebras in fusion 2-categories, a notion which generalizes the definition of a fusion 1-category. In particular, we study the 2-categories of bimodules over rigid algebras. We also consider separable algebras, which are particularly well-behaved rigid algebras, and prove that a rigid algebra in a fusion 2-category is separable if and only if the associated 2-category of bimodules is finite semisimple.

Categorifying the notion of Morita equivalence between fusion 1-categories, we set up the Morita theory of fusion 2-categories. In particular, we give three equivalent characterizations of Morita equivalence between fusion 2-categories based on the 3-category of separable module 2-categories, the dual fusion 2-category, and the fusion 2-category of bimodules.

A crucial property of fusion 1-categories over an algebraically closed field of characteristic zero is that their Drinfeld centers are finite semisimple 1-categories. Categorifying the aforementioned result, we show that the Drinfeld center of any fusion 2-category over an algebraically closed field of characteristic zero is a finite semisimple 2-category. Our proof consists in a careful analysis of the Morita equivalence classes of fusion 2-categories.

Authors: Décoppet, TD

• DOI: 10.5287/ora-jnnrdekyg
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: categorification; topological field theory; fusion category; compact semisimple 2-category; Drinfeld center; Morita theory; separable algebra; finite semisimple 2-category; monoidal 2-category; rigid algebra; fusion 2-category; braided fusion category
• Subjects: Mathematics; Quantum Algebra; Category Theory