Thibault Décoppet

We develop Morita theory for fusion 2-categories, and give examples related to 2-groups and braided fusion 1-categories. In addition, we prove that a fusion 2-category is separable if and only if its Drinfeld center is finite semisimple.

Given a fusion 2-category and a suitable module 2-category, the dual tensor 2-category is the associated 2-category of module 2-endofunctors. In order to study the properties of this 2-category, we begin by proving that the relative tensor product of modules over a separable algebra in a fusion 2-category exists. We use this result to construct the Morita 3-category of separable algebras in a fusion 2-category. Then, we explain how module 2-categories form a 3-category, and we prove that, over a fusion 2-category, the 2-adjoint of a left module 2-functor carries a canonical left module structure. We define separable module 2-categories over a fusion 2-category, and prove that the Morita 3-category of separable algebras is equivalent to the 3-category of separable module 2-categories. Finally, we show that the dual tensor 2-category with respect to a separable module 2-category is a multifusion 2-category.

arXiv 2022.

Given a monoidal 2-category that has right and left adjoints, we prove that the 2-categories of right module and of bimodules over a rigid algebra have right and left adjoints. Given a compact semisimple monoidal 2-category, we use this result to prove that the 2-categories of modules and of bimodules over a separable algebra are compact semisimple. Finally, we define the dimension of a connected rigid algebra in a compact semisimple 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.

arXiv 2022.

We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field.

arXiv 2022.

Working over an arbitrary field, we define compact semisimple 2- categories, and show that every compact semisimple 2-category is equivalent to the 2-category of separable module 1-categories over a finite semisimple tensor 1-category. Then, we prove that, over an algebraically closed field or a real closed field, compact semisimple 2-categories are finite. Finally, we explain how a number of key results in the theory of finite semisimple 2-categories over an algebraically closed field of characteristic zero can be generalized to compact semisimple 2-categories.

arXiv 2021.

Given C a multifusion 2-category. We show that every finite semisimple C-module 2-category is canonically enriched over C. Using this enrichment, we prove that every finite semisimple C-module 2-category is equivalent to the 2-category of modules over an algebra in C.

arXiv 2021.

We prove that the 2-Deligne tensor product of two compact semisimple 2-categories exists. Further, under suitable hypotheses, we explain how to describe the Hom-categories, connected components, and simple objects of a 2-Deligne tensor product. Finally, we prove that the 2-Deligne tensor product of two compact semisimple tensor 2-categories is a compact semisimple tensor 2-category

To appear in the Kyoto Journal of Mathematics, arXiv 2021.

We introduce a weakening of the notion of fusion 2-category given in arXiv:1812.11933. Then, we establish a number of properties of (multi)fusion 2-categories. Finally, we describe the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories.

Published in the Cahier de Topologie et Géométrie Différentielle Catégoriques in 2022, arxiv 2021.

The 3-categories of semisimple 2-categories and of multifusion categories are shown to be equivalent. A weakening of the notion of fusion 2-category is introduced. Multifusion 2-categories are defined, and the fusion rule of the fusion 2-categories associated to certain pointed braided fusion categories is described.

Published in the Journal of Pure and Applied Algebra in 2022, arxiv 2020.