# Thibault Décoppet

I am pursuing a DPhil in Mathematics at the University of Oxford. Broadly, I am interested in (higher) category theory, (higher) algebra, fusion categories, low dimensional topology, and the representation categories of vertex operator algebras. Currently, I am thinking about fusion 2-categories, how to generalize them, and how they can be used to produce 4D TQFTs.

**In Preparation**

## The Morita Theory of Fusion 2-Categories

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.

**Papers**

## Dual Fusion 2-Categories

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.

## Rigid and Separable Algebras in Fusion 2-Categories

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.

## The Relative Deligne Tensor Product over Pointed Braided Fusion Categories

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.

## Compact Semisimple 2-Categories

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.

## Finite Semisimple Module 2-Categories

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.

## The 2-Deligne Tensor Product

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

## Weak Fusion 2-Categories

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.

## Multifusion Categories and Finite Semisimple 2-Categories

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.

**Talks**

**Future**

**06.10.22****.**TBD at UQSL.**06.12.22****.**TBD at the EPFL Topology Seminar.

**Past**

**23.06.22.**"Fusion 2-Categories and Fully Extended 4D TQFTs " at the Defects and Symmetry Meeting.**25.05.22.**"Topological Orders" at the Junior Physics and Geometry Seminar.**06.05.22.**"Once and Twice Categorified Algebra" at the Junior Algebra and Representation Theory Seminar.**16.07.21.**"The Cobordism Hypothesis and Fusion (2-)Categories" at YTM21.**11.06.21.**"Multifusion Categories vs Finite Semisimple 2-Categories" at CaCS2021.**10.06.21.**"Higher Fusion Categories Described by Spaces" at the Junior Topology and Group Theory Seminar.**19.02.21.**"Finite Semisimple 2-Categories and Fusion 2-Categories" at QSSS.**03.06.20.**"An Introduction to Fusion Categories" at the Junior Topology and Group Theory Seminar.

**Teaching**

Tutor for Homological Algebra, MT21.

Tutor for Category Theory, MT21.

TA for Introduction to Schemes, HT21.

TA for Homological Algebra, MT20.

TA for Category Theory, MT19.