# 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.

This term, I am also organizing the topology advanced class on non-semisimple 3D TQFTs.

**In Preparation**

**Papers**

## Gauging Noninvertible Defects: A 2-Categorical Perspective

Joint with Matthew Yu.

We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted by surfaces operators using the framework of condensation in 2-categories. Given a multifusion 2-category, potentially with some additional levels of monoidality, we prove theorems about the structure of the 2-category obtained by condensing a suitable algebra object. We give examples where the resulting category displays grouplike fusion rules and through a cohomology computation, find the obstruction to condensing further to the vacuum theory.

## On The Drinfeld Centers of Fusion 2-Categories

We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence and under taking the 2-Deligne tensor product with an invertible fusion 2-category. We go on to show that the concept of Morita equivalence between connected fusion 2-categories recovers exactly the notion of Witt equivalence between braided fusion 1-categories. Then, we introduce the notion of separable fusion 2-category. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, which is a scalar whose non-vanishing is equivalent to separability. In addition, we prove that a fusion 2-category is separable if and only if its Drinfeld center is finite semisimple. We then establish the separability of every strongly fusion 2-category, that is fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is **Vect** or **SVect**. We proceed to show that every fusion 2-category is Morita equivalent to the 2-Deligne tensor product of a strongly fusion 2-category and an invertible fusion 2-category. Finally, we prove that every fusion 2-category is separable.

## The Morita Theoy of Fusion 2-Categories

We develop the Morita theory of fusion 2-categories. In order to do so, 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 go on to explain how module 2-categories form a 3-category. After that, 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. As a consequence, we show that the dual tensor 2-category with respect to a separable module 2-category, that is the associated 2-category of module 2-endofunctors, is a multifusion 2-category. Finally, we give three equivalent characterizations of Morita equivalence between fusion 2-categories.

## 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**

**24.01****.23.**TBD at the Research Seminar on Quantum Topology and Categorification.**0****2.02****.2****3****.**TBD at the Higher Structures & Field Theory Seminar.

**Past**

**06.12.22.**"Fusion 2-Categories and Fully Extended 4-Dimensional TQFTs" at the EPFL Topology Seminar.**08.11.22.**"Fusion 2-Categories associated to 2-groups" at the Algebraic Topology Seminar at the University of Warwick.**02.11.22****.**"A 2-Categorical Perspective on Braided Fusion 1-Categories" at the Quantum Mathematics Seminar at the University of Nottingham.**06.10.22.**"The Morita Theory of Fusion 2-Categories" at UQSL.**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 Galois Theory, MT22.

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.