Thibault Décoppet

I have recently completed a DPhil in Mathematics at the University of Oxford. I am currently a postdoctoral fellow at Harvard University. Broadly speaking, I am interested in (higher) category theory, (higher) algebra, fusion categories, low dimensional topology, and their application to Physics. My research has been focused on fusion 2-categories, how to generalize them, and how they can be used to produce 4D TQFTs. I am currently working on projects related to non-semisimple tensor 1-categories (in positive characteristic as well as in characteristic zero), as well as fusion 3-categories.

In Preparation

The Homotopy Coherent Classification of Fusion 2-Categories

Papers

On the Dualizability of Fusion 2-Categories

Over an arbitrary field, we prove that the relative 2-Deligne tensor product of two separable module 2-categories over a compact semisimple tensor 2-category exists. This allows us to consider the Morita 4-category of compact semisimple tensor 2-categories, separable bimodule 2-categories, and their morphisms. Categorifying a result of Dualizable Tensor Categories, we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories that, over an algebraically closed field of characteristic zero, every fusion 2-category is a fully dualizable object of the above Morita 4-category. We explain how this can be extended to any field of characteristic zero. Finally, we discuss the field theoretic interpretation of our results.

arXiv 2023.

Local Modules in Braided Monoidal 2-Categories

Joint with Hao Xu.

Given an algebra in a monoidal 2-category, one can construct a 2-category of right modules. Given a braided algebra in a braided monoidal 2-category, it is possible to refine the notion of right module to that of a local module. Under mild assumptions, we prove that the 2-category of local modules admits a braided monoidal structure. In addition, if the braided monoidal 2-category has duals, we go on to show that the 2-category of local modules also has duals. Furthermore, if it is a braided fusion 2-category, we establish that the 2-category of local modules is a braided multifusion 2-category. We examine various examples. For instance, working within the 2-category of 2-vector spaces, we find that the notion of local module recovers that of braided module 1-category. Finally, we examine the concept of a Lagrangian algebra, that is a braided algebra with trivial 2-category of local modules. In particular, we completely describe Lagrangian algebras in the Drinfeld centers of fusion 2-categories, and we discuss how this result is related to the classifications of topological boundaries of (3+1)d topological phases of matter.

Accepted for publication in J. Math. Phys., arXiv 2023.


Fiber 2-Functors and Tambara-Yamagami Fusion 2-Categories

Joint with Matthew Yu.

We introduce group-theoretical fusion 2-categories, a strong categorification of the notion of a group-theoretical fusion 1-category. Physically speaking, such fusion 2-categories arise by gauging subgroups of a global symmetry. We show that group-theoretical fusion 2-categories are completely characterized by the property that the braided fusion 1-category of endomorphisms of the monoidal unit is Tannakian. Then, we describe the underlying finite semisimple 2-category of group-theoretical fusion 2-categories, and, more generally, of certain 2-categories of bimodules. We also partially describe the fusion rules of group-theoretical fusion 2-categories, and investigate the group gradings of such fusion 2-categories. Using our previous results, we classify fusion 2-categories admitting a fiber 2-functor. Next, we study fusion 2-categories with a Tambara-Yamagami defect, that is Z/2-graded fusion 2-categories whose non-trivially graded factor is 2Vect. We classify these fusion 2-categories, and examine more closely the more restrictive notion of Tambara-Yamagami fusion 2-categories. Throughout, we give many examples to illustrate our various results.

arXiv 2023.


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.

Published in Letters in Mathematical Physics in 2023, arXiv 2022.


Drinfeld Centers and Morita Equivalence Classes of Fusion 2-Categories

We prove that the Drinfeld center of a fusion 2-category is invariant under Morita equivalence. 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. A strongly fusion 2-category is a fusion 2-category whose braided fusion 1-category of endomorphisms of the monoidal unit is Vect or SVect. We prove 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. We proceed to show that every fusion 2-category is Morita equivalent to a connected fusion 2-category. As a consequence, we find that every rigid algebra in a fusion 2-category is separable. This implies in particular that every fusion 2-category is separable. Conjecturally, separability ensures that a fusion 2-category is 4-dualizable. We define the dimension of a fusion 2-category, and prove that it is always non-zero. Finally, we show that the Drinfeld center of any fusion 2-category is a finite semisimple 2-category. 

arXiv 2022.

The Morita Theory 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. 

Published in Higher Structures in 2023, arXiv 2022.

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.

Published in Advances in Mathematics in 2023, arXiv 2022.

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.

Published in the Journal of Algebra in 2023, arXiv 2022.

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. 

Published in Transactions of the American Mathematical Society in 2023, arXiv 2021.

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.

arXiv 2021.

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 

Published in the Kyoto Journal of Mathematics, arXiv 2021.

Weak Fusion 2-Categories

We introduce a weakening of the notion of fusion 2-category given in Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds. 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

We give a 3-universal property for the Karoubi envelope of a 2-category. Using this, we show that the 3-categories of finite semisimple 2-categories (as introduced in Fusion 2-Categories and a State-Sum Invariant for 4-Manifolds) and of multifusion categories are equivalent. 

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

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