# Thibault Décoppet

I have recently completed a DPhil in Mathematics at the University of Oxford. I am currently a postdoctoral fellow and lecturer 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 and in characteristic zero), as well as fusion 3-categories.

In Preparation

## The Homotopy Coherent Classification of Fusion 2-Categories

Joint with Peter Huston, Theo Johnson-Freyd, Dave Penneys, Julia Plavnik, Dmitri Nikshych, David Reutter, and Matthew Yu.

Collaboration supported by the American Institute of Mathematics.

## Compact Semisimple Tensor 2-Categories are Morita Connected

Joint with Christopher Douglas and Sean Sanford.

Papers

## Fusion 3-Categories for Duality Defects

Joint with Lakshya Bhardwaj, Sakura Schäfer-Nameki, and Matthew Yu.

We study the fusion 3-categorical symmetries for quantum theories in (3+1)-dimensions with self-duality defects. Such defects have been realized physically by half-space gaug- ing in theories with one-form symmetries A[1] for an abelian group A, and have found applications in the continuum and the lattice. These fusion 3-categories will be called (generalized) Tambara-Yamagami fusion 3-categories (3TY), in analogy to the TY fusion 1-categories. We consider the Brauer-Picard and Picard 4-groupoids to construct these categories using a 3-categorical version of the extension theory introduced by Etingof, Nikshych and Ostrik. These two 4-groupoids correspond to looking at the construction of duality defects either directly from the 4d point of view, or from the point of view of the 5d Symmetry Topological Field Theory (SymTFT). At this categorical level, the Witt group of non-degenerate braided fusion 1-categories naturally appears in the aforementioned 4- groupoids and represents enrichments of standard duality defects by (2+1)d TFTs. Our main objective is to study graded extensions of the fusion 3-category 3Vect(A[1]) for some finite abelian group A, which is the symmetry category associated to a (3+1)d theory with 1-form symmetry A. Firstly, we do so explicitly using invertible bimodule 3-categories and the Brauer-Picard 4-groupoid. Secondly, we use that the Brauer-Picard 4-groupoid of 3Vect(A[1]) can be identified with the Picard 4-groupoid of its Drinfeld center. Moreover, the Drinfeld center of 3Vect(A[1]), which represents topological defects of the SymTFT, is completely described by a sylleptic strongly fusion 2-category formed by topological surface defects of the SymTFT. These are classified by a finite abelian group equipped with an alternating 2-form. We relate the Picard 4-groupoid of the corresponding braided fusion 3-categories with a generalized Witt group constructed from certain graded braided fusion 1-categories using a twisted Deligne tensor product. In tractable examples, we are able to carry out explicit computations so as to understand the categorical structure of the Z/2 and Z/4 graded 3TY categories.

## Higher Verlinde Categories: The Mixed Case

Over a field of characteristic p>0, the higher Verlinde categories Ver p^n are obtained by considering the abelian envelope of a quotient of the category of tilting modules for the algebraic group SL2. These symmetric tensor categories have been introduced in New incompressible symmetric tensor categories in positive characteristic and Monoidal abelian envelopes, and their properties have been extensively studied in the former reference. In SL2 tilting modules in the mixed case, the above construction for SL2 has been generalized to Lusztig's quantum group for sl2 and root of unity ζ, which produces the mixed higher Verlinde categories Ver^ζ p^(n). Inspired by the results of New incompressible symmetric tensor categories in positive characteristic, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of Ver^ζ p^(n), construct a braided embedding of Ver p^n-1 into Ver^ζ p^(n), compute the symmetric center of Ver^ζ p^(n), and identify its Grothendieck ring.

# Extension Theory and Fermionic Strongly Fusion 2-Categories

The updated version will include an appendix joint with Theo Johnson-Freyd.

We study group graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail.

## 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.

## 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.

Published 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.

## 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.

Accepted for publication in Compositio Mathematica, 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.

## 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.

Talks

Future

Past

13.06.24. "The Homotopy Coherent Classification of Fusion 2-Categories" at the Workshop on topology, representation theory and higher structures.

14.03.24. "Fusion (2-)Categories as Higher Symmetries" at the Geometry, Physics, and Representation Theory Seminar at Northeastern.

27.02.24. "Higher Brauer Groups associated to Fusion Categories" at the Algebra Seminar at Indiana University.

26.02.24. "Separable Module Categories have no Deformations" at the Quantum Algebra and Topology Seminar at Indiana University.

08.12.23. "The Homotopy Coherent Classification of Fusion 2-Categories" at the BIRS conference on Subfactors and Fusion (2-)Categories. Recording available here.

28.11.23. "The Homotopy Coherent Classification of Fusion 2-Categories" at the Quantum Symmetry Seminar at the Ohio State University.

19.10.23. "Higher Tambara-Yamagami categories and when they are group-theoretical" at the University of New Hampshire.

16.08.23. "Morita Equivalence Classes of Fusion 2-Categories" at the Conference on Higher Structures in Functorial Field Theory.

12.06.23. "On the Dualizability of Fusion 2-Categories" at the Oxford Topology Seminar.

02.02.23. "Separable Algebras in Fusion 2-Categories" at the Higher Structures & Field Theory Seminar.

24.01.23. "Fusion 2-Categories up to Morita Equivalence" at the Research Seminar on Quantum Topology and Categorification.

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

At Harvard University:

(Fall 2023) MATH 1B: Integration, Series, and Differential Equations

At the University of Oxford:

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.