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Thibault D. Décoppet

Publications and source records attributed to Thibault D. Décoppet.

At least 19 recordsLinked to original sources
arXiv2026

Compact Semisimple Tensor 2-Categories are Morita Connected

In arXiv:2211.04917, it was shown that, over an algebraically closed field of characteristic zero, every fusion 2-category is Morita equivalent to a connected fusion 2-category, that is, one arising from a braided fusion 1-category. This result has recently allowed for a complete classification of fusion 2-categories. Here we establish that compact semisimple tensor 2-categories, which generalize fusion 2-categories to an arbitrary field of characteristic zero, also enjoy this ``Morita connectedness'' property. In order to do so, we generalize to an arbitrary field of characteristic zero many well-known results about braided fusion 1-categories over an algebraically closed field. Most notably, we prove that the Picard group of any braided fusion 1-category is indfinite, generalizing the classical fact that the Brauer group of a field is torsion. As an application of our main result, we derive the existence of braided fusion 1-categories indexed by the fourth Galois cohomology group of the absolute Galois group that represent interesting classes in the appropriate Witt groups.

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arXiv2026

The Classification of 3+1d Symmetry Enriched Topological Order

We use a 2-categorical version of (de-)equivariantization to classify (3+1)d topological orders with a finite $G$-symmetry. In particular, we argue that (3+1)d fermionic topological order with $G$-symmetry correspond to $\mathbf{2SVect}$-enriched $G$-crossed braided fusion 2-categories. We then show that the categorical data necessary to define these theories agrees with that arising from a fermionic generalization of the Wang-Wen-Witten construction of bosonic topological theories with $G$-symmetry saturating an anomaly. More generally, we also explain how 2-categorical (de-) equivariantization yields a classification of all braided fusion 2-categories.

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arXiv2026

Finite Pre-Tensor Categories that are Morita Equivalent to Finite Tensor Categories

A finite pre-tensor category is a finite abelian category equipped with a right exact tensor product for which every projective object has duals. Finite tensor categories, for which every object has duals, are notable examples. More generally, the category of bimodules over an algebra in a finite tensor category is a finite pre-tensor category. In particular, it is natural to extend the notion of Morita equivalence between finite tensor categories to finite pre-tensor categories. We characterize completely those finite pre-tensor categories that are Morita equivalent to finite tensor categories. More precisely, we show that a finite pre-tensor category $\mathcal{C}$ is Morita equivalent to a finite tensor category if and only if the Drinfeld center of $\mathcal{C}$ is a finite tensor category. We also discuss higher algebraic consequences of our characterization.

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arXiv2025

Invariants of 4-Dimensional 2-Handlebodies from the Temperley-Lieb Category in Positive Characteristic

We investigate invariants of 4-dimensional 2-handlebodies associated to the Temperley-Lieb category in characteristic $p>2$ and at a primitive fourth root of unity. These invariants depend additionally on a height parameter $n$, and we focus on the case $n=2$. Provided that $p>3$, we show that the height $n=2$ invariant associated to the Temperley-Lieb category at a primitive fourth root of unity vanishes on $\mathbb{C}P^2$, $\overline{\mathbb{C}P}^2$, and $S^2\times S^2$. In particular, it has the potential to detect exotic smooth structures.

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arXiv2025

Relative Invertibility and Full Dualizability of Finite Braided Tensor Categories

Fix a finite symmetric tensor category $\mathcal{E}$ over an algebraically closed field. We derive an $\mathcal{E}$-enriched version of Shimizu's characterizations of non-degeneracy for finite braided tensor categories. In order to do so, we consider, associated to any $\mathcal{E}$-enriched finite braided tensor category $\mathcal{A}$ satisfying a mild technical assumption, a Hopf algebra $\mathbb{F}_{\mathcal{E}/\mathcal{A}}$ in $\mathcal{A}$. This is a generalization of Lyubashenko's universal Hopf algebra $\mathbb{F}_{\mathcal{A}}$ in $\mathcal{A}$. In fact, we show that there is a short exact sequence $\mathbb{F}_{\mathcal{E}}\rightarrow\mathbb{F}_{\mathcal{A}}\rightarrow\mathbb{F}_{\mathcal{E}/\mathcal{A}}$ of Hopf algebras in $\mathcal{A}$, and that the canonical pairing on $\mathbb{F}_{\mathcal{A}}$ descends to a pairing $ω_{\mathcal{E}/\mathcal{A}}$ on $\mathbb{F}_{\mathcal{E}/\mathcal{A}}$. We prove that $\mathcal{A}$ is $\mathcal{E}$-non-degenerate, i.e.\ its symmetric center is exactly $\mathcal{E}$, if and only if the pairing $ω_{\mathcal{E}/\mathcal{A}}$ is non-degenerate. We then use the above characterization to show that an $\mathcal{E}$-enriched finite braided tensor category is invertible in the Morita 4-category of $\mathcal{E}$-enriched pre-tensor categories if and only if it is $\mathcal{E}$-non-degenerate. As an application of our relative invertibility criterion, we extend the full dualizability result of Brochier-Jordan-Snyder by showing that a finite braided tensor category is fully dualizable as an object of the Morita 4-category of braided pre-tensor categories if its symmetric center is separable.

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arXiv2024

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

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 $\mathbb{Z}/2$-graded fusion 2-categories whose non-trivially graded factor is $\mathbf{2Vect}$. 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.

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arXiv2024

The Classification of Fusion 2-Categories

We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of $\mathrm{B}\mathbb{Z}/2$-equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.

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arXiv2024

Higher Verlinde Categories: The Mixed Case

Over a field of characteristic $p>0$, the higher Verlinde categories $\mathrm{Ver}_{p^n}$ are obtained by taking the abelian envelope of quotients of the category of tilting modules for the algebraic group $\mathrm{SL}_2$. These symmetric tensor categories have been introduced in arXiv:2003.10499 & arXiv:2003.10105, and their properties have been extensively studied in the former reference. In arXiv:2105.07724, the above construction for $\mathrm{SL}_2$ has been generalized to Lusztig's quantum group for $\mathfrak{sl}_2$ and root of unity $ζ$, which produces the mixed higher Verlinde categories $\mathrm{Ver}^ζ_{p^{(n)}}$. Inspired by the results of arXiv:2003.10499, 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 $\mathrm{Ver}^ζ_{p^{(n)}}$, construct a braided embedding $\mathrm{Ver}_{p^n}\hookrightarrow \mathrm{Ver}^ζ_{p^{(n+1)}}$, compute the symmetric center of $\mathrm{Ver}^ζ_{p^{(n)}}$, and identify its Grothendieck ring.

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arXiv2024

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 arXiv:1312.7188, we prove that separable compact semisimple tensor 2-categories are fully dualizable objects therein. In particular, it then follows from the main theorem of arXiv:2211.04917 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.

math.CT↗
arXiv2024

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 $\mathbf{Vect}$ or $\mathbf{SVect}$. 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.

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arXiv2023

Local Modules in Braided Monoidal 2-Categories

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.

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arXiv2023

Rigid and Separable Algebras in Fusion 2-Categories

Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include $G$-graded fusion 1-categories, and $G$-crossed fusion 1-categories. We explore the properties of the 2-categories of modules and of bimodules over a rigid algebra, by giving a criterion for the existence of right and left adjoints. Then, we consider separable algebras, which are particularly well-behaved rigid algebras. Specifically, given a fusion 2-category, we prove that the 2-categories of modules and of bimodules over a separable algebra are finite semisimple. Finally, we define the dimension of a connected rigid algebra in a fusion 2-category, and prove that such an algebra is separable if and only if its dimension is non-zero.

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arXiv2023

Finite Semisimple Module 2-Categories

Let $\mathfrak{C}$ be a multifusion 2-category. We show that every finite semisimple $\mathfrak{C}$-module 2-category is canonically enriched over $\mathfrak{C}$. Using this enrichment, we prove that every finite semisimple $\mathfrak{C}$-module 2-category is equivalent to the 2-category of modules over a rigid algebra in $\mathfrak{C}$.

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arXiv2023

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.

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arXiv2022

Gauging Noninvertible Defects: A 2-Categorical Perspective

We generalize the notion of an anomaly for a symmetry to a noninvertible symmetry enacted by surface 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.

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arXiv2022

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.

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arXiv2021

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.

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