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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., Comment: 50 pages

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 $\zeta$, which produces the mixed higher Verlinde categories $\mathrm{Ver}^{\zeta}_{p^{(n)}}$. 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 $\mathrm{Ver}^{\zeta}_{p^{(n)}}$, construct a braided embedding $\mathrm{Ver}_{p^n}\hookrightarrow \mathrm{Ver}^{\zeta}_{p^{(n+1)}}$, compute the symmetric center of $\mathrm{Ver}^{\zeta}_{p^{(n)}}$, and identify its Grothendieck ring., Comment: Minor corrections

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., Comment: Minor corrections

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., Comment: Comments welcome!

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.

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., Comment: 26 pages, v2 a new theorem about symmetric fusion 2-categories is added

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., Comment: New lemma 4.2.3

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., Comment: Minor corrections

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., Comment: A missing hypothesis was added to the statement of theorem 3.2.4

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., Comment: Minor corrections

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., Comment: Added references

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 $\mathfrak{C}$., Comment: Minor Revision

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.

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., Comment: Minor corrections. previously part of arXiv:2012.15774v1

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 arXiv:1812.11933) and of multifusion categories are equivalent., Comment: Minor corrections