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 an algebra in $\mathfrak{C}$.Comment: Many figure

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.Comment: Added reference

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.Comment: Many figures. Added new section on the Morita theory of fusion 2-categorie

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

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

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.Comment: Minor corrections. previously part of arXiv:2012.15774v

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