Related papers: Compact Semisimple Tensor 2-Categories are Morita …

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…

We give criteria for when finitely generated local modules over a commutative algebra $A$ in the ind-completion $\widehat{\mathcal{C}}$ of a braided tensor category $\mathcal{C}$ inherit the structure of a (rigid, braided, ribbon) tensor…

We introduce a ramified monoid, attached to each Brauer--type monoid, that is, to the symmetric group, to the Jones and Brauer monoids among others. Ramified monoids correspond to a class of tied monoids which arise from knot theory and are…

We axiomatize the extended operators in topological orders (possibly gravitationally anomalous, possibly with degenerate ground states) in terms of monoidal Karoubi-complete $n$-categories which are mildly dualizable and have trivial…

Presentations for unbraided, braided and symmetric pseudomonoids are defined. Biequivalences characterising the semistrict bicategories generated by these presentations are proven. It is shown that these biequivalences categorify results in…

Let $(H,R)$ be a quasitriangular weak Hopf algebra over a field $k$. We show that there is a braided monoidal equivalence between the Yetter-Drinfeld module category $^H_H\mathscr{YD}$ over $H$ and the category of comodules over some…

We abstract the study of irreducible characters of finite groups vanishing on all but two conjugacy classes, initiated by S. Gagola, to irreducible characters of fusion rings whose kernel has maximal rank. These near-integral fusion rings…

For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum…

We prove that braid group representations associated to braided fusion categories and mapping class group representations associated to modular fusion categories are always semisimple. The proof relies on the theory of extensions in…

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard $2$-categorical groups and realize them as fixed points of natural $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$…

We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various…

In this work, we introduce a family of new equivalence relations among fusion categories that are less refined than the usual Morita equivalence. We obtain abelian groups by quotienting these new equivalence relations from the commutative…

The paper is devoted to prove a version of Milnor-Moore Theorem for connected braided bialgebras that are infinitesimally cocommutative. Namely in characteristic different from 2, we prove that, for a given connected braided bialgebra $A$…

We consider the possibility of semisimple tensor categories whose fusion rule includes exactly one noninvertible simple object. Conditions are given for the existence or nonexistence of coherent associative structures for such fusion rules,…

Just as links may be algebraically described as certain morphisms in the category of tangles, compact surfaces smoothly embedded in R^4 may be described as certain 2-morphisms in the 2-category of `2-tangles in 4 dimensions'. In this…

Let $\mathcal{A}$ be a completely rational local M\"obius covariant net on $S^1$, which describes a set of chiral observables. We show that local M\"obius covariant nets $\mathcal{B}_2$ on 2D Minkowski space which contains $\mathcal{A}$ as…

We introduce a tensor product for symmetric monoidal categories with the following properties. Let SMC denote the 2-category with objects small symmetric monoidal categories, arrows symmetric monoidal functors and 2-cells monoidal natural…

A Morita context is constructed for any comodule of a coring and, more generally, for an $L$-$\cC$ bicomodule $\Sigma$ for a pure coring extension $(\cD:L)$ of $(\cC:A)$. It is related to a 2-object subcategory of the category of $k$-linear…

We initiate the study of non-semisimple algebras in fusion categories by establishing the framework of $\mathcal{C}$-species -- analogous to the framework of species and quivers used in the study of Artin algebras. Under the (necessary)…

We present a new model for continuous tensor categories as algebra objects in the Morita bicategory of $\mathrm{C}^*$-algebras. In this setting, we generalize the construction of Tambara-Yamagami tensor categories from finite abelian groups…