Related papers: Galois Theory for Braided Tensor Categories and th…

We first generalize the logarithmic tensor category theory of Huang-Lepowsky-Zhang to the more general case that the module category for a vertex operator algebra $V$ (more generally a M\"{o}bius vertex algebra) might not be closed under…

A theory of monoids in the category of bicomodules of a coalgebra $C$ or $C$-rings is developed. This can be viewed as a dual version of the coring theory. The notion of a matrix ring context consisting of two bicomodules and two maps is…

We interpret Galois covers in terms of particular monoidal functors, extending the correspondence between torsors and fiber functors. As applications we characterize tame $G$-covers between normal varieties for finite and \'etale group…

We associate a rigid C*-tensor category $C$ to a totally disconnected locally compact group $G$ and a compact open subgroup $K < G$. We characterize when $C$ has the Haagerup property or property (T), and when $C$ is weakly amenable. When…

A classical tensor product $A \,\otimes\, B$ of complete lattices $A$ and $B$, consisting of all down-sets in $A \times B$ that are join-closed in either coordinate, is isomorphic to the complete lattice $Gal(A,B)$ of Galois maps from $A$…

We give complete descriptions of the tracial states on both the universal and reduced crossed products of a C*-dynamical system consisting of a unital C*-algebra and a discrete group. In particular, we also answer the question of when the…

The Hodge Galois group is the Tannakian Galois group of the category of real mixed Hodge structures. It has a subgroup, called the twistor Galois group, which is the Galois group of the category of mixed twistor structures, defined by C.…

Given a ring object $A$ in a symmetric monoidal category, we investigate what it means for the extension $\mathbb{1}\rightarrow A$ to be (quasi-)Galois. In particular, we define splitting ring extensions and examine how they occur.…

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…

Let V be a simple vertex operator algebra satisfying the following conditions: (i) The homogeneous subspaces of V of weights less than 0 are 0, the homogeneous subspace of V of weight 0 is spanned by the vacuum and V' is isomorphic to V as…

Let $\mathsf{PreOrd}(\mathbb C)$ be the category of internal preorders in an exact category $\mathbb C$. We show that the pair $(\mathsf{Eq}(\mathbb C), \mathsf{ParOrd}(\mathbb C))$ is a pretorsion theory in $\mathsf{PreOrd}(\mathbb C)$,…

We extend our previous results on generalized Dixmier-Douady theory to graded $C^*$-algebras, as means for explicit computations of the invariants arising for bundles of ungraded $C^*$-algebras. For a strongly self-absorbing $C^*$-algebra…

For an algebraically closed field $\mathbb{K}$, we consider a Galois $G$-covering $\mathcal{B} \to \mathcal{A}$ between locally bounded $\mathbb{K}$-categories given by bound quivers, where $G$ is torsion-free and acts freely on the objects…

In this paper we study the tensor category structure of the module category of the restricted quantum enveloping algebra associated to $\mathfrak{sl}_2$. Indecomposable decomposition of all tensor products of modules over this algebra is…

Let $G$ be a group acting on a category $\mathcal{C}$. We give a definition for a functor $F\colon \mathcal{C} \to \mathcal{C}'$ to be a $G$-covering and three constructions of the orbit category $\mathcal{C}/G$, which generalizes the…

Galois categories can be viewed as the combinatorial analog of Tannakian categories. We introduce the notion of pre-Galois category, which can be viewed as the combinatorial analog of pre-Tannakian categories. Given an oligomorphic group…

We apply the yoga of classical homotopy theory to classification problems of G-extensions of fusion and braided fusion categories, where G is a finite group. Namely, we reduce such problems to classification (up to homotopy) of maps from BG…

We show that a finite-dimensional tame division algebra D over a Henselian field F has a maximal subfield Galois over F if and only if its residue division algebra has a maximal subfield Galois over the residue field of F. This generalizes…

Given a presentably symmetric monoidal $\infty$-category $\mathcal{C}$ and an $\mathbb{E}_{\infty}$-monoid $M$, we introduce and classify twisted graded categories, which generalize the Day convolution structure on $\mathrm{Fun}(M,…

We show that the non-trivially associated tensor category constructed from left coset representatives of a subgroup of a finite group is a modular category. Also we give a definition of the character of an object in a ribbon category which…