Related papers: Equivalence classes of exact module categories ove…

We prove, over any base ring, that the infinity-category of strictly unital A-infinity-categories (and strictly unital functors) is equivalent to the infinity-category of unital A-infinity-categories (and unital functors). We also identify…

In a perfect category every object has a minimal projective resolution. We give a criterion for the category of modules over a categorygraded algebra to be perfect.

Categorical aspects of the theory of modules over trusses were studied in recent years. The snake lemma and the nine lemma in categories of modules over trusses are formulated in this paper.

We define equivariantization of tensor categories under tensor group scheme actions and give necessary and sufficient conditions for an exact sequence of tensor categories to be an equivariantization under a finite group or finite group…

Noncommutative near-group fusion categories were completely classified in the previous work of the first named author by using an operator algebraic method (and hence under the assumption of unitarity), and they were shown to be group…

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…

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,…

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an…

We study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We…

In this paper, we study fusion categories which contain a proper fusion subcategory with maximal rank. They can be viewed as generalizations of near-group fusion categories. We first prove that they admit spherical structure. We then…

We establish some relations between the orders of simple objects in a fusion category and the structure of its universal grading group. We consider fusion categories which have a faithful simple object and show that its universal grading…

In this study, internal categories in the category of the crossed modules are characterized and it has been shown that there is a natural equivalence between the category of the crossed modules over crossed modules, i.e. crossed squares,…

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and…

We develop further the techniques presented in [M. Mombelli. On the tensor product of bimodule categories over Hopf algebras. Preprint arXiv:1111.1610 ] to study bimodule categories over the representation categories of arbitrary…

We show an equivalence of categories, over general $p$-adic bases, between finite locally $p^n$-torsion commutative group schemes and $\Int/p^n\Int$-modules in perfect $F$-gauges of Tor amplitude $[-1,0]$ with Hodge-Tate weights $0,1$. By…

We establish a set of general results to study how the Galois action on modular tensor categories interacts with fusion subcategories. This includes a characterization of fusion subcategories of modular tensor categories which are closed…

Given a tensor functor between tensor categories $\mathcal{C}$ and $\mathcal{D}$, we give criteria that, under certain assumptions, the Frobeniusness of $\mathcal{C}$ or $\mathcal{D}$ implies the Frobeniusness of the other one. We also give…

This is the second in a series of papers intended to set up a framework to study categories of modules in the context of non-commutative geometries. In \cite{mem} we introduced the basic DG category $\Pc_{\A^\bullet}$, the perfect category…

We show that every unitarizable fusion category, and more generally every semisimple C*-tensor category, admits a unique unitary structure. Our proof is based on a categorified polar decomposition theorem for monoidal equivalences between…

In this paper, we extend the study of graded equivalences to the case of general idempotent graded rings. We prove that the existence of a graded equivalence between two categories of graded torsion-free unital modules may be characterized…