Related papers: Pre-modular categories of rank 3

We show that ribbon rational homology cobordism is a partial order within the class of irreducible 3-manifolds. This makes essential use of the methods recently employed by Ian Agol to show that ribbon knot concordance is a partial order.

We form tricategories and the homomorphisms between them into a bicategory, whose 2-cells are certain degenerate tritransformations. We then enrich this bicategory into an example of a three-dimensional structure called a locally cubical…

We classify irreducible homogeneous almost Hermite-Lorentz spaces of complex dimension 3.

We describe the semisimplification of the monoidal category of tilting modules for the algebraic group GL_n in characteristic p > 0. In particular, we compute the dimensions of the indecomposable tilting modules modulo p.

Starting with a self-dual Hopf algebra H in a braided monoidal category S we construct a Z/2Z-graded monoidal category C = C_0 + C_1. The degree zero component is the category Rep_S(H) of representations of H and the degree one component is…

Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not…

The central object studied in this paper is a multiplier bimonoid in a braided monoidal category C. Adapting the philosophy of Janssen and Vercruysse, and making some mild assumptions on the category C, we consider a category M whose…

Modular categories are a well-known source of quantum 3-manifold invariants. In this paper we study structures on modular categories which allow to define refinements of quantum 3-manifold invariants involving cohomology classes or…

The aim of this paper is to solve a problem proposed by Dominique Bourn: to provide a categorical-algebraic characterisation of groups amongst monoids and of rings amongst semirings. In the case of monoids, our solution is given by the…

Starting from an abelian category A such that every object has only finitely many subobjects we construct a semisimple tensor category T. We show that T interpolates the categories Rep(Aut(p),K) where p runs through certain projective…

We determine the structure of the semisimple group algebra of certain groups over the rationals and over those finite fields where the Wedderburn decompositions have the least number of simple components. We apply our work to obtain similar…

We introduce a monoidal category whose morphisms are finite partial orders, with chosen minimal and maximal elements as source and target respectively. After recalling the notion of presentation of a monoidal category by the means of…

This note relates axioms for partial semigroups and monoids with those for small object-free categories, either with multiple monoidal units or with source and target maps. We discuss the adjunction of a zero element to both kinds of…

We classify certain subcategories in quotients of exact categories. In particular, we classify the triangulated and thick subcategories of an algebraic triangulated category, i.e. the stable category of a Frobenius category.

We first introduce the notion of Doi Hom-Hopf modules and find the sufficient condition for the category of Doi Hom-Hopf modules to be monoidal. Also we obtain the condition for the monoidal Hom-algebra and monoidal Hom-coalgebra to be…

Let $H$ be a finite dimensional bialgebra. In this paper, we prove that the category of Yetter-Drinfeld-Long bimodules is isomorphic to the Yetter-Drinfeld category over the tensor product bialgebra $H\o H^*$ as monoidal category. Moreover…

Let $\mathcal{A}$ and $\mathcal{B}$ be subcategories of tensor categories $\mathcal{C}$ and $\mathcal{D}$, respectively, both of which are abelian categories with finitely many isomorphism classes of simple objects. We prove that if their…

We define a monad $T_n^{\operatorname{D^s}}$ whose operations are encoded by simple string diagrams and we define $n$-sesquicategories as algebras over this monad. This monad encodes the compositional structure of $n$-dimensional string…

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical…

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…