Related papers: Tensor categories

In this paper we provide an overview of category theory, focussing on applications in physics. The route we follow is motivated by the final goal of understanding anyons and topological QFTs using category theory. This entails introducing…

Following Nori's original idea we here provide certain motivic categories with a canonical tensor structure. These motivic categories are associated to a cohomological functor on a suitable base category and the tensor structure is induced…

We prove Heyneman-Radford Theorem in the framework of Monoidal Categories.

We study internal structures in regular categories using monoidal methods. Groupoids in a regular Goursat category can equivalently be described as special dagger Frobenius monoids in its monoidal category of relations. Similarly,…

We develop the idea of a supersymmetric monoidal supercategory, following ideas of Kapranov. Roughly, this is a monoidal category in which the objects and morphisms are ${\bf Z}/2$-graded, equipped with isomorphisms $X \otimes Y \to Y…

We explore the category of internal categories in the usual category of (right) group-sets, whose objects are referred to as categorified group-sets. More precisely, we develop a new Burnside theory, where the equivalence relation between…

The notion of a tensor captures three great ideas: equivariance, multilinearity, separability. But trying to be three things at once makes the notion difficult to understand. We will explain tensors in an accessible and elementary way…

This paper gives an explicit description of the categorical operad whose algebras are precisely symmetric monoidal categories. This allows us to place the operad in a sequence of four, and therefore a sequence of four successively stricter…

Classical diagram categories and monoids, including the Temperley--Lieb, Brauer, and partition cases, arise as special instances of the category of two dimensional cobordisms and admit additional twists that produce a large new family of…

We describe the origins of the theory of tannakian categories, and summarize its main results.

We extend the notion of the nerve of a category for a small class of crossed simplicial groups, explicitly describing them using generators and relations. We do this by first considering a generalised bar construction of a group before…

These notes are meant to provide a rapid introduction to triangulated categories. We start with the definition of an additive category and end with a glimps of tilting theory. Some exercises are included.

This work is the first one in a series, in which we develop a mathematical theory of enriched (braided) monoidal categories and their representations. In this work, we introduce the notion of the $E_0$-center ($E_1$-center or $E_2$-center)…

We construct a symmetric monoidal closed category of polynomial endofunctors (as objects) and simulation cells (as morphisms). This structure is defined using universal properties without reference to representing polynomial diagrams and is…

This paper gives an introduction to some results on monodromy groupoids and the monodromy principle, and then develops the notion of monodromy groupoid for group groupoids.

This paper is a contribution to the construction of non-semisimple modular categories. We establish when M\"uger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which…

We give a natural-deduction-style type theory for symmetric monoidal categories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler…

We develop layered monoidal theories -- a generalisation of monoidal theories combining formal descriptions of a system at different levels of abstraction. Via their representation as string diagrams, monoidal theories provide a graphical…

We review the complete definition of monoidal 2-categories and recover Kapranov and Voevodsky's definition from the algebraic definition of weak 3-category(or tricategory).

We introduce the theory of enrichment over an internal monoidal category as a common generalization of both the standard theories of enriched and internal categories. The aim of the paper is to justify and contextualize the new notion by…