Related papers: Categories for the practising physicist
We construct an exact tensor functor from the category $\mathcal{A}$ of finite-dimensional graded modules over the quiver Hecke algebra of type $A_\infty$ to the category $\mathscr C_{B^{(1)}_n}$ of finite-dimensional integrable modules…
We provide a necessary and sufficient condition for a simple object in a pivotal k-category to be ambidextrous. In turn, these objects imply the existence of nontrivial trace functions in the category. These functions play an important role…
Unitary Ribbon Fusion Categories (URFC) formalize anyonic theories. It has been widely assumed that the same category formalizes a topological quantum computing model. However, in previous work, we addressed and resolved this confusion and…
We discuss relations between some category-theoretical notions for a finite tensor category and cointegrals on a quasi-Hopf algebra. Specifically, for a finite-dimensional quasi-Hopf algebra $H$, we give an explicit description of…
The content of this paper can be roughly organized into a three-level hierarchy of generality. At the first, most general level, we introduce a new language which allows us to express various categorical structures in a systematic and…
We show that, with some technical conditions, an abelian category can be embedded into the category of bimodules over a ring. The case of semisimple rigid monoidal categories is studied in more detail.
We introduce the notion of meromorphic tensor category and illustrate it in several examples. They include representations of quantum affine algebras, chiral algebras of Beilinson and Drinfeld, G-vertex algebras of Borcherds, and…
In this paper, we introduce the notion of a four-angle Hopf module for a Hom-Hopf algebra $(H,\beta)$ and show that the category $\!^{H}_{H}\mathfrak{M}^{H}_{H}$ of four-angle Hopf modules is a monoidal category with either a Hom-tensor…
We axiomatise the dagger category of complex Hilbert spaces and bounded linear maps, using exclusively purely categorical conditions. Our axioms are chosen with the aim of an easy interpretability: two of them describe the composition of…
In this short note, we classify linear categorified open topological field theories in dimension two by pivotal Grothendieck-Verdier categories, a type of monoidal category equipped with a weak, not necessarily rigid duality. In combination…
Motivated by potential applications to theoretical computer science, in particular those areas where the Curry-Howard correspondence plays an important role, as well as by the ongoing search in pure mathematics for feasible approaches to…
We use geometric ideas coming from certain classic algebraic constructions to associate, to every classical field theory, a symmetric monoidal double functor from the double category of cobordisms with corners to a certain symmetric…
A modular tensor category provides the appropriate data for the construction of a three-dimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are…
Motivated by ideas from string theory and quantum field theory new invariants of knots and 3-dimensional manifolds have been constructed from complex algebraic structures such as Hopf algebras (Reshetikhin and Turaev), monoidal categories…
We use non-standard analysis to define a category $^\star\!\operatorname{Hilb}$ suitable for categorical quantum mechanics in arbitrary separable Hilbert spaces, and we show that standard bounded operators can be suitably embedded in it. We…
An n-category is some sort of algebraic structure consisting of objects, morphisms between objects, 2-morphisms between morphisms, and so on up to n-morphisms, together with various ways of composing them. We survey various concepts of…
Monoidal computer is a categorical model of intensional computation, where many different programs correspond to the same input-output behavior. The upshot of yet another model of computation is that a categorical formalism should provide a…
We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $\mathfrak{sl}_2$ at a root of unity $q$ of odd order. Our approach is diagrammatic, and it relies on…
We generalize the notion of nuclear maps from functional analysis by defining nuclear ideals in tensored *-categories. The motivation for this study came from attempts to generalize the structure of the category of relations to handle what…
A braided monoidal category may be considered a $3$-category with one object and one $1$-morphism. In this paper, we show that, more generally, $3$-categories with one object and $1$-morphisms given by elements of a group $G$ correspond to…