Related papers: A lifting theorem for Grothendieck-Verdier categor…
For a functor $Q$ from a category $C$ to the category $Pos$ of ordered sets and order-preserving functions, we study liftings of various kind of structures from the base category $C$ to the total(or Grothendieck) category $\int Q$. That…
A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic ($\mathsf{MELL}$), known as a \emph{linear category}, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known…
Huang, Lepowsky and Zhang have developed a module theory for vertex operator algebras that endows suitably chosen module categories with the structure of braided monoidal categories. Included in the theory is a functor which assigns to…
We construct a separable Frobenius monoidal functor from $\mathcal{Z}\big(\mathsf{Vect}_H^{\omega|_H}\big)$ to $\mathcal{Z}\big(\mathsf{Vect}_G^\omega\big)$ for any subgroup $H$ of $G$ which preserves braiding and ribbon structure. As an…
Grothendieck-Verdier duality is a powerful and ubiquitous structure on monoidal categories, which generalises the notion of rigidity. Hopf algebroids are a generalisation of Hopf algebras, to a non-commutative base ring. Just as the…
We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely weak monoidal pseudofunctors to the 2-category of categories. In doing so, we…
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…
Various monoidal categories, including suitable representation categories of vertex operator algebras, admit natural Grothendieck-Verdier duality structures. We recall that such a Grothendieck-Verdier category comes with two tensor products…
We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples…
Let $A$ be a commutative algebra in a braided monoidal category $\mathcal{C}$; e.g., $A$ could be an extension of a vertex operator algebra (VOA) $V$ in a category $\mathcal{C}$ of $V$-modules. We study when the category $\mathcal{C}_A$ of…
Grothendieck-Verdier categories (also known as $\ast$-autonomous categories) generalize rigid monoidal categories, with notable representation-theoretic examples including categories of bimodules, modules over Hopf algebroids, and modules…
We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally…
We set the foundations of a theory of Grothendieck $(\infty,2)$-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an $(\infty,2)$-category as well as…
We develop the theory of module categories over a Grothendieck-Verdier category, i.e. a monoidal category with a dualizing object and hence a duality structure more general than rigidity. Such a category C comes with two monoidal structures…
Given an operad $\mathcal{O}$, we define a notion of weak $\mathcal{O}$-monoids -- which we term $\mathcal{O}$-pseudomonoids -- in a 2-category. In the special case with the 2-category in question is the 2-category $\mathsf{Cat}$ of…
In this article, we characterize convexity in terms of algebras over a PROP, and establish a tensor-product-like symmetric monoidal structure on the category of convex sets. Using these two structures, and the theory of $\scr{O}$-monoidal…
Liftable pairs of adjoint functors between braided monoidal categories in the sense of \cite{GV-OnTheDuality} provide auto-adjunctions between the associated categories of bialgebras. Motivated by finding interesting examples of such pairs,…
The bicategorical point of view provides a natural setting for many concepts in the representation theory of monoidal categories. We show that centers of twisted bimodule categories correspond to categories of 2-dimensional natural…
By building on the notions of internal projective and injective objects in a module category introduced by Douglas, Schommer-Pries, and Snyder, we extend the reconstruction theory for module categories of Etingof and Ostrik. More…
Grothendieck's theory of fibred categories establishes an equivalence between fibred categories and pseudo functors. It plays a major role in algebraic geometry and categorical logic. This paper aims to show that fibrations are also very…