Related papers: On Frobenius exact symmetric tensor categories

We develop theory and examples of monoidal functors on tensor categories in positive characteristic that generalise the Frobenius functor from \cite{Os, EOf, Tann}. The latter has proved to be a powerful tool in the ongoing classification…

We develop a theory of Frobenius functors for symmetric tensor categories (STC) $\mathcal{C}$ over a field $\bf k$ of characteristic $p$, and give its applications to classification of such categories. Namely, we define a twisted-linear…

A symmetric tensor category $\mathcal D$ over an algebraically closed field $k$ is incompressible if every tensor functor out of $\mathcal D$ is an embedding. E.g., the categories $Vec$ and $sVec$ of (super)vector spaces are incompressible.…

We study properties of symmetric fusion categories in characteristic $p$. In particular, we introduce the notion of a super Frobenius-Perron dimension of an object $X$ of such a category, and derive an explicit formula for the Verlinde…

We propose a method of constructing abelian envelopes of symmetric rigid monoidal Karoubian categories over an algebraically closed field $\bf k$. If ${\rm char}({\bf k})=p>0$, we use this method to construct generalizations ${\rm…

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a…

A Tannakian category is an abelian tensor category equipped with a fiber functor and additional structures which ensure that it is equivalent to the category of representations of some affine groupoid scheme acting on the spectrum of a…

This paper is a complement to the paper "On $p$-adic differential equations on semistable varieties" written by V. Di Proietto. Given an open variety over a DVR with semistable reduction, the author constructed in that paper a fully…

We define a differential Tannakian category and show that under a natural assumption it has a fibre functor. If in addition this category is neutral, that is, the target category for the fibre functor are finite dimensional vector spaces…

Under certain conditions, a scheme can be reconstructed from its category of quasi-coherent sheaves. The Tannakian reconstruction theorem provides another example where a geometric object can be reconstructed from an associated category, in…

The principal aim of this note is to give an elementary proof of the fact that any two fiber functors of a Tannakian category are locally isomorphic. This builds on an idea of Deligne concerning scalar extensions of Tannakian categories and…

We propose a conjectural extension to positive characteristic case of a well known Deligne's theorem on the existence of super fiber functors. We prove our conjecture in the special case of semisimple categories with finitely many…

Given a semi-simple pre-Tannakian category over a finite field, we show that (a slight modification of) its linearization over a field of characteristic 0 is also semi-simple and pre-Tannakian. The key input is a result of Kuhn on the…

Here we study bounds on the Frobenius-Schur exponent of spherical fusion categories based on their global dimension generalizing bounds from the representation theory of finite-dimensional quasi-Hopf algebras. Our main result is that if the…

These notes give an exposition of Deligne's theorem on the existense of super fiber functor.

Using a variety of methods developed in the literature (in particular, the theory of weak Hopf algebras), we prove a number of general results about fusion categories in characteristic zero. We show that the global dimension of a fusion…

Let G be a group and V a finite dimensional representation of G over an algebraically closed field k of characteristic p>0. Let $d_n(V)$ be the number of indecomposable summands of $V^{\otimes n}$ of nonzero dimension mod p. It is easy to…

In this paper we generalize Tannakian formalism to fiber functors over general tensor categories. We will show that (under some technical conditions) if the fiber functor has a section, then the source category is equivalent to the category…

We characterize a natural class of modular categories of prime power Frobenius-Perron dimension as representation categories of twisted doubles of finite p-groups. We also show that a nilpotent braided fusion category C admits an analogue…

We define tensor categories ${\sf Ver}_{p^n}(G)$ in characteristic $p$ for connected reductive groups $G$ and positive integers $n$, generalising the semisimple Verlinde categories ${\sf Ver}_p(G)$ originating from Gelfand-Kazhdan and the…