Tannakian formalism for fiber functors over tensor categories

TL;DR

This paper extends Tannakian formalism to fiber functors over general tensor categories, establishing conditions under which the source category is equivalent to comodules over a Hopf algebra, and describing this algebra via framed objects.

Contribution

It generalizes Tannakian formalism to broader tensor categories and characterizes the associated Hopf algebra using framed objects.

Findings

Source category equivalent to comodules over a Hopf algebra under certain conditions

Provides a description of the Hopf algebra using framed objects

Establishes a generalized Tannakian formalism for tensor categories

Abstract

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 of comodules over a Hopf algebra in the target category. We will also give a description of this Hopf algebra using the notion of framed objects.