Uniqueness of fiber functors and universal Tannakian categories

TL;DR

This paper provides an elementary proof that any two fiber functors of a Tannakian category are locally isomorphic, using basic algebraic properties and ideas from Deligne and Grothendieck, and explores characterizations and constructions of Tannakian categories.

Contribution

It offers a new elementary proof of fiber functor uniqueness and characterizes Tannakian categories within weakly Tannakian categories, also constructing universal Tannakian categories from given categories.

Findings

Elementary proof of fiber functor local isomorphism

Characterization of Tannakian categories among weakly Tannakian categories

Construction of universal Tannakian categories from certain monoidal categories

Abstract

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 implements a proof strategy which Deligne attributes to Grothendieck. Besides categorical generalities, the proof merely relies on basic properties of exterior powers and the classification of finitely generated modules over a principal ideal domain. Using related ideas (but less elementary means) we also present an alternative characterization of Tannakian categories among the more general weakly Tannakian categories. As an application of this result we can construct from any right exact symmetric monoidal abelian category with simple unit object a universal Tannakian category associated to it.