Model theory and the Tannakian formalism

TL;DR

This paper connects model theory with the Tannakian formalism, showing how model theoretic internality concepts can derive fundamental results in Tannakian categories and their differential analogs.

Contribution

It introduces a novel approach by associating first order theories to Tannakian categories, deriving key results via model theoretic internality, and extends this to differential tensor categories.

Findings

Derived Tannakian formalism results using model theory

Formulated differential tensor categories and extended Tannakian results

Unified approach to classical and differential Tannakian theories

Abstract

We draw the connection between the model theoretic notions of internality and the binding group on one hand, and the Tannakian formalism on the other. More precisely, we deduce the fundamental results of the Tannakian formalism by associating to a Tannakian category a first order theory, and applying the results on internality there. We also formulate the notion of a differential tensor category, and a version of the Tannakian formalism for differential linear groups, and show how the same techniques can be used to deduce the analogous results in that context.