Model Theory of Proalgebraic Groups

TL;DR

This paper develops a model theoretic framework for proalgebraic groups, using tannakian philosophy and tensor categories, to analyze their structure and classify their theories.

Contribution

It introduces a first order axiomatization of proalgebraic groups based on tannakian categories and analyzes the theory of diagonalizable proalgebraic groups in detail.

Findings

The theory of a diagonalizable proalgebraic group is determined by the base field and its character group.

A tensor analog of skeletal categories is used to model neutral tannakian categories.

Initial steps are taken towards understanding types in this model theoretic setting.

Abstract

We lay the foundations for a model theoretic study of proalgebraic groups. Our axiomatization is based on the tannakian philosophy. Through a tensor analog of skeletal categories we are able to consider neutral tannakian categories with a fibre functor as many-sorted first order structures. The class of diagonalizable proalgebraic groups is analyzed in detail. We show that the theory of a diagonalizable proalgebraic group $G$ is determined by the theory of the base field and the theory of the character group of $G$. Some initial steps towards a comprehensive study of types are also made.