Model Theory of Proalgebraic Groups

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.