Tannakian formalism over fields with operators

We develop a theory of tensor categories over a field endowed with abstract operators. Our notion of a "field with operators", derived from the work of Moosa and Scanlon, includes the familiar cases of differential and difference fields, Hasse-Schmidt derivations, and their combinations. We develop a corresponding Tannakian formalism, describing the category of representations of linear groups defined over such fields. This paper extends the previously known (classical) algebraic and differential algebraic Tannakian formalisms.