Tannakian formalism over fields with operators

Given an algebraic group (or more generally, a group scheme) $G$ over a field $K$, one may consider the category $C$ of $K$-linear representations of $G$. The classical Tannakian formalism describes the structure of such a category $C$ as an abstract (tensor) category. In particular, it shows that $G$ can be recovered from its category of representations.

If the field $K$ is given with some operators, such as a derivation or an automorphism, one could consider groups $G$ and representations definable with the extra operators. The collection of all such representations again forms a tensor category $C$, and it is natural to ask for an analogue of the Tannakian formalism. I will outline such a formalism in a general context, developed by Moosa and Scanlon, that includes the cases of differential and difference fields and their combination.