Classical Tannaka duality is a duality between groups and their categories of representations. It answers two basic questions: can we recover the group from its category of representations, and can we characterize categories of representations abstractly? These are often called the reconstruction problem and the recognition problem. In the context of affine group schemes over a field, the recognition problem was solved by Saavedra and Deligne using the notion of a (neutral) Tannakian category.
In this talk I will explain how this theory can be generalized to the context of certain algebraic stacks and their categories of coherent sheaves (using the notion of a weakly Tannakian category). On Tuesday [in the Quantum Algebras seminar] I will talk about work in progress to construct universal weakly Tannakian categories and some of their applications. The aim is to interpret various constructions on stacks (for example fiber products) in terms of the corresponding weakly Tannakian categories.