Abstract: Tannakian duality as developed by Grothendieck, Saavedra, Deligne and Milne is a duality between geometric objects (affine group schemes, gerbes) and tensor categories (Tannakian categories). A Tannakian category is a tensor category equipped with additional structure which ensures that it is equivalent to the category of representations of an affine group scheme or gerbe. In characteristic zero Deligne has found a particularly simple description of Tannakian categories. This description is convenient since it only involves talking about properties of the tensor category. The properties in question are enough to construct the required additional structure.

Tannakian duality can be extended to a broader class of geometric objects including schemes and certain algebraic stacks. The corresponding tensor categories (that is, the categories of coherent sheaves on these objects) are the weakly Tannakian categories. I will review the notion of weakly Tannakian category, and I will talk about work in progress to generalize Deligne's description of Tannakian categories in characteristic zero to an intrinsic description of weakly Tannakian categories in characteristic zero.