Which abelian tensor categories are geometric?

Understand

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories.

• The notion of weakly Tannakian categories introduced by the author gives a characterization of tensor categories in the image of this embedding.
• However, this notion requires additional structure to be present, namely a fiber functor.
• For the case of classical Tannakian categories in characteristic zero, Deligne has found intrinsic properties - expressible entirely within the language of tensor categories - which are necessary and sufficient for the existence of a fiber functor.