  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. In this paper we generalize Deligne's result to
weakly Tannakian categories in characteristic zero. The class of geometric
objects whose tensor categories of quasi-coherent sheaves can be recognized in
this manner includes both the gerbes arising in classical Tannaka duality and
more classical geometric objects such as projective varieties over a field of
characteristic zero.
  Our proof uses a different perspective on fiber functors, which we formalize
through the notion of geometric tensor categories. A second application of this
perspective allows us to describe categories of quasi-coherent sheaves on fiber
products.
