Constructing colimits by gluing vector bundles

As already observed by Gabriel, coherent sheaves on schemes obtained by gluing affine open subsets can be described by a simple gluing construction. An example due to Ferrand shows that this fails in general for pushouts along closed immersions, though the gluing construction still works for flat coherent sheaves. We show that by further restricting this gluing construction to vector bundles, we can construct pushouts along arbitrary morphisms (and more general colimits) of certain algebraic stacks called Adams stacks. The proof of this fact uses generalized Tannaka duality and a variant of Deligne's argument for the existence of fiber functors which works in arbitrary characteristic. We use this version of Deligne's existence theorem for fiber functors as a novel way of recognizing stacks which have atlases. It differs considerably from Artin's algebraicity results and their generalizations: rather than studying conditions on the functor of points which ensure the existence of an atlas, our theorem identifies conditions on the category of quasi-coherent sheaves of the stack which imply that an atlas exists.