Fiber functors, monoidal sites and Tannaka duality for bialgebroids

What are the fiber functors on small additive monoidal categories C which are not abelian? We give an answer which leads to a new Tannaka duality theorem for bialgebroids generalizing earlier results by Phung Ho Hai. The construction reveals a sheaf theoretic interpretation in so far as the reconstructed bialgebroid H has comodule category equivalent to the category of T-sheaves w.r.t. a monoidal Grothendieck topology on C. We also prove an existence theorem for fiber functors on small additive monoidal categories with bounded fusion and weak kernels. For certain autonomous categories a generalized Ulbrich Theorem can be formulated which relates fiber functors to Hopf algebroid Galois extensions.