  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.
