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What are the fiber functors on small additive monoidal categories C which arenot abelian? We give an answer which leads to a new Tannaka duality theorem forbialgebroids generalizing earlier results by Phung Ho Hai. The constructionreveals a sheaf theoretic interpretation in so far as the reconstructedbialgebroid H has comodule category equivalent to the category of T-sheavesw.r.t. a monoidal Grothendieck topology on C. We also prove an existencetheorem for fiber functors on small additive monoidal categories with boundedfusion and weak kernels. For certain autonomous categories a generalizedUlbrich Theorem can be formulated which relates fiber functors to Hopfalgebroid Galois extensions.

Fiber functors, monoidal sites and Tannaka duality for bialgebroids