Hopf algebras are closely related to monoidal categories. More precise,
k-Hopf algebras can be characterized as those algebras whose category of
finite dimensional representations is an autonomous monoidal category such that
the forgetful functor to k-vectorspaces is a strict monoidal functor. This
result is known as the Tannaka reconstruction theorem (for Hopf algebras).
Because of the importance of both Hopf algebras in various fields, over the
last last few decades, many generalizations have been defined. We will survey
these different generalizations from the point of view of the Tannaka
reconstruction theorem.Comment: Survey paper to appear in to appear in C. Heunen, M. Sadrzadeh, E.
Grefenstette (eds.): Compositional methods in quantum physics and
linguistics, Oxford University Press, 201
