Tannaka duality describes the relationship between algebraic objects in a given category and functors into that category; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful ``fibre functors'' to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras with weak Hopf algebras and strong monoidal functors with separable Frobenius monoidal functors; second, we replace the category of vector spaces with an arbitrary braided monoidal category. To accomplish this goal, we make use of a graphical notation for functors between monoidal categories, using string diagrams with coloured regions. Not only does this notation extend our capacity to give simple proofs of complicated calculations, it makes plain some of the connections between Frobenius monoidal or separable Frobenius monoidal functors and the topology of the axioms defining certain algebraic structures. Finally, having generalized Tannaka duality to an arbitrary base category, we briefly discuss the functoriality of the construction as this base is varied.
Keywords: Tannaka duality, Tannaka reconstruction, bialgebras, Hopf algebras, weak bialgebras, weak Hopf algebras, separable Frobenius monoidal functors, graphical methods
2000 MSC: 18D10
Theory and Applications of Categories, Vol. 26, 2012, No. 9, pp 233-280.