The Tannaka-Krein reconstruction of a separable Frobenius functor is a weak Hopf algebra

Micah Blake McCurdy – 29 October 2008

If H is a Hopf object in a monoidal category V, then we can consider its category of representations, which comes equipped with a forgetful functor to V, in this way, one obtains a functor Hopf(V)-->MonCat/V, where we take as arrows in MonCat the strong monoidal functors. One can ask if there is a left adjoint to this functor, there certainly is in the case when we restrict MonCat (and hence V) to categories having left duals, namely, Tannaka-Krein reconstruction. This talk discusses a generalization of the above to the case where H is replaced with a weak Hopf algebra and we restrict the arrows in MonCat to be the Frobenius separable ones. Existing work has focussed on the case where V = Vector Spaces and giving sufficient (and sometimes necessary) conditions for the counit of the adjunction to be an isomorphism, that is, recognising when a category is (isomorphic to) a category of representations of a Hopf algebra. There is a small novel component in the above, but the main purpose of the talk is to illustrate a graphical notation for calculations involving separable Frobenius functors, so I will be giving (graphical) proofs in more detail than normal.