The Tannaka construction in general monoidal categories, Part Two

Micah Blake McCurdy – 1 April 2009

Let F:A-->B be a functor between monoidal categories. When B is braided and suitably complete or cocomplete, we construct from F an algebraic object in B, which today we write as Tan(F). Previously, we have seen that, when F is a separable Frobenius functor, Tan(F) is a weak Hopf algebra, generalizing the known result that strong such functors give rise to Hopf algebras. We pursue this notion further, cataloguing the methods by which one can obtain further structure on Tan(F) from structures on A and B. We shall discuss three such: If A is braided then we construct a braided structure on Tan(F) If A and B are both ribbon, then we can construct a ribbon strucure on Tan(F) If A and B are both cyclic, then we can construct a cyclic structure on Tan(F) The notions of 1 and 2, namely, of braided and ribbon structures, are generalizations of existing notions (with the same names) for ordinary Hopf algebras. The third notion, of a cyclic structure on a a weak bi or Hopf algebra, is new, but appears to be related to existing notions of "special grouplike elements".