Weak morphisms of weak Hopf algebras

Micah Blake McCurdy – 5 August 2009

Hopf algebras are defined with a built in symmetry between algebra and colagebra structures, and the usual notion of morphism of Hopf algebras is similarly symmetric: namely, a moprhism which strictly preserves the units, counits, multiplications, and comultiplications. Let us call such morphisms "strong" morphisms of Hopf algebras. As Kornel Szlachanyi has pointed out, although weak Hopf algebras are also symmetrically defined, the usual notion of strong moprhism is not quite suitable when taken between weak Hopf algebras. A slight modification of the weak morphisms he suggests are useful to our purposes, specifically: We have described before a generalized Tannaka construction which takes a separable Frobenieus functor F:A--->B and produces a weak Hopf algebra in B, this generalizes the known construction which takes a strong monoidal such functor and produces a Hopf algebra. It is also known that this construction provides a left adjoint tan:Aut-Strong/B ----> Hopf(B) to the representation functor. By extending to weak morphisms between weak Hopf algebras, we obtain another "reconstruction-representation" adjunction, between the categories SepFrob/B and weak Hopf algebras in B.