Hopf comonads on naturally Frobenius map-monoidales (Q908323)

There are several ways to distinguish (classical) Hopf algebras among bialgebras: the existence of an antipode; the fact that the monad \(A\otimes -\) on the category of vector spaces is a left Hopf monad; the fact that the comonad \(A\otimes -\) is a left Hopf comonad; the fact that the category of Hopf modules over \(A\) is equivalent to the category of vector spaces; etc. These results are extended to monoidal comonads on a naturally Frobenius map-monoidale \(M\) in a monoidal bicategory \(\mathcal{M}\), seen as bimonoids in the duoidal hom-category \(\mathcal{M}(M,M)\). In particular, a notion of antipode is introduced. Under technical restrictions, it is shown that these conditions are equivalent. As examples, one obtains Hopf monoids in a braided monoidal category, small groupoids, Hopf algebroids over commutative algebras, weak Hopf algebras, Bruguières and Virelizier's Hopf monads.