Monads on tensor categories

A Hopf monad is a monad on a tensor category, equipped with comparison maps relating the monad structure to the tensor structure. We study some general properties of such Hopf monads, their algebras and their Hopf algebras. In particular, we prove that for a given monad, there is a one-to-one correspondence between additional data making the monad into a Hopf monad on the one hand, and liftings of the tensor to the category of algebras on the other.