# ct.category theory – Does the notion of a Poisson monad exist?

Starting with a monoidal category with duals $$C$$, one may consider the category $$End(C)$$ of endofunctors of $$C$$. A Hopf monad on $$C$$ is a bimonad on $$C$$ with (a generalised notion of the) antipode. Under suitable assumptions for the Hopf monad, this Hopf monad applied on the unit object of the category $$C$$ is a Hopf algebra in the centre of $$C$$.

The question is if the notion of Poisson monad in $$End(C)$$ has been defined somewhere? In the case that such a notion exists, does it return a Poisson algebra object in the category?