I am reading Awodey's text "Category Theory". In section 10.4 he talks about comonads and monads that occur together and interact and he says that "possibility" and "necessity" in propositional modal logic is an example. Can anyone give a few more interesting examples of interacting monads and comonads?
In particular, I am interested to know if interacting (co)monads form a frobenius algebra. Can ee define a frobenius algebra with a suitable monad comonad pair? Also, if anyone has seen a cstar algebra presented as a monad comonad pair, please let me know.