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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.

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Ben, I do not know anything on the subject, but another example of this kind is the duality between quantifiers $\exists$ and $\forall$, and more generally, you will find a lot of such examples in almost any interesting fibration (with comonad $f^* \circ \Pi_f$ and monad $f^* \circ \Sigma_f$). However, I do not understand how such a monad-comonad pair could form a Frobenius algebra, as the later requires both monoidal and comonoidal structure to operate on a single carrier, and in our examples the carriers are just far different (i.e. the functors are different). – Michal R. Przybylek May 27 '12 at 12:55
Thanks Michal! The set theory example sounds very interesting. Perhaps you are right about the necessity for a single carrier (which I assume is the endofunctor). Are you saying that one could not ever have a comanad and monad from the same endofunctor? – Ben Sprott May 27 '12 at 17:39
Any identity functor is both a monad and a comonad. What I was to say, is that in our examples this is not the case. I still do not know what is that mysterious "interaction" between monads and comonads that you are refering to. However, it is easy to check, that if a monad/comonad pair arises from an adjoint triple $\Sigma_f \dashv f \dashv \prod_f$ and $\Sigma_f \approx \prod_f$ then the pair form a frobenius algebra. – Michal R. Przybylek May 28 '12 at 11:13
Hi Michal, Could you also see how to create a Cstar algebra from the frobenius algebra? – Ben Sprott May 29 '12 at 18:06

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