From monads to comonads and back

Updated: 2021-01-29

I was recently recommended the paper Monads from Comonads, Comonads from Monads: An exercise in program transformation by Ralf Hinze. It’s a nice read, and I love the results.

I was taken aback by the title, because I thought that this couldn’t be true in general. I mean, due to the duality between monads and comonads for every monad there is a comonad and vice versa, but they will not be equivalent necessarily.

One example where a monad and a comonad differ can be found in linear logic. Take the of-course modality \(!A\) which is a comonad \(! : \cat{M} \mto{} \cat{M}\) on a symmetric monoidal category which adds the structural rules to linear logic. Take it’s dual and we arrive at the why-not modality \(? : \cat{M} \mto{} \cat{M}\). These two are not equivalent. I don’t know of any monad that is equivalent to \(!\) nor a comonad that is equivalent to \(?\); and I suspect there aren’t any. So in order for the results of this paper to work, there must be an additional assumption on the monad/comonad.

It turns out that there is an additional assumption. Hinze shows that given an adjunction \(\cat{A} : \func{F} \dashv \func{G} : \cat{A}\) where \(\func{G} : \cat{A} \mto{} \cat{A}\) is also a monad, then \(\func{F} : \cat{A} \mto{} \cat{A}\) is also a comonad. The first thing to notice is that this is not a general adjunction, because the functors must be endo-functors in order for the adjuncts to be a monad and a comonad respectively. So this is not true for every monad and comonad, but only those of which have left and right adjoints respectively. This means that for the of-course comonad there is no equivalent monad, because it does not have a right adjoint. The case is similar for the why-not monad.

But, if your monad or comonad can be shown to have left and right adjoints, then those adjoints are an equivalent–in terms of expressivity–comonad or monad respectively. This is a nice result. In addition, Hinze shows that the adjunction induces program transformations that translate the monad to its equivalent comonad and vice versa. Thus allowing one to program using which ever structure is easiest, and then obtaining the other through a transformation that could be automated.

I recommend reading the paper for the details.

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