Nuclear monads and their uses

Every adjunction induces a monad and a comonad, and hence a category of algebras and a category of coalgebras. I will describe a canonical adjunction between algebras and coalgebras, and prove that the right adjoint is always monadic, whereas the left adjoint is always comonadic. Such an adjunction, and the induced monad and comonad, are called nuclear. If the monad is induced by some constructors of computational effects, then the comonad displays the canonical destructors of the same effects. The fact that they are nuclear over each other's resolutions makes them into a curious phenomenon: a pseudomonad over adjunctions, monads, or comonads, that is idempotent on the nose. It disproves the claim that there is an infinite sequence of constructions of "algebras over coalgebras over algebras...", which was consistently repeated since the seminal categorical algebra publications from the 60s, as well as in recent literature.

There are at least three interesting consequences. On the practical side, the result gives a convenient version of the the Eilenberg-Moore construction of categories of algebras or coalgebras, where the split coequalizers become idempotents. (This was presented at LICS 2017.) On the theoretical side, it resolves the Lambek-Isbell conundrum of the Dedekind completion of a category, also going back to the 60s, and uncovers the categorical versions of limit inferior and superior. (The latter was presented at CALCO 2016.) On the meta-level, it shows that the familiar process, whereby solving categorical problems helps with practical applications, has a feedback loop, whereby practical applications can help solving categorical problems.