Strong Dinatural Transformations and Generalised Codensity Monads

We introduce dicodensity monads: a generalisation of pointwise codensity monads generated by functors to monads generated by mixed-variant bifunctors. Our construction is based on the notion of strong dinaturality (also known as Barr dinaturality), and is inspired by denotational models of certain types in polymorphic lambda calculi - in particular, a form of continuation monads with universally quantified variables, such as the Church encoding of the list monad in System F. Extending some previous results on Cayley-style representations, we provide a set of sufficient conditions to establish an isomorphism between a monad and the dicodensity monad for a given bifunctor. Then, we focus on the class of monads obtained by instantiating our construction with hom-functors and, more generally, bifunctors given by objects of homomorphisms (that is, internalised hom-sets between Eilenberg--Moore algebras). This gives us, for example, novel presentations of monads generated by different kinds of semirings and other theories used to model ordered nondeterministic computations.