On Graded Monads, Distributive Laws and Costrong Functors

Abstract

Strong functors and monads are ubiquitous in Computer Science. More recently, (strong) comonads have demonstrated their use in structuring context-dependent notions of computation. However, the dualisation of ``being strong'' property passed somehow unobserved so far. We argue that ``being costrong'' gives a different understanding of how functors can interact with monoidal structures. We shall see that the well-known correspondence between distributive laws $F T \to T F$ of an endofunctor $F$ over a monad $T$, on one hand, and extensions of $F$ to the Kleisli category of that monad, on the other hand, generalises from ordinary monads to graded ones. The gist here is to recognise that the costrength of a costrong functor is nothing but a ``graded'' distributive law. As such, ``being costrong'' is a structure that a functor may have. Examples of costrong functors with respect to different graded monads are provided, with emphasis to the cartesian case, and applications to optics and coalgebras are given.