On Graded Monads, Distributive Laws and Costrong Functors
arXiv:2509.13026 · doi:10.1016/j.jlamp.2026.101148
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 of an endofunctor over a monad , on one hand, and extensions of 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.
Change of title; the current version (arXiv:2509.11877v2) is a revised and significantly extended version of arXiv:2509.11877v1 (which was published in FROM 2025 proceedings, EPTCS 427, 2025, pp. 141-154, DOI: 10.4204/EPTCS.427.10). In particular, Section 3, devoted to graded monads and distributive laws, is entirely new and did not appear in arXiv:2509.11877v1