Abstract Kleisli Structures

Adrian Miranda – 23 March 2022

Abstract Kleisli Structures are used in computer science to model call by value programming languages with effects. As the name suggests, they abstract away certain structure which is always present on the Kleisli category of a monad. When the unit of the monad satisfies a certain equalising condition, the original monad can be recovered from the abstract Kleisli structure on its Kleisli category. In the literature, this is made precise by exhibiting abstract kleisli structures as objects of a full reflective subcategory of a category whose objects are monads.

As far as I have been able to find in the literature though, only strict morphisms between monads are considered and neither of the two possible notions of 2-cells of monads are considered. In the first part of this talk I will review the existing literature and fill these gaps. In the second part of this talk I will define abstract kleilsi structures on 2-categories and show how the results generalise to the pseudomonad setting.