The theory of localisable monads

Nuiok Dicaire – 7 June 2023

Monads have many useful applications both in mathematics and in computer science. Notably they provide a convenient way to describe computational side-effects. An important question is how to handle several instances of such side-effects or a graded collection of them. The usual approach consists in defining many “small” monads and combining them together using distributive laws.

In this talk, we take a different approach and look for a pre-existing internal structure to a monoidal category that allows us to develop a fine-graining of monads. We call the monads obtained "localisable". This uses techniques from tensor topology and provides an intrinsic theory of local computational effects without needing to know how the constituent effects interact beforehand. In this talk we introduce notions in tensor topology, define localisable monads, and show how they are equivalent to monads in a specific 2-category. To motivate the talk, we will also consider an application to the modelling of concurrent systems.