Theory of localisable monads

Abstract

Monads are used in programming semantics to govern computational side-effects such as reading and writing to a memory store, non-determinism, exceptions, and interactive inputs and outputs among others. Monads are also useful in mathematics where they provide a way of studying algebras at the levels of theories rather than specific structures. An important question that arises is how to combine monads that, for example, describe several instances of side-effects or graded collections of them. The general approach consists in defining many “small” monads and combining them together using distributive laws.

Here 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. The internal structure in question is a lattice of central idempotents. Central idempotents are a generalisation of idempotent ideals in modules over rings. They can also be thought of as open sets of a hidden base space over which a monoidal category decomposes as a sheaf of local categories.

In this thesis we ask the following question: given a monad on a monoidal category, when does it decompose as a presheaf of monads? We provide an answer in terms of the existence of a strength-like map for each central idempotent. We go on to show that localisable monads are equivalent to a specific type of formal monads in an appropriate presheaf 2-category. We then demonstrate how localisable monads can interpret the base space as locations in a computer memory and as time in stochastic processes.

Next, we study how the structure of localisability can be added to monoid-graded monads. This is inspired by a recent paper by Orchard et al. that showed that graded monads and parameterised monads are unified under a generalisation of 2-category-graded monads. We extend this to localisable 2-category-graded monads and recover equivalence theorems similar to those for localisable monads.

Lastly, we explore how localisable monads can be used to describe the contribution of a given agent in a network of interacting agents acting concurrently. We discuss how central idempotents in a monoidal category are insufficient to capture this scenario and show it is instead described by defining localisable monads on categories with an action by central idempotents.