Localisable Monads

Monads govern side effects in programming semantics and, when a computation uses more than one effect, the corresponding monads can be combined using distributive laws into a single monad. A related use of monads is to have several layers of access to a computational effect, as modelled by indexed and graded monads. This is usually conceived in a 'bottom-up' fashion, where one specifies the behaviour at each level and then adds interplay between the levels.

In this talk, the opposite, 'top-down' approach will be taken: starting with a single monad on a category with some structure we will ask when and how is that monad a combination of constituent monads. This is done using the mathematical formalism of tensor topology, which associates a notion of base space to a given monoidal category. The category then decomposes as a sheaf of categories over its associated base space. Thus, the main question is when and how a monad respects this decomposition.

This work is a first step towards an intrinsic theory of computational effects, which does not need to specify in detail how constituent effects have to interact in advance.