Within the study of the semantics of programming languages, computational effects may be modelled with monads, and weak distributive laws between monads are then a tool to combine two such effects. The first part of the talk will be dedicated to introducing (monotone) (weak) distributive laws.
In both the category of sets and the category of compact Hausdorff spaces, there is a monotone weak distributive law that combines two layers of non-determinism. Noticing the similarity between these two laws, we study in the second part of the talk whether the latter can be obtained automatically as some sort of lifting of the former.
More specifically, we show how a framework for constructing monotone weak distributive laws in regular categories lifts to categories of algebras, giving a full characterization for the existence of monotone weak distributive laws therein. We then exhibit such a law, combining probabilities and non-determinism, in compact Hausdorff spaces; but we also show how such laws do not exist in a lot of other cases.