Giry and the Machine

We present a general method - the Machine - to analyse and characterise in finitary terms natural transformations between well-known functors in the category Pol of Polish spaces. The method relies on a detailed analysis of the structure of Pol and a small set of categorical conditions on the domain and codomain functors. We apply the Machine to transformations from the Giry and positive measures functors to combinations of the Vietoris, multiset, Giry and positive measures functors. The multiset functor is shown to be defined in Pol and its properties established. We also show that for some combinations of these functors, there cannot exist more than one natural transformation between the functors, in particular the Giry monad has no natural transformations to itself apart from the identity. Finally we show how the Dirichlet and Poisson processes can be constructed with the Machine.