Commutative codensity monads and probability bimeasures

Several well-studied probability monads have been expressed as codensity monads over small categories of stochastic maps, giving a limit description of spaces of probability measures. In this paper we show how properties of probability monads such as commutativity and affineness can arise from their codensity presentation. First we show that their codensity presentation is closely related to another characterisation of probability monads as terminal endofunctors admitting certain maps into the Giry monad, which allows us to generalise a result by Van Breugel on the Kantorovich monad. We then provide sufficient conditions for a codensity monad to lift to $\bf{MonCat}$, and give a characterisation of exactly pointwise monoidal codensity monads; codensity monads that satisfy a strengthening of these conditions. We show that the Radon monad is exactly pointwise monoidal, and hence give a description of the tensor product of free algebras of the Radon monad in terms of Day convolution. Finally we show that the Giry monad is not exactly pointwise monoidal due to the existence of probability bimeasures that do not extend to measures, although its restriction to standard Borel spaces is. We introduce the notion of a $*$-monad and its Kleisli monoidal op-multicategory to describe the categorical structure that organises the spaces of probability polymeasures on measurable spaces.