Commutativity and Kleisli laws of codensity monads of probability measures

Abstract

Several monads of probability measures have been shown to have presentations as codensity monads over small categories of stochastic maps. This paper studies how three key properties of these probability monads, relevant to categorical approaches to probability, can arise from their codensity presentations. We first derive the existence of a Kleisli law into the Giry monad, which provides a formal connection to measurable probability. In particular, from their codensity presentations, we prove a novel universal property of several probability monads as terminal liftings of the Giry monad. This generalises a result by Van Breugel on the Kantorovich monad, and proves the existence of such Kleisli laws. We additionally provide sufficient conditions for a codensity monad to be lax monoidal and affine, which provides a connection to the theory of Markov categories. In particular, we introduce the condition for a codensity monad to be exactly pointwise monoidal, which is then lax monoidal, and prove a characterisation of this condition in terms of Day convolution. We show that the Radon monad is exactly pointwise monoidal, and use our characterisation to 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 only exactly pointwise monoidal when restricted to standard Borel spaces, due to the existence of probability bimeasures that do not extend to measures.