[The Giry Monad](https://ncatlab.org/nlab/show/Giry+monad) captures probability measures. What is the adjunction that generates the Giry Monad? To narrow this down, perhaps we can talk about the adjunction between the category of Polish spaces and the Kleisli category for the Giry monad. Is there anything that can be said about *all* the adjunctions that generate the Monad?



What are the adjunctions that generate the Giry Monad?
It could be argued that this isn't quite the right question, or at least, not the most interesting one. 


Any adjunction gives rise to a monad, as you know. But more generally, any *functor* (subject to some mild conditions) gives rise to a monad, its **codensity monad**. If the functor you start with has a left adjoint, then the monad you end up with is the one generated by the adjunction. But the codensity monad is defined in many situations where the functor has no left adjoint, and that gets particularly interesting. [See here for a gentle explanation.](https://golem.ph.utexas.edu/category/2012/09/where_do_monads_come_from.html) 


For instance, let $\mathbf{3}$ be the subcategory of $\mathbf{Set}$ consisting of a three-element set only and all endomorphisms of it. Then the codensity monad of the inclusion $\mathbf{3} \hookrightarrow \mathbf{Set}$ is the ultrafilter monad.


So, you could ask more generally: which functors have the Giry monad as their codensity monad? [Tom Avery gave an answer.](https://arxiv.org/abs/1410.4432) In fact, Avery described *several* functors whose codensity monad is the Giry monad. Here we're dealing with the Giry monad on the category $\mathbf{Meas}$ of measurable spaces. 


For instance, let $d\_0$ be the set of all sequences in $[0, 1]$ converging to $0$, with its evident measurable structure, and let $D$ be the subcategory of $\mathbf{Meas}$ consisting of $d\_0$ only and all affine endomorphisms of it. Then the codensity monad of the inclusion $D \hookrightarrow \mathbf{Meas}$ is the Giry monad. This is Avery's Proposition 5.10(ii).


(Kirk Sturtz [also attempted](https://arxiv.org/abs/1406.6030v2) to describe the Giry monad as a codensity monad, but the attempt was wrong, as detailed in the appendix of Avery's paper. I don't know whether Sturtz succeeded in fixing it — the link just given is to v2 of an arXiv paper that later went up to v4, as well as undergoing a change of title.) 


