What are the adjunctions that generate the Giry Monad?

Question

The 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?

Answer 1

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.

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. 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 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.)

Answer 2

Theorem 7.2 of Kirk Sturtz, The factorization of the Giry monad and convex spaces as an extension of the Kleisi category, provides one answer for measurable spaces.

Answer 3

Recall, for any monad, the Kleisi category of the monad is initial while the Eilenberg-Moore (EM) category of the monad is final among all ways of factoring that monad as an adjoint pair. (To see how this works, see, for example, Toposes, Triples, and Theories by Barr & Wells (BW), p89-90. ) Loosely speaking, we say the Kleisi category is the smallest factorization and the Eilenberg-Moore category (of the monad) is the largest factorization possible. Generally, we are interested in knowing the largest factorization (we can always recover the smallest factorization by taking the ``free objects'' in the Eilenberg-Moore category, i.e., the Kleisi category is embedded in the EM category as the image... which is Prop. 2.2 BW.).

The Eilenberg-Moore category of the Giry monad is equivalent to the category of convex spaces. The proof of the equivalence of the Eilenberg-Moore category of the Giry monad and the category of convex spaces is given in my article referenced by David above. (The refereed journal version should be available later this year.)

Note that for applications, the abstract presentation of the EM category is often not so useful - and the problem is to characterize the EM category as equivalent to a better known category. By ``not so useful'' I simply mean we cannot use our intuition (or existing knowledge) to quickly understand the properties of the category.)

The category of (abstract) convex spaces is the affine part of the category of $K$-modules. (See Meng's thesis for Lawvere's development of this viewpoint of convex spaces.) Alternatively, see S. Gudder's paper Convex Structures and Operational Quantum Mechanics for the (now standard) axiomatic definition.

The Giry algebras are simply convex spaces, and the theory of probability can be extended to the category of convex spaces. By this I mean that every convex space has a canonical $\sigma$-algebra associated with it. For example, the canonical $\sigma$-algebra of the real line is the Borel $\sigma$-algebra (as one would expect). Now the whole point (at least from my perspective) of using the largest factorization of the Giry monad is that all the concepts of (Bayesian) probability theory can now be placed within the framework of the category of convex spaces, $\mathbf{Cvx}$, wherein the inference maps can be explicitly calculated. (One cannot do this using the Kleisi category of the Giry monad - there it is necessary to place additional restrictions on the spaces involved to be able to construct the inference maps.) In fact, if you analyze the category of convex spaces you are ``doing (Bayesian) probability theory''. I speculate that by generalizing this factorization from the category of measurable spaces, $\mathbf{Meas}$, to say the category of Orthomodular Lattices, one should encounter various aspects of Quantum probability, i.e., the Giry monad is a commutative monad with respect to the (standard) product monoidal structure. However , that product monoidal structure does not yield a symmetric monoidal closed category, and it is necessary to invoke the tensor product monoidal structure to obtain a SMCC. Under this monoidal structure, coupled with non Boolean nature of orthomodular lattices, one may encounter noncommutative Probability theory. (See Anders Kock's definition of commutative monads in Monads on Symmetric Monoidal Closed Categories.) Indeed, it is the tensor product monoidal structure of the two categories, $\mathbf{Meas}$ and $\mathbf{Cvx}$, both generated by an identical process using the constant graph functions, which makes the factorization of the Giry monad rather elementary once that connection is recognized.

Answer 4

Answer to Roberts question: That's correct - the obvious choice of an equivalence does not work. Take the adjunction between $\mathbf{Meas}$ and $\mathbf{Cvx}$ which yields the Giry monad. Using the notation in my paper, $\mathcal{P} \dashv \mathbf{\Sigma}$, $\mathcal{G} = \Sigma \mathcal{P}$. To obtain the convex space corresponding to any Giry algebra $\mathcal{G}(X) \rightarrow X$. You take the coequalizer of the pair shown in the diagram on page 34 - so (the object) $Coeq$, which is a convex space, corresponds to that Giry algebra $h: \mathcal{G}(X) \rightarrow X$. (True for any measurable space $X$, including the real line $\mathbb{R}$.) Conversely, given a convex space $C$ the Giry algebra is obtained by taking the counit of the adjunction, $\epsilon_{C}: \mathcal{P}\Sigma(C) \rightarrow C$. This is explained on the bottom of page 35. Now, to make it a measurable map (and a Giry algebra) just apply the functor $\Sigma$ to that counit, which endows those two convex spaces (domain & codomain) with a $\sigma$-algebra structure.