# Hopf monads in categorical probability theory

1. Context. According to [1], probability monads are arguably the most important concept in categorical probability theory. In [2] Fritz and Perrone argue that "in order for a monad to really count as a probability monad, it should be a bimonoidal monad." Here, a bimonoidal monad is a monad whose underlying functor is a bilax monoidal functor, and whose unit and multiplication are both lax and oplax monoidal natural transformations. In particular, a bimonoidal monad has an underlying oplax monoidal monad, also called a bimonad.

In [3], Bruguières, Lack and Virelizier introduced the notion of a Hopf monad on a monoidal category. A left (resp. right) Hopf monad on a monoidal category is an opmonoidal monad whose left (resp. right) fusion operator is invertible. Hopf monads are generalizations of Hopf algebras, and more generally Hopf algebroids.

2. Question. This leads to the following question:

3. Remarks. Left (resp. right) Hopf monads are precisely those opmonoidal monads $$T$$ on a left (resp. right) closed monoidal category $$\mathcal{C}$$ such that their Eilenberg-Moore category $$\mathcal{C}^T$$ is left (resp. right) closed monoidal and the forgetful functor $$\mathcal{C}^T\rightarrow \mathcal{C}$$ is left (resp. right) closed. This is Theorem 3.6. in [3]. It is claimed here that the Eilenberg-Moore category of a probability monad is less useful than its Kleisli category since it "lacks copying morphisms" and "non-free algebras of a probability monad rarely come up". Thus, it would surprise me to find an intricate connection between probability monads and Hopf monads, but these are just heuristic considerations.

• A suggestion - in some natural examples, the Kleisli category is equivalent to the category of comonoids in the Eilenberg-Moore category... Commented Feb 8 at 10:38

Since the idea of a probability monad is not a fully formal concept, there can be no fully formal answer to this question. Nevertheless, I will argue below that no nontrivial probability monad is a Hopf monad.

Probability monads: The idea behind this concept is that such a monad consists of the following:

• The underlying symmetric monoidal category is some kind of category of "spaces", where the tensor product is a version of the Cartesian product of spaces. (Which is not necessarily the categorical product; metric spaces with $$\ell^1$$-product are an example where it isn't.)
• For every space $$X$$, the object $$TX$$ is the space of probability measures on $$X$$, and the unit $$\delta_X : X \to TX$$ assigns to every point $$x$$ the Dirac delta $$\delta_x$$.
• The multiplication $$T^2 X \to TX$$ takes a probability measure on probability measures—often called a random measure—and maps it to its expectation value.
• The monoidal structure maps $$TX \otimes TY \to T(X \otimes Y)$$ implement the formation of product measures.
• The opmonoidal structure maps $$T(X \otimes Y) \to TX \otimes TY$$ implement the formation of marginals by mapping every measure on a product space to the pair of its marginals.

Hopf monads: As per the nLab, an opmonoidal monad is a Hopf monad if the left fusion operator $$T(X \otimes TY) \to TX \otimes T^2 Y \to TX \otimes TY$$ is invertible, and similarly for the right fusion operator.

The actual answer: Let's see now what the left fusion operator does in a probability context. An element of $$T(X \otimes TY)$$ is a joint distribution of an $$X$$-valued random variable and a random measure on $$Y$$. Here are two examples: $$\frac{1}{2} \delta_{\left(x,\delta_y\right)} + \frac{1}{2} \delta_{\left(x',\delta_{y'}\right)}, \qquad \frac{1}{2} \delta_{\left(x,\delta_{y'}\right)} + \frac{1}{2} \delta_{\left(x',\delta_y\right)}$$ for points $$x,x' \in X$$ and $$y, y' \in Y$$. Both are measures which correlate a point in $$X$$ with a probability measure on $$Y$$. Now the first map above is $$T(X \otimes TY) \to TX \otimes T^2Y$$, which is one of the opmonoidal structure maps and therefore amounts to taking the pair of marginals. This gives the pair $$\left( \frac{1}{2} \delta_x + \frac{1}{2} \delta_{x'}, \frac{1}{2} \delta_{\delta_y} + \frac{1}{2} \delta_{\delta_{y'}} \right)$$ for both of the elements of $$T(X \otimes TY)$$ above. Therefore our monad is not a Hopf monad, already because the first map is not injective. In fact, the second map is not injective either, and we could construct other pairs of elements of $$T(X \otimes TY)$$ where the invertibility fails at that step.

There are two more general and abstract ways to see what goes wrong:

• In a probability monad, the monoidal structure maps $$TX \otimes TY \to T(X \otimes Y)$$ are right inverse to the opmonoidal structure maps $$T(X \otimes Y) \to TX \otimes TY$$. In probability terms, this expresses the fact that the two marginals of a product measure are exactly the original factors. (See our paper for more details.) Whenever you have this property, then the first map above is a split epimorphism; so if $$T$$ is a Hopf monad as well, then this opmonoidal structure map must be an isomorphism, and therefore $$T$$ is even strong monoidal, at least on all pairs of objects of the form $$(X, TY)$$. But strong monoidality trivializes the probabilistic interpretation, because it says that every probability measure is uniquely determined by its marginals. Since probability theory is all about how things are correlated, there is no nontrivial flavour of probability theory which satisfies this, and therefore no nontrivial probability monad that is a Hopf monad.

• A probability monad is affine, meaning that $$TI \cong I$$ canonically, because there is exactly one probability measure on a one-point space. When you plug $$X = I$$ into the definition of Hopf monad above and use $$TI \cong I$$, you'll see that $$T$$ being a Hopf monad amounts to the multiplication $$T^2 Y \to TY$$ being invertible. Thus an affine monad is a Hopf monad if and only if it is idempotent. Once again this does not happen for any nontrivial probability monad, because it amounts to saying that the formation of expectation values is a bijection.

• It might be off base what I am going to ask, but, since being an algebra for a typical probability monad usually does not exactly mean being a probability space in traditional sense - could it happen that the latter (i. e. traditional probability spaces) still do possess the properties one would expect of an algebra over a Hopf monad? Commented Feb 11 at 9:50
• @მამუკაჯიბლაძე, what relation do you see between a probability space and an algebra of a probability monad at all? Such an algebra is a kind of space with a notion of integral, i.e. on which every probability measure has a designated expectation. An abstract probability space doesn't have this structure. It has a designated probability measure instead, which an algebra of a probability monad doesn't have. Commented Feb 11 at 13:31
• A suitable way to talk about probability spaces in categorical terms is to use Markov categories. (But even then, probability spaces are surprisingly redundant for the purposes of probability theory.) Commented Feb 11 at 13:32
• Thank you for the clarification, and for pointing out the notion of Markov category which I did not know! I will think about it. Commented Feb 11 at 16:17