Synthetic probability is an attempt to isolate the essential features of probability theory (in the same way that, say, the axioms of a topological space are an attempt to isolate the essential features of Euclidean space that make it possible to study continuity). To be a bit more concrete, the prototypical "probability theory" is usually some variant of the category \mathsf {Meas} of measurable spaces, and so-called Giry monad G on it, which carries a measurable space X to the set of probability measures on it, equipped with the \sigma -algebra generated by evaluation at each measurable subset of X. We are asking which properties or what structure of this pair (\mathsf {Meas},G) make it suitable for probability theory.

Let me say a few words about why this is a relevant question to ask (after all, we already have one perfectly serviceable model). There are a few reasons this could be useful:

1. A pedagogical reason: by pulling out the essential features of probability theory, and separating them from incidental details, we can make proofs more understandable, and easier to find: if we only allow ourselves a few "moves" in any given situation, it's easier to find the right way. This is obviously subjective, but in my opinion this has been a great success of the school of Markov categories. However it's not the main advantage I want to focus on in these lectures - in fact we will have to prove a number of things in an incredibly annoying way because of the minimal setting I will place us in.
2. A philosophical reason: it is interesting to think about why probability theory is the way it is. In some sense it's about modeling natural phenomena that involve uncertainty or frequency, but there certainly aren't any \sigma -algebras out in the world. Studying the foundations from this synthetic point of view might give us insight into which parts are really necessary.
3. A technical reason: there might be other "models of probability theory" that are useful in themselves. In particular from a programming languages point of view, we may be interested in characterizing the initial model of probability theory (in the presence of some other constructions) - this will be, in some sense, the syntax of probabilistic programming.

We will primarily be concerned by this third point.

Lawvere was the first to consider the monad G as such, and understand that its Kleisli category was the essential thing to understand probability theory. He presented this at a seminar in '62 but never published his work, and so this monad has been named after Michele Giry, who reinvented the same idea in '82.

A Kelisli morphism for the Giry monad is (of course) a measurable function f: X \to GY assigning to each x: X a probability measure on Y. The measurability constraint says that for every measurable subset U \subseteq Y, the assignment f(-)(U): X \to \mathbb {R} is itself measurable (for the usual Borel \sigma -algebra on \mathbb {R}). This is precisely what is known as a Markov kernel to probability theorists.

We may regard a Markov kernel as a sort of "detached" conditional probability distribution---that is, it's the type of thing a conditional distribution is. It tells us the distribution of y : Y given x : X. (This is a bit misleading, as conditional distributions are only defined up to a certain equivalence relation, but we won't discuss that now).

We can also view a Markov kernel as a "probabilistic mapping" (this the term Lawvere used). That is, it is something which computes an element y : Y given x : X, but may do so in a random way. In this sense it is like an effectful function in programming.

Let us start being a little bit precise:

Definition Giry Monad

• May 6, 2026
• Eigil Fjeldgren Rischel

Let \mathsf {Meas} denote the category whose objects are measurable spaces, and whose maps are measurable functions. Given a measurable space X=(X,\Sigma ), let GX denote the measurable space whose elements are probability measures on X, and whose \sigma -algebra is the one generated by the maps \operatorname {ev}_U : \mu \mapsto \mu (U) : GX \to \mathbb {R}. For f: X \to Y measurable, let Gf: GX \to GY be defined as the image measure. Then G is an endofunctor on \mathsf {Meas}.

GX carries a monad structure defined as follows

1. The unit X \to GX carries each point x: X to the Dirac measure \delta _x(U) = [x \in U] (that is, it is one on those sets that contain x and zero otherwise)
2. The multiplication GGX \to GX carries a measure \nu on probability measures to the measure defined by U \mapsto \int _{\mu \in GX} \mu (U) d\nu . (That is, the expected value of \mu (U) if \mu is sampled according to \nu )

(Of course somebody has to verify that this is associative and unital, that the integral is actually a measurable operation, that it results in a probability measure, and that it is natural. I won't do this.)

What is the composition in this Kleisli category? Given f: X \to GY, Y \to GZ we want to obtain X \to GZ by saying "sample y from f(x), then z from g(y)".