References:
A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics
Markov category gives a way to write probability theory that is ‘decoupled’ from a particular implementation (i.e. discrete distributions, continuous ones, or general measure theoretic ones). It is an interesting abstraction where you use morphisms to talk about random variables.
A markov category is a semi-cartesian symmetric monoidal category \((\mathcal{C}, \otimes, 1)\) that supplies cocommutative comonoids.
The means:
semi-cartesian: The category has counit \(\varepsilon: X\to 1\) and comultiplication \(\Delta_X:X\to X\otimes X\) for each \(X\). Typically the counit is called the ‘discard’ map (and is denoted by exclamation mark, or bang \(\!\)) and comultiplication is called the copy map.
monoidal: A monoidal structure is a ‘tensor product’ \(\otimes:\mathcal{C}\times \mathcal{C}\to \mathcal{C}\) and a unit \(1\in \mathcal{C}\) such that associativity and unit laws are satisfied. In principle this would need several gluing morphisms to describe the commutative diagrams involved.
symmetric: The monoidal structure is commutative.
supplying: A symmetric monoidal category \(\mathcal{C}\) supplies a property \(P\) if every object has a \(P\) structure that is compatible with tensor product.