A causal Markov category with Kolmogorov products

TL;DR

This paper constructs a causal Markov category with Kolmogorov products using Stone spaces and Radon monads, addressing a problem in categorical probability theory.

Contribution

It provides a new example of a causal Markov category with all small Kolmogorov products, linking Stone spaces, Boolean algebras, and effect algebras.

Findings

Deterministic subcategory is the category of Stone spaces.

Kernels correspond to a restricted class of Kleisli arrows for the Radon monad.

Dual perspectives via pro-completions and duality with Boolean and effect algebras.

Abstract

In Fritz & Rischel, Infinite products and zero-one laws in categorical probability, the problem was posed of finding an interesting Markov category which is causal and has all (small) Kolmogorov products (there Problem 6.7). Here we give an example where the deterministic subcategory is the category of Stone spaces (i.e. the dual of the category of Boolean algebras) and the kernels correspond to a restricted class of Kleisli arrows for the Radon monad. We look at this from two perspectives. First via pro-completions and Stone spaces directly. Second via duality with Boolean and algebras and effect algebras.