The Universal Property of Measure-Theoretic Probability
Eigil Fjeldgren Rischel
TL;DR
The paper develops a universal property framework for probability in Markov categories by introducing a canonical coinflip (unbiased binary choice) structure, modeled as a midpoint algebra, and proving that the standard Borel stochastic category BorelStoch is initial among countably extensive Boolean coinflip Markov categories with Kolmogorov products. It then shows how discrete probability and standard Borel probability lift to this universal setting via iterated sampling constructions, yielding essentially unique Markov functors from base categories (e.g., Set^{<kappa} and Borel_{Delta}) into a target category. A key outcome is a monad-level correspondence: any bicommutative affine monad with coinflip and iterable midpoint structures admits a unique monoidal monad morphism from the Giry monad, reflecting the universality of measure-theoretic probability in this framework. The results offer a principled categorical foundation for probability and probabilistic programming syntax, with potential generalizations to wider classes of measurable spaces and related stochastic categories.
Abstract
Building on work of Chen, we give a universal property of the Markov category BorelStoch of standard Borel spaces and Markov kernels between them. To do this, we introduce a new notion of *coinflip*, or unbiased binary choice, in a Markov category. These are unique if they exist, and automatically preserved by all Markov functors which preserve coproducts. We also provide universal characterizations of various Markov categories of discrete kernels.