The Universal Property of Measure-Theoretic Probability

TL;DR

This paper establishes a universal property for the Markov category of standard Borel spaces, introducing a new concept of coinflip, and characterizes various Markov categories of discrete kernels.

Contribution

It introduces a new notion of coinflip in Markov categories and provides a universal property for the category of standard Borel spaces, advancing the theoretical understanding of measure-theoretic probability.

Findings

Universal property of BorelStoch category established

New notion of coinflip introduced and characterized

Universal characterizations of Markov categories of discrete kernels provided

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.