On the occasion of my return to Mathstodon, a short note about my most recent preprint, The Universal Property of Measure-Theoretic Probability (https://arxiv.org/abs/2512.15485).
To summarize, I prove the following theorem: Among Markov categories \(\mathcal{C}\) such that
- The deterministic subcategory \(\mathcal{C}_\mathrm{det}\) is countably complete, (countably) extensive and Boolean,
- the inclusion \(\mathcal{C}_\mathrm{det} \hookrightarrow \mathcal{C}\) preserves the countable coproducts, pullbacks along coproduct inclusions, and carries the countable products to *Kolmogorov products* (in the sense of my previous paper https://arxiv.org/abs/1912.02769 with Tobias Fritz),
- there exists a morphism \(1 \to 1+1\) satisfying some equations identifying it as a "fair coinflip" (such a morphism is unique if it exists),
(and Markov functors which preserve this structure in an obvious sense), the usual Markov category \(\mathsf{BorelStoch}\) of standard Borel measurable spaces and measurable Markov kernels is (bi-) initial.
This uses a previous result by Chen, that the category of standard Borel spaces and measurable functions is initial among categories satisfying the requirements for \(\mathcal{C}_\mathrm{det}\) above. Thus the really interesting part is that, combined with the assumptions about how the stochastic maps interact with the limits and colimits (point 2 above), adjoining a single binary stochastic map suffices to generate all the probability measures, and moreover to prove all the equations between them!
