Dilations and information flow axioms in categorical probability

We study the positivity and causality axioms for Markov categories as properties of dilations and information flow and also develop variations thereof for arbitrary semicartesian monoidal categories. These help us show that being a positive Markov category is merely an additional property of a symmetric monoidal category (rather than extra structure). We also characterize the positivity of representable Markov categories and prove that causality implies positivity, but not conversely. Finally, we note that positivity fails for quasi-Borel spaces and interpret this failure as a privacy property of probabilistic name generation.

Automated summary

This paper studies the positivity and causality axioms for Markov categories as properties of dilations and information flow. It also develops variations for arbitrary semicartesian monoidal categories. The research shows that being a positive Markov category is just an additional property of a symmetric monoidal category, not extra structure. The positivity of representable Markov categories is characterized, and it is proven that causality implies positivity. Furthermore, the paper finds that positivity fails for quasi-Borel spaces and interprets this failure as a privacy property of probabilistic name generation.