Dilations and information flow axioms in categorical probability

Abstract

We study the positivity and causality axioms for Markov categories as properties of dilations and information flow in Markov categories, and in 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.

49 pages. v2: The published version. v3: A correction to the erroneous Remark 2.3. v4: Added missing diagram in eq. (126)

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Cited by in corpus (7)

• Markov Categories and Entropy
• Bayesian inversion and the Tomita-Takesaki modular group
• Bipartite quantum states admitting a causal explanation
• Combs, Causality and Contractions in Atomic Markov Categories
• The Aldous$\unicode{x2013}$Hoover Theorem in Categorical Probability
• Categories of abstract and noncommutative measurable spaces
• Categorical algebra of conditional probability