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.

翻译：我们研究马尔可夫范畴的正性与因果性公理，将其视为马尔可夫范畴及其在任意半笛卡尔幺半范畴变体中的扩张与信息流性质。这些研究帮助我们证明：成为正性马尔可夫范畴仅是对称幺半范畴的附加性质（而非额外结构）。我们进一步刻画了可表示马尔可夫范畴的正性特征，并证明因果性蕴含正性，但反之不成立。最后，我们指出拟Borel空间不满足正性公理，并将此失效现象解释为概率性名称生成的隐私特性。