Lemma [efr-198L]

Let $\mathcal {C}$ be a representable Markov category, and let $P$ be the associated distribution monad. Then

If $\mathcal {C}_\mathrm {det}$ is $\kappa$ -extensive and $P: \mathcal {C}_\mathrm {det} \to \mathcal {C}_\mathrm {det}$ preserves pullbacks along coproduct inclusions, then $\mathcal {C}$ is $\kappa$ -extensive.
If $\mathcal {C}_\mathrm {det}$ is Boolean, then $\mathcal {C}$ is Boolean.
If $\mathcal {C}_\mathrm {det}$ is (finitely) extensive, Boolean, and $\kappa$ -complete, and $P: \mathcal {C}_\mathrm {det} \to \mathcal {C}_\mathrm {det}$ preserves the cofiltered limits $\prod _{i \in J} A_i = \lim _{F \subseteq J \mathrm { finite}} \prod _{i \in F} A_i$ for $|J| < \kappa$ , then $\mathcal {C}$ is $\kappa$ -extensive (and Boolean and admits $\kappa$ -small Kolmogorov products).

The Universal Property of Measure-Theoretic Probability › Distributive and extensive Markov Categories [efr-MGSE]