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Totality of the sampling morphism in representable quasi-Markov categories

Determine whether, for every object X in any representable quasi-Markov category, the sampling morphism samp_X: PX → X is total (i.e., defined for all inputs) without assuming observational representability. Equivalently, ascertain conditions under which samp_X is guaranteed to be total in the general representable quasi-Markov setting.

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Background

In representable quasi-Markov categories, each object X has a distribution object PX and associated unit δ_X: X → PX and counit (sampling morphism) samp_X: PX → X. The unit δ_X is monic and hence total, but the totality of samp_X is not established in general.

The authors note that totality of samp_X can be ensured under the stronger assumption of observational representability. The open question is whether this totality holds universally in representable quasi-Markov categories without imposing observational representability.

References

One of the triangle identities is that each δ_X is a section of samp_X. As a consequence, δ_X is monic, and is thus total by \cref{lem:mono_total}. We leave open the question as to whether samp_X is total in general. At present, we only know how to ensure this fact under the stronger assumption of observational representability introduced in \cref{def:obs_rep} below.

— Empirical Measures and Strong Laws of Large Numbers in Categorical Probability (2503.21576 - Fritz et al., 27 Mar 2025) in Section 2.4 (Representable Quasi-Markov Categories)