Partializations of Markov categories

Abstract

The present work develops a construction of a CD category of partial kernels from a particular type of Markov category called a partializable Markov category. These are a generalization of earlier models of categories of partial morphisms such as p-categories, dominical categories, restriction categories, etc. to a non-deterministic/non-cartesian setting. Here all morphisms are quasi-total, with a natural poset enrichment corresponding to one morphism being a restriction of the other. Furthermore, various properties important to categorical probability are preserved, such as positivity, representability, conditionals, Kolmogorov products, and splittings of idempotents. We additionally discuss an alternative notion of Kolmogorov product suitable for partial maps, as well as partial algebras for probability monads. The primary example is that of the partialization of the category of standard Borel spaces and Markov kernels. Other examples include variants where the distributions are finitely supported, or where one considers multivalued maps instead.