Order in Partial Markov Categories

TL;DR

This paper explores order relations in partial Markov categories, showing their canonical preorder enrichment, the role of codiagonal maps, and a synthetic Cauchy--Schwarz inequality that demonstrates updating increases validity.

Contribution

It introduces a canonical preorder enrichment for partial Markov categories, links codiagonal maps to order properties, and develops a synthetic Cauchy--Schwarz inequality for probabilistic updates.

Findings

Partial Markov categories are canonically preorder-enriched.

Existence of codiagonal maps relates to order properties.

Updating increases validity, as shown by the synthetic Cauchy--Schwarz inequality.

Abstract

Partial Markov categories are a recent framework for categorical probability theory that provide an abstract account of partial probabilistic computation with updating semantics. In this article, we discuss two order relations on the morphisms of a partial Markov category. In particular, we prove that every partial Markov category is canonically preorder-enriched, recovering several well-known order enrichments. We also demonstrate that the existence of codiagonal maps (comparators) is closely related to order properties of partial Markov categories. Finally, we introduce a synthetic version of the Cauchy--Schwarz inequality and, from it, we prove that updating increases validity.