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.
@article{ordermarkov25,
author = {Elena Di Lavore and
Mario Rom{\'{a}}n and
Pawel Sobocinski and
M{\'{a}}rk Sz{\'{e}}les},
title = {Order in Partial Markov Categories},
journal = {CoRR},
volume = {abs/2507.19424},
year = {2025},
url = {https://doi.org/10.48550/arXiv.2507.19424},
doi = {10.48550/ARXIV.2507.19424},
eprinttype = {arXiv},
eprint = {2507.19424},
timestamp = {Thu, 21 Aug 2025 15:51:32 +0200},
biburl = {https://dblp.org/rec/journals/corr/abs-2507-19424.bib},
bibsource = {dblp computer science bibliography, https://dblp.org}
}