Oxford Seminar / Accepting Normalization in Markov Categories
Normalization of probability distributions is not a distributive law, and normalized kernel composition does not define a category. This talk aims to highlight these limitations and take an alternative route: we will introduce distributive sesquilaws and, from their axioms, derive some of the properties of normalization. In particular, we will see how normalized kernels form a unital magmoid with an action of the category of subdistributions, and we will recap its possibilistic analogue: the action from may-must relations into the category of relations. In this sense, the magmoid of normalized kernels is the stochastic analog of the category of relations. The talk is based on joint work with Elena Di Lavore, Paweł Sobociński, and Márk Széles (arxiv.org/pdf/2502.03477, arxiv.org/pdf/2510.01131).