The Category of Markov Kernels

Abstract

Markov kernels are fundamental objects in probability theory. One can define a category based on Markov kernels which has many of the formal properties of the ordinary category of relations. In the present paper we will examine the categorical properties of Markov kernels and stress the analogies and differences with the category of relations. We will show that this category has partially-additive structure and, as such, supports basic constructs like iteration. This allows one to give a probabilistic semantics for a language with while loops in the manner of Kozen. The category in question was originally defined by Giry following suggestions of Lawvere.