Categories for probability theory

Category theory was initially developed to address some structural questions in algebraic topology. Shortly after it was extended to algebraic geometry, logic, universal algebra, and more recently, theoretical computer science. Each of these subjects was heavily influenced by category theory, and in turn, the development of category theory was prominently shaped by the strucures and problems arising in these fields. In the past few years there has been a growing interest in applying categorical techniques to fields such as probability, statistics and information theory, to study their structures and to find patterns in their techniques. Perhaps surprisingly, it turns out that these fields present a rich and principled structure when addressed categorically, with functors and universal properties arising everywhere. However, most of the time, new category theory is needed to study these subjects, as they are quite far from the algebra and geometry for which category theory was initially developed. Two of the current most prolific environments to study probability categorically are Markov categories and dagger categories. In this talk we will give an introduction to both, show their similarities, differences and connections, and use them to prove some core theorems of probability.