CT2023, Paolo Perrone: Universal properties in probability theory скачать в хорошем качестве

CT2023, Paolo Perrone: Universal properties in probability theory

International Category Theory Conference 2023 03-08/07/2023 Speaker: Paolo Perrone (University of Oxford) Title: Universal properties in probability theory Abstract: see https://sites.uclouvain.be/ct2023/img... In its early days, category theory was developed primarily for algebra, geometry, and topology. Several constructions and results in algebra and geometry have since been expressed in terms of universal prop- erties. This point of view has provided us with a much deeper understanding of these structures, and of algebra and geometry in general. Today it seems almost impossible to imagine, for example, how the field of algebraic topology would look without category theory, as well as the other way around. In the last few years we have experienced a rise in applications of category theory to domains such as probability, statistics, and information theory. What universal properties do we see in these contexts? The universal constructions of relevance here are less known in the community than their algebraic and geometric counterparts, but they can give us just as well a deep understanding of some aspects of probability. In this talk we will focus on three key examples (or more, depending on time and interest): We will show how finite products are fundamentally incompatible with the idea of uncertainty and correlations; We will show how the incompatibility above vanishes ”at infinity”, by expressing the Kolmogorov extension theorem in terms of a cofiltered limit; We will formulate De Finetti’s theorem as a particular equalizer construction, which only works in categories where the arrows have randomness. We will use the language of probability monads and *Markov categories*, which we will briefly introduce, and for which we will assume no previous knowledge. Website: https://sites.uclouvain.be/ct2023/