Random Variables, Independence Structures and Dagger Categories of Relations

This colloquium talk from the Topos Institute explores the mathematical foundations of probability theory through the lens of category theory. Delve into how random variables, traditionally defined with reference to sample spaces, can be understood using sheaves over sample spaces as pioneered by Alex Simpson. Discover how sample spaces can be generalized to encompass various probability-like phenomena including nondeterminism and fresh name generation. Follow Dario Stein as he presents a tour of abstract samples and examines the relationship between sample spaces and couplings, demonstrating how these categories determine each other with couplings functioning as relations over sample spaces that compose via independent pullback. The presentation represents joint work with Chris Heunen, Matthew Di Meglio, and Paolo Perrone, offering insights into independence structures and dagger categories of relations in the context of probability theory.