Metric Spaces, Entropic Spaces and Convexity in Category Theory

Explore advanced mathematical concepts in this 59-minute Topos Institute Colloquium talk that delves into metric spaces, entropic spaces, and convexity. Learn how convexity of sets can be captured through convexity or barycentric monads, and understand their relationship to probability monads. Examine the work of various mathematicians including Mardare-Panangaden-Plotkin and Fritz-Perrone on convexity/probability monads in metric spaces categories. Discover how metric spaces can be viewed as categories enriched over the quantale of extended non-negative real numbers. Study the application of convexity monads to R-categories and explore Lawvere's concept of entropic spaces in thermodynamics. Investigate the connection between Lawvere's thermodynamic approach and the recent work of Baez-Lynch-Moeller involving convex spaces and concave maps. Cover topics including rational convexity monads, optimal transport, classical metric spaces, convex metrics, and underlying category convexity through detailed mathematical analysis and theoretical frameworks.