Kleisli categories and probability - 01 - The Giry monad
Channel: Arthur Parzygnat
Date: 2020-03-19

This video introduces the Giry monad. A monad on a category consists of three data. In this case, it gives an endo-functor on the category of measurable spaces and measurable maps. The functor takes a measurable space to the space of probability measures and it takes a map to the pushforward of measures. In addition, there is a natural transformation that computes the barycenter of a probability measure on the space of probability measures. Finally, there is a natural transformation that views each point of a space as the Dirac delta distribution on that point. 

Note: I forgot to mention that the second pair of diagrams in the definition of a monad both equal the identity natural transformation for T. 

Brief history: In 1962, Bill Lawvere introduced the category of probabilistic maps. In 1982, Michele Giry viewed Lawvere's construction as the Kleisli category of a particular monad. 

This video is part of the playlist on Categorical probability theory: https://www.youtube.com/playlist?list=PLSx1kJDjrLRSKKHj4zetTZ45pVnGCRN80

The next video in this series is on the Kleisli category of a monad: https://youtu.be/m9KdSXHrHD8

These videos were created during the 2019 Spring/Summer semester at the UConn CETL Lightboard Room.