Codensity and the Giry monad Tom Avery 19th June 2015 Tom Avery Codensity and the Giry monad Structure of this talk Introduction The Giry monad Codensity monads Main result Tom Avery Codensity and the Giry monad Need a probability measure on X . Write TX for the 'space' of probability measures on X; then we want a point of TX. To be fair we should choose this at random too. To do this, we need a probability measure on TX , i.e. an element of TTX . So, given ρ 2 TTX , we can 1 choose π 2 TX at random according to ρ, 2 choose x 2 X at random according to π. This is a way of choosing an element of X `at random' i.e. a probability measure on X . So we have a map TTX ! TX Given x 2 X we have a way of choosing a point of X at random: \always choose x". So we have X ! TX : Introduction Let X be a `space'. Want to choose a point of X at random. Tom Avery Codensity and the Giry monad To be fair we should choose this at random too. To do this, we need a probability measure on TX , i.e. an element of TTX . So, given ρ 2 TTX , we can 1 choose π 2 TX at random according to ρ, 2 choose x 2 X at random according to π. This is a way of choosing an element of X `at random' i.e. a probability measure on X . So we have a map TTX ! TX Given x 2 X we have a way of choosing a point of X at random: \always choose x". So we have X ! TX : Introduction Let X be a `space'. Want to choose a point of X at random. Need a probability measure on X . Write TX for the 'space' of probability measures on X; then we want a point of TX. Tom Avery Codensity and the Giry monad So, given ρ 2 TTX , we can 1 choose π 2 TX at random according to ρ, 2 choose x 2 X at random according to π. This is a way of choosing an element of X `at random' i.e. a probability measure on X . So we have a map TTX ! TX Given x 2 X we have a way of choosing a point of X at random: \always choose x". So we have X ! TX : Introduction Let X be a `space'. Want to choose a point of X at random. Need a probability measure on X . Write TX for the 'space' of probability measures on X; then we want a point of TX. To be fair we should choose this at random too. To do this, we need a probability measure on TX , i.e. an element of TTX . Tom Avery Codensity and the Giry monad This is a way of choosing an element of X `at random' i.e. a probability measure on X . So we have a map TTX ! TX Given x 2 X we have a way of choosing a point of X at random: \always choose x". So we have X ! TX : Introduction Let X be a `space'. Want to choose a point of X at random. Need a probability measure on X . Write TX for the 'space' of probability measures on X; then we want a point of TX. To be fair we should choose this at random too. To do this, we need a probability measure on TX , i.e. an element of TTX . So, given ρ 2 TTX , we can 1 choose π 2 TX at random according to ρ, 2 choose x 2 X at random according to π. Tom Avery Codensity and the Giry monad Given x 2 X we have a way of choosing a point of X at random: \always choose x". So we have X ! TX : Introduction Let X be a `space'. Want to choose a point of X at random. Need a probability measure on X . Write TX for the 'space' of probability measures on X; then we want a point of TX. To be fair we should choose this at random too. To do this, we need a probability measure on TX , i.e. an element of TTX . So, given ρ 2 TTX , we can 1 choose π 2 TX at random according to ρ, 2 choose x 2 X at random according to π. This is a way of choosing an element of X `at random' i.e. a probability measure on X . So we have a map TTX ! TX Tom Avery Codensity and the Giry monad Introduction Let X be a `space'. Want to choose a point of X at random. Need a probability measure on X . Write TX for the 'space' of probability measures on X; then we want a point of TX. To be fair we should choose this at random too. To do this, we need a probability measure on TX , i.e. an element of TTX . So, given ρ 2 TTX , we can 1 choose π 2 TX at random according to ρ, 2 choose x 2 X at random according to π. This is a way of choosing an element of X `at random' i.e. a probability measure on X . So we have a map TTX ! TX Given x 2 X we have a way of choosing a point of X at random: \always choose x". So we have X ! TX : Tom Avery Codensity and the Giry monad Can vary what we mean by `space': sets, measurable spaces, topological spaces, Polish spaces, etc. and what notion of `measure' we are interested in: measures taking values in f0; 1g; [0; 1]; [0; 1]; R; C; etc. finitely vs. countably additive distributions of compact support etc. We get the Giry monad when we take `measurable spaces' and `probability measures'. Introduction So we might expect space X 7! probability measures on X to form a monad. Tom Avery Codensity and the Giry monad and what notion of `measure' we are interested in: measures taking values in f0; 1g; [0; 1]; [0; 1]; R; C; etc. finitely vs. countably additive distributions of compact support etc. We get the Giry monad when we take `measurable spaces' and `probability measures'. Introduction So we might expect space X 7! probability measures on X to form a monad. Can vary what we mean by `space': sets, measurable spaces, topological spaces, Polish spaces, etc. Tom Avery Codensity and the Giry monad We get the Giry monad when we take `measurable spaces' and `probability measures'. Introduction So we might expect space X 7! probability measures on X to form a monad. Can vary what we mean by `space': sets, measurable spaces, topological spaces, Polish spaces, etc. and what notion of `measure' we are interested in: measures taking values in f0; 1g; [0; 1]; [0; 1]; R; C; etc. finitely vs. countably additive distributions of compact support etc. Tom Avery Codensity and the Giry monad Introduction So we might expect space X 7! probability measures on X to form a monad. Can vary what we mean by `space': sets, measurable spaces, topological spaces, Polish spaces, etc. and what notion of `measure' we are interested in: measures taking values in f0; 1g; [0; 1]; [0; 1]; R; C; etc. finitely vs. countably additive distributions of compact support etc. We get the Giry monad when we take `measurable spaces' and `probability measures'. Tom Avery Codensity and the Giry monad Introduction Question: Is there a unified categorical description of all these variations? Tom Avery Codensity and the Giry monad analogous to Theorem (Gildenhuys, Kennison) The ultrafilter monad (on Set) is the codensity monad of FinSet ,! Set. (The ultrafilter monad is a measure monad: An ultrafilter on a set X is the same as a finitely additive probability measure defined on the power set of X taking values in f0; 1g) Introduction Codensity monads provide one tool. Theorem The Giry monad and finitely additive Giry monad arise as codensity monads (of certain functors). Tom Avery Codensity and the Giry monad Introduction Codensity monads provide one tool. Theorem The Giry monad and finitely additive Giry monad arise as codensity monads (of certain functors). analogous to Theorem (Gildenhuys, Kennison) The ultrafilter monad (on Set) is the codensity monad of FinSet ,! Set. (The ultrafilter monad is a measure monad: An ultrafilter on a set X is the same as a finitely additive probability measure defined on the power set of X taking values in f0; 1g) Tom Avery Codensity and the Giry monad Introduction Notation: Meas is the category of measurable spaces and maps. I = [0; 1] (with Borel σ-algebra). If S is a set, X an object of a category, then [S; X ] is the `S-th power of X '. Tom Avery Codensity and the Giry monad Unit: If ! 2 Ω and A ⊆ Ω, is measurable then ( 1 if ! 2 A η(!)(A) = 0 otherwise. Multiplication: If ρ 2 GGΩ and A ⊆ Ω, then Z µ(ρ)(A) = π(A) dρ(π) π2GΩ replacing `probability measures' with ‘finitely additive probability measures' gives the finitely additive Giry monad F = (F ; η; µ). Note that G is a submonad of F. The Giry Monad The Giry monad, G = (G; η; µ) consists of: Endofunctor: G : Meas ! Meas, sends Ω to the set of probability measures on Ω, with a suitable σ-algebra. Tom Avery Codensity and the Giry monad Multiplication: If ρ 2 GGΩ and A ⊆ Ω, then Z µ(ρ)(A) = π(A) dρ(π) π2GΩ replacing `probability measures' with ‘finitely additive probability measures' gives the finitely additive Giry monad F = (F ; η; µ).