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
Introduction Let X be a ‘space’. Want to choose a point of X at random. 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. 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 . 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 ∈ TTX , we can 1 choose π ∈ TX at random according to ρ , 2 choose x ∈ X at random according to π . 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 ∈ TTX , we can 1 choose π ∈ TX at random according to ρ , 2 choose x ∈ 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 ∈ TTX , we can 1 choose π ∈ TX at random according to ρ , 2 choose x ∈ 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 ∈ 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
Introduction So we might expect space X �→ probability measures on X to form a monad. Tom Avery Codensity and the Giry monad
Introduction So we might expect space X �→ 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
Introduction So we might expect space X �→ 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 { 0 , 1 } , [0 , 1] , [0 , ∞ ] , 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 �→ 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 { 0 , 1 } , [0 , 1] , [0 , ∞ ] , 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
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 { 0 , 1 } ) 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
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
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. Unit: If ω ∈ Ω and A ⊆ Ω, is measurable then � 1 if ω ∈ A η ( ω )( A ) = 0 otherwise. Tom Avery Codensity and the Giry monad
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. Unit: If ω ∈ Ω and A ⊆ Ω, is measurable then � 1 if ω ∈ A η ( ω )( A ) = 0 otherwise. Multiplication: If ρ ∈ GG Ω and A ⊆ Ω, then � µ ( ρ )( A ) = π ( A ) d ρ ( π ) π ∈ G Ω Tom Avery Codensity and the Giry monad
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. Unit: If ω ∈ Ω and A ⊆ Ω, is measurable then � 1 if ω ∈ A η ( ω )( A ) = 0 otherwise. Multiplication: If ρ ∈ GG Ω and A ⊆ Ω, then � µ ( ρ )( A ) = π ( A ) d ρ ( π ) π ∈ G Ω 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 . Tom Avery Codensity and the Giry monad
The Giry Monad Let C be the category with objects I n for n ∈ N , morphisms are affine maps (i.e. maps that preserve convex combinations) Tom Avery Codensity and the Giry monad
The Giry Monad Let C be the category with objects I n for n ∈ N , morphisms are affine maps (i.e. maps that preserve convex combinations) Then C is in fact a Lawvere theory. A C -algebra can be described as a set A equipped with for each r ∈ I , a map + r : A × A → A (think of this as x + r y = rx + (1 − r ) y ), for each r ∈ I , an element ¯ r ∈ A , (’constant at r ’), a unary operation i : A → A (think of i ( a ) as ’1 − a ’) satisfying some axioms. Tom Avery Codensity and the Giry monad
The Giry Monad Let C be the category with objects I n for n ∈ N , morphisms are affine maps (i.e. maps that preserve convex combinations) Then C is in fact a Lawvere theory. A C -algebra can be described as a set A equipped with for each r ∈ I , a map + r : A × A → A (think of this as x + r y = rx + (1 − r ) y ), for each r ∈ I , an element ¯ r ∈ A , (’constant at r ’), a unary operation i : A → A (think of i ( a ) as ’1 − a ’) satisfying some axioms. Examples: I and Meas (Ω , I ) for any Ω ∈ Meas . Tom Avery Codensity and the Giry monad
The Giry Monad Let C be the category with objects I n for n ∈ N , morphisms are affine maps (i.e. maps that preserve convex combinations) Then C is in fact a Lawvere theory. A C -algebra can be described as a set A equipped with for each r ∈ I , a map + r : A × A → A (think of this as x + r y = rx + (1 − r ) y ), for each r ∈ I , an element ¯ r ∈ A , (’constant at r ’), a unary operation i : A → A (think of i ( a ) as ’1 − a ’) satisfying some axioms. Examples: I and Meas (Ω , I ) for any Ω ∈ Meas . Tom Avery Codensity and the Giry monad
The Giry Monad Consider C -algebra homomorphisms Meas (Ω , I ) → I . Call such a homomorphism continuous if it preserves pointwise limits of sequences. That is φ ∈ C - Alg ( Meas (Ω , I ) , I ) is continuous if, for every sequence f n : Ω → I with f n → f pointwise as n → ∞ , we have φ ( f n ) → φ ( f ) as n → ∞ . Write C - Alg cts ( Meas (Ω , I ) , I ) for the set of continuous homomorphisms. Tom Avery Codensity and the Giry monad
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