Codensity and the Giry Monad

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 ∈ 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 .

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 ∈ 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 .

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 ∈ 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 .

Introduction

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 ∈ X we have a way of choosing a point of X at random: “always choose x”. So we have X → TX .

Introduction

Tom Avery Codensity and the Giry monad Given x ∈ X we have a way of choosing a point of X at random: “always choose x”. So we have X → TX .

Introduction

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 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’.

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 {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’.

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 {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 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 {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 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})

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 Unit: If ω ∈ Ω and A ⊆ Ω, is measurable then ( 1 if ω ∈ A η(ω)(A) = 0 otherwise.

Multiplication: If ρ ∈ GGΩ and A ⊆ Ω, then Z µ(ρ)(A) = π(A) dρ(π) π∈GΩ

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 ∈ GGΩ and A ⊆ Ω, then Z µ(ρ)(A) = π(A) dρ(π) π∈GΩ

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. 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 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

Multiplication: If ρ ∈ GGΩ and A ⊆ Ω, then Z µ(ρ)(A) = π(A) dρ(π) π∈GΩ

Tom Avery Codensity and the Giry monad The Giry Monad

Multiplication: If ρ ∈ GGΩ and A ⊆ Ω, then Z µ(ρ)(A) = π(A) dρ(π) π∈GΩ

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 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 elementr ¯ ∈ 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.

The Giry Monad

Let C be the category with n objects I for n ∈ N, morphisms are aﬃne maps (i.e. maps that preserve convex combinations)

Tom Avery Codensity and the Giry monad Examples: I and Meas(Ω, I ) for any Ω ∈ Meas.

The Giry Monad

Let C be the category with n objects I 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

Tom Avery Codensity and the Giry monad The Giry Monad

Tom Avery Codensity and the Giry monad Proposition (correction of Sturtz) Let Ω ∈ Meas. As sets, we have ∼ ∼ C-Alg(Meas(Ω, I ), I ) = F Ω and C-Algcts (Meas(Ω, I ), I ) = GΩ.

(Compare: An ultraﬁlter on a set S is an element of Bool(Set(S, 2), 2).)

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 fn :Ω → I with fn → f pointwise as n → ∞, we have φ(fn) → φ(f ) as n → ∞.

Write C-Algcts (Meas(Ω, I ), I ) for the set of continuous homomorphisms.

Tom Avery Codensity and the Giry monad (Compare: An ultraﬁlter on a set S is an element of Bool(Set(S, 2), 2).)

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 fn :Ω → I with fn → f pointwise as n → ∞, we have φ(fn) → φ(f ) as n → ∞.

Write C-Algcts (Meas(Ω, I ), I ) for the set of continuous homomorphisms. Proposition (correction of Sturtz) Let Ω ∈ Meas. As sets, we have ∼ ∼ C-Alg(Meas(Ω, I ), I ) = F Ω and C-Algcts (Meas(Ω, I ), I ) = GΩ.

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 fn :Ω → I with fn → f pointwise as n → ∞, we have φ(fn) → φ(f ) as n → ∞.

(Compare: An ultraﬁlter on a set S is an element of Bool(Set(S, 2), 2).)

Tom Avery Codensity and the Giry monad Given a ﬁnitely additive probability measure π, deﬁne φ: Meas(Ω, I ) → I by Z φ(f ) = f dπ Ω for f ∈ Meas(Ω, I ).

The Giry Monad

Proposition (correction of Sturtz) Let Ω ∈ Meas. As sets, we have ∼ ∼ C-Alg(Meas(Ω, I ), I ) = F Ω and C-Algcts (Meas(Ω, I ), I ) = GΩ.

Given φ: Meas(Ω, I ) → I , deﬁne a ﬁnitely additive probability measure π by

π(A) = φ(χA),

for A ⊆ Ω measurable, with characteristic function χA.

Tom Avery Codensity and the Giry monad The Giry Monad

Proposition (correction of Sturtz) Let Ω ∈ Meas. As sets, we have ∼ ∼ C-Alg(Meas(Ω, I ), I ) = F Ω and C-Algcts (Meas(Ω, I ), I ) = GΩ.

Given φ: Meas(Ω, I ) → I , define Given a finitely additive a finitely additive probability probability measure π, define measure π by φ: Meas(Ω, I ) → I by

π(A) = φ(χ ), Z A φ(f ) = f dπ Ω for A ⊆ Ω measurable, with characteristic function χA. for f ∈ Meas(Ω, I ).

Tom Avery Codensity and the Giry monad U U U U U We obtain η : idM → T and µ: T T → T making T into a monad via:

U U B / M B / M κ Ò U = T ks idM U η U × & M & M and U U B / M B / M T U κ Ö T U U |Ô κ # κ T U ks M = ÐØ ) M U µ U T U { T U $ M M.

Codensity Monads

Let U : B → M be a functor. The codensity monad of U (if it exists) is the right Kan extension T U : M → M of U along itself. Write κ: T U U → U for the universal natural transformation.

Tom Avery Codensity and the Giry monad Codensity Monads

Let U : B → M be a functor. The codensity monad of U (if it exists) is the right Kan extension T U : M → M of U along itself. Write κ: T U U → U for the universal natural transformation. U U U U U We obtain η : idM → T and µ: T T → T making T into a monad via:

U U B / M B / M κ Ò U = T ks idM U η U × & M & M and U U B / M B / M T U κ Ö T U U |Ô κ # κ T U ks M = ÐØ ) M U µ U T U { T U $ M M. Tom Avery Codensity and the Giry monad The end formula for right Kan extensions gives Z T U X =∼ [M(X , Ub), Ub] b∈B for X ∈ M.

Example: If B is a Lawvere theory and U : B → M is a model with underlying object R, say, then T U is characterised by

U ∼ M(Y , T X ) = B-Alg(M(X , R), M(Y , R)).

Codensity Monads

Remark: If U is a right adjoint, then T U is the monad induced by the adjunction.

Tom Avery Codensity and the Giry monad Example: If B is a Lawvere theory and U : B → M is a model with underlying object R, say, then T U is characterised by

U ∼ M(Y , T X ) = B-Alg(M(X , R), M(Y , R)).

Codensity Monads

Remark: If U is a right adjoint, then T U is the monad induced by the adjunction. The end formula for right Kan extensions gives Z T U X =∼ [M(X , Ub), Ub] b∈B for X ∈ M.

Tom Avery Codensity and the Giry monad Codensity Monads

Remark: If U is a right adjoint, then T U is the monad induced by the adjunction. The end formula for right Kan extensions gives Z T U X =∼ [M(X , Ub), Ub] b∈B for X ∈ M.

Example: If B is a Lawvere theory and U : B → M is a model with underlying object R, say, then T U is characterised by

U ∼ M(Y , T X ) = B-Alg(M(X , R), M(Y , R)).

Tom Avery Codensity and the Giry monad Deﬁne the category D with objects: the objects of C together with

∞ d0 = {(xn)n=0 | xn ∈ I and xn → 0 as n → ∞}

morphisms: aﬃne maps. There is a similar functor V : D → Meas. (But D is not a Lawvere theory)

Main Result

Recall C is the category (Lawvere theory, in fact) with n objects: I , n ∈ N. morphisms: aﬃne maps. The objects of C have natural (Borel) σ-algebras so there is a functor U : C → Meas. This exhibits I ∈ Meas as a C-algebra.

Tom Avery Codensity and the Giry monad Main Result

Deﬁne the category D with objects: the objects of C together with

∞ d0 = {(xn)n=0 | xn ∈ I and xn → 0 as n → ∞}

morphisms: aﬃne maps. There is a similar functor V : D → Meas. (But D is not a Lawvere theory)

Tom Avery Codensity and the Giry monad Main Result

Theorem (TA) 0 1 2 The codensity monad of U : C = {I , I , I ,...} → Meas is the finitely additive Giry monad F. 0 1 The codensity monad of V : D = {I , I ,..., d0} → Meas is the Giry monad G. Sketch proof: The forgetful functor Meas → Set is representable. Hence for Ω ∈ Meas, the underlying set of T U Ω is ∼ C-Alg(Meas(Ω, I ), I ) = F Ω. Then check the isomorphisms are compatible with measurable space and monad structure. Similarly one can see that the underlying set of T V Ω is ∼ C-Algcts (Meas(Ω, I ), I ) = GΩ, and check compatibility. Tom Avery Codensity and the Giry monad 0 2 2 0 C the monoid of affine endomorphisms of I (or ∆ ), and U the action on I 2 ∈ Meas (or ∆2 ∈ Meas respectively); 0 C the category of all bounded convex subsets of Euclidean space with affine maps, and U0 the forgetful functor.

Then the codensity monad of U0 is still the ﬁnitely additive Giry monad F.

Related Results

0 0 Let U : C → Meas be any of the following: 0 n n C the category of simplices ∆ ⊆ R and aﬃne maps between them, and U0 the obvious forgetful functor;

Tom Avery Codensity and the Giry monad 0 C the category of all bounded convex subsets of Euclidean space with aﬃne maps, and U0 the forgetful functor.

Then the codensity monad of U0 is still the ﬁnitely additive Giry monad F.

Related Results

0 0 Let U : C → Meas be any of the following: 0 n n C the category of simplices ∆ ⊆ R and aﬃne maps between them, and U0 the obvious forgetful functor; 0 2 2 0 C the monoid of aﬃne endomorphisms of I (or ∆ ), and U the action on I 2 ∈ Meas (or ∆2 ∈ Meas respectively);

Tom Avery Codensity and the Giry monad Then the codensity monad of U0 is still the ﬁnitely additive Giry monad F.

Related Results

Tom Avery Codensity and the Giry monad Related Results

Then the codensity monad of U0 is still the ﬁnitely additive Giry monad F.

Tom Avery Codensity and the Giry monad F Ω is a measurable space, and f.a. probability measures can be pushed forward along measurable maps, n every f.a. probability measure on a convex bounded C ⊆ R has a ’barycentre’ in C, pushforward along aﬃne maps between convex bounded sets respects barycentres, and F is characterised by being terminal such.

Related Results

Tom Avery Codensity and the Giry monad Related Results

0 n Consider the last of these: C = {bounded convex subsets of R }, U0 = forgetful. Here’s an interpretation of the result in this case. The collection F Ω of finitely additive probability measures on Ω ∈ Meas satisfies the following: F Ω is a measurable space, and f.a. probability measures can be pushed forward along measurable maps, n every f.a. probability measure on a convex bounded C ⊆ R has a ’barycentre’ in C, pushforward along affine maps between convex bounded sets respects barycentres, and F is characterised by being terminal such.

Tom Avery Codensity and the Giry monad 0 ω 0 D the monoid of aﬃne endomorphisms of d0 (or ∆ ), and V ω the action on d0 ∈ Meas (or∆ respectively); 0 D the category of closed, convex, bounded subsets of the vector space

Related Results

0 0 Let V : D → Meas be any of the following 0 D the category of simplices, together with ω ∞ P∞ ∆ = {(xn)n=0 | xn ∈ I , n=0 xn = 1} and aﬃne maps, and V 0 forgetful;

Tom Avery Codensity and the Giry monad 0 D the category of closed, convex, bounded subsets of the vector space

Related Results

0 0 Let V : D → Meas be any of the following 0 D the category of simplices, together with ω ∞ P∞ ∆ = {(xn)n=0 | xn ∈ I , n=0 xn = 1} and aﬃne maps, and V 0 forgetful; 0 ω 0 D the monoid of aﬃne endomorphisms of d0 (or ∆ ), and V ω the action on d0 ∈ Meas (or∆ respectively);

Tom Avery Codensity and the Giry monad Related Results

∞ c0 = {(xn)n=0 | xn ∈ R, xn → 0} with the sup norm, with aﬃne maps between them, and V 0 forgetful. 0 Then the codensity monad of V is still the Giry monad G.A similar interpretation of the last of these holds as on the previous slide. Tom Avery Codensity and the Giry monad There is a monad on Top of ’Scott-continuous probability valuations’. This can be realised as the codensity monad of a certain forgetfull functor C → Top, where C is a category whose objects are spaces of (transﬁnite) sequences in I .

Related monads

n Recall C is the category with objects I and aﬃne maps (WITHOUT d0). There is an obvious functor from C to CHaus, the category of compact Hausdorﬀ spaces. The codensity monad of this sends a space X to the space of regular Borel probability measures on X .

Tom Avery Codensity and the Giry monad Related monads

There is a monad on Top of ’Scott-continuous probability valuations’. This can be realised as the codensity monad of a certain forgetfull functor C → Top, where C is a category whose objects are spaces of (transﬁnite) sequences in I .

Tom Avery Codensity and the Giry monad T. Avery. Codensity and the Giry monad. arXiv:1410.4432, 2014. M. Giry. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis, volume 915 of Lecture Notes in Mathematics, pages 65–85. Springer 1982. K. Sturtz. The Giry monad as a codensity monad. arXiv:1406.6030, 2014. D. Gildenhuys and J. F. Kennison. Equational completion, model induced triples and pro-objects. Journal of Pure and Applied Algebra, 1:317–346, 1971. T. Leinster. Codensity and the ultraﬁlter monad. Theory and Applications of Categories, 28(13):332270, 2013.

Tom Avery Codensity and the Giry monad