Download 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
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 7→ probability measures on X
to form a monad.
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.
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.
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
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
Z
µ(ρ)(A) =
π(A) dρ(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
Z
µ(ρ)(A) =
π(A) dρ(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 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
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 ) ∼
= G Ω.
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 → ∞.
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 ) ∼
= G Ω.
(Compare: An ultrafilter on a set S is an element of
Bool(Set(S, 2), 2).)
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 ) ∼
= G Ω.
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 ) ∼
= G Ω.
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
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 ).
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.
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.
We obtain η : idM → T U and µ : T U T U → T U making T U into a
monad via:
/M
U
B
κ
U
TU
ks
idM
η
& B
/M
U
=
U
&
M
M
and
/M
U
B
B
TU
| κ
U
TU
{
M
ks
#
µ
M
=
κ
U
TU
) κ
M
U
$
TU
Tom Avery
/M
U
Codensity and the Giry monad
U
T
M.
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
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
TUX ∼
[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
TUX ∼
[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
M(Y , T U X ) ∼
= B-Alg(M(X , R), M(Y , R)).
Tom Avery
Codensity and the Giry monad
Main Result
Recall C is the category (Lawvere theory, in fact) with
objects: I n , 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
Recall C is the category (Lawvere theory, in fact) with
objects: I n , 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.
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)
The codensity monad of U : C = {I 0 , I 1 , I 2 , . . .} → Meas is the
finitely additive Giry monad F.
The codensity monad of V : D = {I 0 , I 1 , . . . , 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 ) ∼
= G Ω,
and check compatibility.
Tom Avery
Codensity and the Giry monad
Related Results
Let U 0 : C0 → Meas be any of the following:
C0 the category of simplices ∆n ⊆ Rn and affine maps
between them, and U 0 the obvious forgetful functor;
Tom Avery
Codensity and the Giry monad
Related Results
Let U 0 : C0 → Meas be any of the following:
C0 the category of simplices ∆n ⊆ Rn and affine maps
between them, and U 0 the obvious forgetful functor;
C0 the monoid of affine endomorphisms of I 2 (or ∆2 ), and U 0
the action on I 2 ∈ Meas (or ∆2 ∈ Meas respectively);
Tom Avery
Codensity and the Giry monad
Related Results
Let U 0 : C0 → Meas be any of the following:
C0 the category of simplices ∆n ⊆ Rn and affine maps
between them, and U 0 the obvious forgetful functor;
C0 the monoid of affine endomorphisms of I 2 (or ∆2 ), and U 0
the action on I 2 ∈ Meas (or ∆2 ∈ Meas respectively);
C0 the category of all bounded convex subsets of Euclidean
space with affine maps, and U 0 the forgetful functor.
Tom Avery
Codensity and the Giry monad
Related Results
Let U 0 : C0 → Meas be any of the following:
C0 the category of simplices ∆n ⊆ Rn and affine maps
between them, and U 0 the obvious forgetful functor;
C0 the monoid of affine endomorphisms of I 2 (or ∆2 ), and U 0
the action on I 2 ∈ Meas (or ∆2 ∈ Meas respectively);
C0 the category of all bounded convex subsets of Euclidean
space with affine maps, and U 0 the forgetful functor.
Then the codensity monad of U 0 is still the finitely additive Giry
monad F.
Tom Avery
Codensity and the Giry monad
Related Results
Consider the last of these: C0 = {bounded convex subsets of Rn },
U 0 = 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:
Tom Avery
Codensity and the Giry monad
Related Results
Consider the last of these: C0 = {bounded convex subsets of Rn },
U 0 = 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,
every f.a. probability measure on a convex bounded C ⊆ Rn
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
Related Results
Let V 0 : D0 → Meas be any of the following
D0 the category of simplices, together with
P∞
∆ω = {(xn )∞
n=0 | xn ∈ I ,
n=0 xn = 1}
and affine maps, and V 0 forgetful;
Tom Avery
Codensity and the Giry monad
Related Results
Let V 0 : D0 → Meas be any of the following
D0 the category of simplices, together with
P∞
∆ω = {(xn )∞
n=0 | xn ∈ I ,
n=0 xn = 1}
and affine maps, and V 0 forgetful;
D0 the monoid of affine endomorphisms of d0 (or ∆ω ), and V 0
the action on d0 ∈ Meas (or∆ω respectively);
Tom Avery
Codensity and the Giry monad
Related Results
Let V 0 : D0 → Meas be any of the following
D0 the category of simplices, together with
P∞
∆ω = {(xn )∞
n=0 | xn ∈ I ,
n=0 xn = 1}
and affine maps, and V 0 forgetful;
D0 the monoid of affine endomorphisms of d0 (or ∆ω ), and V 0
the action on d0 ∈ Meas (or∆ω respectively);
D0 the category of closed, convex, bounded subsets of the
vector space
c0 = {(xn )∞
n=0 | xn ∈ R, xn → 0}
with the sup norm, with affine maps between them, and V 0
forgetful.
Then the codensity monad of V 0 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
Related monads
Recall C is the category with objects I n 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
Recall C is the category with objects I n 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 .
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