thirty yearsof coalgebra what havewe learned · 2015-12-01 · i had to engage with areas of...

Thirty Years of Coalgebra:

WhatHave we Learned?

Larry Moss

Indiana University, Bloomington

Stanford Logic Seminar

October 27, 2015

1/51

History

During 1984-85,

Peter Aczel visited CSLI here at Stanford.

He was working on his book Non-Well-Founded Sets.

The book can be read in several ways:

◮ a solution to a semantic problem coming from theoretical

computer science

◮ a contribution to the foundational aspects of set theory.

The set-theoretic aspect was by far more original and

memorable.

In fact, Aczel proposed changing the very axioms of set

theory(!),

replacing the Foundation Axiom with the Anti-Foundation

Axiom.2/51

next chapter in the history

Jon Barwise got interested in subject due to connections with

circularity.

He wrote the introduction to Aczel’s book.

Later he recruited me to co-author a book which makes circular

phenomena the central theme.

He wanted the book to be “popular”, so the key decisions were.

◮ to emphasize the Anti-Foundation Axiom.

◮ to take the few category-theoretic ideas in Aczel’s book,

and replace them with more familiar and more concrete

ideas.

3/51

next chapter in the history

Aczel’s hardest technical result were called

The Final Coalgebra Theorems

but we didn’t know what a final coalgebra was.

One of the theorems talked about

Functors preserving weak pullbacks.

We didn’t understand that, either.

Worst of all, we could understand the book and even extend

the ideas without understanding a long list of unfamiliar ideas.

4/51

What happened to me later

The overall issues of circularity are still with me.

I certainly have not learned all that I want to about the subject.

When I would give talks on the book, people would say

◮ Larry, this is nice, but where is the computation?

◮ If this is about circularity, where is the circle?

What about fractals?

And there were mathematical topics related to circularity that I

just couldn’t answer.

5/51

Harsanyi type spaces

Consider two-person games of imperfect information.

Each player has beliefs about the world and about the other.

In fact, the type of each player is “just” a probability measure on

possible ways the world can be × possible types

6/51

Harsanyi type spaces

Consider two-person games of imperfect information.

Each player has beliefs about the world and about the other.

In fact, the type of each player is “just” a probability measure on

possible ways the world can be × possible types

But what are these types?

Dropping the two-player aspects, we seem to be asking to solve

M = measurable space of probability measures on (R ×M)

And if that isn’t hard enough, we want the biggest solution.

6/51

History 3

About the same time as we were working on Vicious Circles,

researchers in Europe got interested in Aczel’s book.

But their interest was in many ways the opposite of ours.

For them, the category-theoretic aspects of the book were

highly suggestive,

mostly because they could see connections to other topics that

they were interested in.

7/51

What happened next

In order for me to continue my work on circularity,

I basically had to join the coalgebra community.

I had to engage with areas of mathematics that I had never

thought about.

Jon Barwise held that work on non-wellfounded sets was

“lovely mathematics,”

Coalgebra has been even prettier, and this has kept my interest.

8/51

twenty coalgebraists

Jirı Adamek

Filippo Bonchi

Corina Cirstea

H. Peter Gumm

Helle Hansen

Ichiro Hasuo

Bart Jacobs

Bartek Klin

Barbara Konig

Dexter Kozen

Alexander Kurz

Paul Blain Levy

Stefan Milius

Daniela Petrisan

Dirk Pattinson

John Power

Jan Rutten

Lutz Schroder

Alexandra Silva

Yde Venema

9/51

The conceptual comparison chartThis chart is my real goal; everything else is secondary

set with algebraic set with transitions

operations and observations

algebra for a functor coalgebra for a functor

initial algebra final coalgebra

least fixed point greatest fixed point

congruence relation bisimulation equivalence rel’n

equational logic modal logic

recursion: map out of corecursion: map into

an initial algebra a final coalgebra

Foundation Axiom Anti-Foundation Axiom

iterative conception of set coiterative conception of set

useful in syntax useful in semantics

bottom-up top-down

10/51

Algebras and coalgebras

LetA be a category, and let F :A→A be a functor.

An F-algebra is (A , a : FA → A).An F-coalgebra is (A , a : A → FA).

In both cases, A is the carrier and a the structure.

Example: deterministic automata

(S , s : S → 2 × SA )

are coalgebras of 2 × XA ,

as are Kripke models for modal logic.

11/51

Morphisms of algebras and coalgebras

Let (A , a : FA → A) and (B , b : FB → B) be algebras.

A morphism is f : A → B in the same underlying category so that

FAa //

Ff��

A

f��

FBb

// B

commutes.

Let (A , a : A → FA) and (B , b : B → FB) be coalgebras.

A morphism is f : A → B in the same underlying category so that

Aa //

f��

FA

Ff��

Bb

// FB

commutes.12/51

Example: FX = 1 + (X × X)?

Iterate it to get a fixed point

µF =⋃

i∈ω

F i(∅)

F1∅ : •

F2∅ : •,• •��

////

//

F3∅ : •,• •��

////

//

, •

• •

��

////

///

��

////

// , •

• •

////

//

��

��

////

// ,

• • • •

////

///

��

��

))))

))

��

))))

))

µF ≈ 1 + (µF × µF) = F(µF).

13/51


Recursion Principle for Finite Trees

For all sets X , all x ∈ X , all f : X × X → X ,

there is a unique ϕ : µF → X

so that ϕ is

one-point tree • 7→ x

t u

////

///

��

��

))))

))

��

))))

)) 7→ f(ϕ(t), ϕ(u))

A fixed point is both an algebra and a coalgebra.

13/51


Recursion Principle for Finite Trees

For all sets X , all f : 1 + (X × X)→ X ,

there is a unique ϕ : µF → X so that

F(µF)id //

1+(ϕ×ϕ)

��

µF

ϕ

��FX

f// X

commutes, where (ϕ × ϕ)(t , u) = (ϕ(t), ϕ(u)).

13/51

Recursion on N is tantamount to Initiality

Recursion on N: For all sets A , all a ∈ A , and all f : A → A ,

there is a unique ϕ : N → A so that

ϕ(0) = a, and ϕ(n + 1) = f(ϕ(n)) for all n.

Initiality of N: For all (A , a), there is a unique homomorphism

ϕ : (N, ν)→ (A , a):

1 + Nid //

1+ϕ

��

N

ϕ

��1 + A a

// A

Recursion is often, but not always,

about maps out of an initial algebra.

14/51

Review: initial algebras and final coalgebras

initial algebra FAa //

Ff

��

A

f

��FB

b// B

Aa //

f

��

FA

Ff

��B

b// FB final coalgebra

15/51

Initial algebras and final coalgebras

of various set functors

initial algebra

1 + X ∗ // 0 // 1 // 2 // · · ·

1 + (X × X) finite binary trees

{a, b , c} × X ∅

PfinX Vω = hereditarily finite sets

final coalgebra

1 + X ∗ 0oo 1oo 2oo · · ·oo ∞ ff1 + (X × X) finite and infinite binary trees

{a, b , c} × X infinite streams on {a, b , c}

PfinX finitely branching graphs with a ‘top’ point,

modulo bisimulation

16/51

Initial algebras and final coalgebras

of various set functors

initial algebra

1 + X ∗ // 0 // 1 // 2 // · · ·


{a, b , c} × X ∅

PfinX Vω = hereditarily finite sets

final coalgebra

1 + X ∗ 0oo 1oo 2oo · · ·oo ∞ ff1 + (X × X) finite and infinite binary trees

{a, b , c} × X infinite streams on {a, b , c}

PfinX finitely branching graphs with a ‘top’ point,

modulo bisimulation

Query: is there any relation between these charts?

And how do we get these examples, anyway?

16/51

How can we get our hands on µF?Answer: generalize Kleene’s Theorem

Kleene’s Theorem

Let (A ,≤) be poset with a least element 0 and with the property

that every countable chain C ⊆ A has a least upper bound∨

C.

Let F : A → A be monotone and ω-continuous.

Then the least upper bound of

0 ≤ F0 ≤ F20 ≤ · · ·

has the property that F(µF) ≤ µF ,

and indeed is the least fixed point of F .

17/51

The category-theoretic generalization“Preorders are the poor person’s category”

order-theoretic concept categorification

preorder (A ,≤) categoryA

x ≤ y and y ≤ x A and B are isomorphic objects

least element 0 initial object 0

monotone F : A → A functor F :A→A

pre-fixed point: Fx ≤ x F-algebra: f : FA → A

countable chain functor from (ω,<) to A

F is ω-continuous F preserves ω-colimits

least pre-fixed point: Fx ≤ x initial F-algebra: f : FA → A

18/51

A generalization of Kleene’s Theorem

Kleene’s Theorem

Let (A ,≤) be poset with a least element 0 and with the property

that every countable chain C ⊆ A has a least upper bound∨

C.

Let F : A → A be monotone and ω-continuous.

Then the least upper bound of

0 ≤ F0 ≤ F20 ≤ · · ·

has the property that F(µF) ≤ µF ,

and indeed is the least fixed point of F .

Adamek 1974

LetA be a category with initial object 0

and with the property that every ω-chain in A has a colimit.

Let F : A→A preserve ω-colimits,

let µF be the colimit of the initial sequence of F :

0! // F0

F! // F20F2! // · · · Fn−1! // Fn0

Fn! // Fn+10Fn+1! // · · ·

There is a canonical m : F(µF)→ µF

and (µF ,m) is an initial F-algebra.

19/51

Where does it apply?

FX initial algebra µF


PfinX HF

Pκ Hκ

1 + X natural numbers

signature functor terms on the signature

bag functor finite unordered trees

Adamek’s generalization of Kleene’s Theorem is not the only

way to get an initial algebra, but it is the most common.

20/51

Lambek’s Lemma

Lambek’s Lemma

Let P be a poset.

Let f : P→ P be a monotone function.

And let x be a minimal pre-fixed point:

fx ≤ x, and x minimal like this.

Then: fx = x.

Proof.

Note that f(fx) ≤ fx.

And so x ≤ fx, too. �

21/51

Lambek’s Lemma

Categorification of this fact:

Lambek’s Lemma

The structure morphisms of initial algebras

are always categorical isomorphisms.

21/51

Lambek’s Lemma

Categorification of this fact:

Lambek’s Lemma

The structure morphisms of initial algebras

are always categorical isomorphisms.

In particular, P has no initial algebra on Set,

as we see from Cantor’s Theorem.

21/51

Review/Example

initial algebra 1 + Nt //

id1+ϕ

��

N

ϕ

��1 + A a

// A

N is an initial algebra of 1 + X on Set

Coalgebras of 2 × XA are deterministic automata

Ss //

ϕ

��

2 × SA

id2×ϕA

��L

l// 2 × LA final coalgebra

The set L of formal languages is a final coalgebra.

The map ϕ takes a state to the language accepted there.

22/51

Where we are






23/51

Where we are






In some ways, the mathematics of transitions and observations

is less familiar than that of sets and operations.

Coalgebra is trying to be the general mathematics of transitions

and observations.

23/51

Final Coalgebras: why and what?

Final coalgebras are like the most abstract collections of

“transitions” or “observations”.

Again, given F , does the initial algebra exist?

Does the final coalgebra?

How can we get our hands on them?

24/51

The main existence theorem for initial

algebras/final coalgebras

Adamek 1974

Assume thatA has enough structure

to take the colimit µF of the initial sequence

0! // F0

F! // F20F2! // · · · Fn−1! // Fn0

Fn! // · · ·

and that F : A→ A preserves this ω-colimit.

There is a canonical morphism m : F(µF)→ F such that

(µF ,m)

is an initial F-algebra.

25/51

The main existence theorem for initial

algebras/final coalgebras

Barr 1993

Assume thatA has enough structure

to take the limit νF of the initial sequence

1 F1!oo F21

F!oo F2!oo · · · Fn1Fn−1!oo · · ·Fn!oo

and that F : A→ A preserves this ω-limit.

There is a canonical morphism m : F → F(νF) such that

(νF ,m)

is a final F-coalgebra.

25/51

Example: streams over 2 = {0, 1}

Here our functor is FX = 2 × X .

1 is any one point set, say {∗}.

So F1 = 2 × 1 = {(0, ∗), (1, ∗)}.

F21 = 2 × F1 = {(0, (0, ∗)), (0, (1, ∗)), (1, (0, ∗)), (1, (1, ∗))}.

1 F1!oo F21

F!oo · · · Fn1 Fn+11Fn!oo · · ·

L

ln

OO

26/51

Example: streams over 2 = {0, 1}

Here our functor is FX = 2 × X .

1 F1!oo F21

F!oo · · · Fn1 Fn+11Fn!oo · · ·

L

ln

OO

A representation that you have seen:

take L = 2N,

ln : L → Fn1 is f 7→ (f(0), (f(1), (f(2), . . . f(n))))).

The coalgebra structure m : 2N → 2 × 2N is a little easier:

m(f) = (f(0), n 7→ f(n + 1)).

26/51

Corecursion on streams is tantamount to finality

Corecursion on 2N: For all sets A , all f : A → 2 × A ,

there is a unique ϕ : A → 2N so that

ϕ(x) = (head f(x), ϕ(tail(f(x))))

Finality of 2N: For all (A , a), there is a unique homomorphism

ϕ : (N, ν)→ (A , a):

Af //

ϕ

��

2 × A

id2×ϕ

��2N

〈head,tail〉// 2 × 2N

Corecursion is about maps into a final algebra.

27/51

Streams: illustration

Consider a coalgebra (A , a : A → R × A), where

A = {α, β, γ, δ}, and

a(α) = (0, β)a(β) = (1, γ)

a(γ) = (0, δ)a(δ) = (−1, α)

What is the map h below?

Aa //

h��

R × A

Fh��

RN

ϕ//R ×RN

28/51

Streams: illustration


A = {α, β, γ, δ}, and

a(α) = (0, β)a(β) = (1, γ)

a(γ) = (0, δ)a(δ) = (−1, α)


Aa //

h��

R × A

Fh��

RN

ϕ//R ×RN

h(α) = (0, h(β)) = (0, 1, 0,−1, 0, 1, . . .)h(β) = (1, h(γ)) = (1, 0,−1, 0, 1, 0, . . .)h(γ) = (0, h(δ)) = (0,−1, 0, 1, 0, 1, . . .)h(δ) = (−1, h(α)) = (−1, 0, 1, 0, 1, 0, . . .)

28/51

Where does the theorem apply?

Finitary Iteration

LetA be a category with final object 1

and with the property that every ωop-chain in A has a limit.

Let F : A→A preserve ωop-limits,

and consider the final ωop-chain of F :

1 F1!oo F21

F!oo · · · Fn1 Fn+11Fn!oo · · ·

Let νF be its limit, and let m : νF → F(νF) be the factorizing

morphism.

Then (νF ,m) is a final F-coalgebra.

We don’t really need all limits, only the one shown.

And this is the only limit we need F to preserve.

29/51


Finitary iteration gives final coalgebras for all functors on Set

built from

◮ the identity functor

◮ constant functors

and using

◮ +, ×, FA for fixed sets A

◮ composition

But it doesn’t work for Pfin or its relatives Pκ.

It doesn’t work for the discrete measure functor ∆,

either.

30/51


Finitary iteration gives final coalgebras for all functors on

compact Hausdorff spaces built from



◮ the Vietoris functor V.

V(X) is the hyperspace of X ,

the set of compact subsets of X , with a certain topology.

For f : X → Y and A ∈ VX ,

(Vf)A = f [A ].

◮ the Borel measure functor B. For f : X → Y and A ∈ VX ,

((Bf)µ)A = µ(f−1(A))

and using

◮ +, ×

◮ composition31/51


Finitary iteration gives final coalgebras for all functors on MS

built from



◮ εPk , the scaled version of the closed set functor Pk ,

using the Hausdorff distance

d(s, t) = max{supx∈s

infy∈t

d(x, y), supx∈s

infy∈t

d(x, y)}.

The distance from ∅ to any other closed set is 1.

ε < 1 scales distances.

◮ using +, ×, and composition.

van Breugel: Pk without scaling has no final coalgebra

on MS.

32/51

Iteration in CPO⊥-enriched categories

A CPO⊥ is a complete partial order with ⊥.

A is CPO⊥-enriched if its homsetsA(X ,Y)carry the structure of a CPO with ⊥

and composition is strict (preserves the least element) and

continuous (preserves ω-joins) in both variables.

F : A→ A is locally continuous if F⊔

fn =⊔

Ffnfor all ω-chains fn ∈ A(X ,Y).

33/51

Iteration in CPO⊥-enriched categories

A CPO⊥ is a complete partial order with ⊥.

A is CPO⊥-enriched if its homsetsA(X ,Y)carry the structure of a CPO with ⊥

and composition is strict (preserves the least element) and

continuous (preserves ω-joins) in both variables.

F : A→ A is locally continuous if F⊔

fn =⊔

Ffnfor all ω-chains fn ∈ A(X ,Y).

Theorem (Adamek, based on Smyth and Plotkin 1982)

Every locally continuous F : A→A has a canonical fixed point:

there is an initial algebra and it is the inverse of a final

coalgebra.

This result is at the core of Dana Scott’s construction of

D � [D → D]

giving a model of the lambda calculus.33/51


SB = standard Borel spaces,

measurable spaces which use the Borel subsets of a Polish

space

∆ : SB→ SB takes M to the set of its Borel probability

measures with σ-algebra generated by

{Bp(E) | p ∈ [0, 1],E ∈ Σ},

where

Bp(E) = {µ ∈ ∆(M) | µ(E) ≥ p}.

One uses the Kolmogorov Consistency Theorem to see

that the functor preserves the limit.

∆ : SB→ SB has a final coalgebra, as does a functor like

FX = ∆(X × [0, 1])

34/51

What about ∆ : Meas→ Meas?

Viglizzo 2005

The functor ∆ : Meas→ Meas does not preserve limits of

ωop-chains.

So it looks like we’re out of luck!

35/51

Pavlovic and Pratt 2002

Consider FX = N × X on Set

Final coalgebra is the Baire space B = irrationals in [0, 1],with structure

〈β, γ〉 : B → N × B

where

β(x) =

⌊

1

x

⌋

− 1 and γ(x) =

(

1

x

)

mod 1

γ is called the Gauss map.

The set Nω of streams on N is also a final coalgebra,

and the isomorphism

Nω

ϕ

��

〈head,tail〉 // N × Nω

N×ϕ

��B

〈β,γ〉// N × B

is given by continued fractions.

36/51

[0, 1] as a final coalgebra

Let BiP be the category of bi-pointed sets.

These are (X ,⊤,⊥) with ⊤,⊥ ∈ X and ⊤ , ⊥.

The bipointed set {⊤,⊥} is initial, but there is no final object.

37/51





X XX X7→⊤⊥ ⊤⊥

identify ⊤ of left with ⊥ of right

Initial algebras and final coalgebras of F : BiP→ BiP

0 is the two point space {0, 1} with d(0, 1) = 1.

F0 is {0, 12 , 1} with evident distances.

F20 is {0, 14, 1

2, 3

4, 1}.

Initial algebra of F on BiP is dyadic rationals.

37/51





Initial algebras and final coalgebras of F : BiP→ BiP

Freyd 1999:

νF can be taken to be the (set of points in) unit interval [0, 1],with a structure

i : [0, 1]→ F [0, 1]

as shown below.

37/51

The map i : [0, 1]→ F [0, 1]

d 12dd 1

2d→10 10

a < 12

7→ 2a on left12

7→ midpoint

a > 12

7→ 2a on right

Note that i is an isometry.

38/51

Proof of Freyd’s Theorem

Let e : X → FX be any morphism of BiP.

Recall that [0, 1] is a complete metric space.

Regard the set X a (discrete) space.

The space

S = homBiP(X , [0, 1]).

is a closed subspace of homCMS(X , [0, 1]).

F : homCMS(X , [0, 1])→ homCMS(FX ,F [0, 1])

is a contracting map: for f , g : X → [0, 1],

d(Ff ,Fg) ≤1

2d(f , g).

39/51

Proof of Freyd’s Theorem

Recall

i : [0, 1]→ F [0, 1].

This map is a bijection and an isometry.

We have a contracting endofunction ψ : S → S given by

ψ(f) = i−1 · Ff · e.

By the Contraction Mapping Thm., there’s a unique f = ψ(f).

f is exactly a coalgebra morphism (X , e)→ ([0, 1], i).

39/51

The Sierpinski Gasket is a final coalgebra, even

with the metric

With the topology but not the metric,

this was done in the seminal paper of Leinster.

With 3 IU graduate students, we got it with the metric,

using tripointed sets.

There are further connections to be made with metric geometry

and with self-similar groups.

40/51

Finality at work: FX = R × X

RA is the set of function f : R→ R such that

for all n, fn(0) exists, and f agrees with its

Taylor series on an interval containing 0.

The final coalgebra is (RA , ϕ) where

ϕ : RA → R × RA is f 7→ (f(0), f ′)

41/51






ϕ : RA → R × RA is f 7→ (f(0), f ′)


A = {α, β, γ, δ}, and

a(α) = (0, β)a(β) = (1, γ)

a(γ) = (0, δ)a(δ) = (−1, α)


Aa //

h��

R × A

Fh��

RA ϕ//R × RA

41/51






ϕ : RA → R × RA is f 7→ (f(0), f ′)


A = {α, β, γ, δ}, and

a(α) = (0, β)a(β) = (1, γ)

a(γ) = (0, δ)a(δ) = (−1, α)


Aa //

h��

R × A

Fh��

RA ϕ//R × RA

α 7→ sin x, β 7→ cos x, γ 7→ − sin x, γ 7→ − cos x

Pavlovic and Escardo 1998, “Calculus in Coinductive Form”41/51

Bisimulation and the final coalgebra of Pfin

Let (G,→) be a graph.

A relation R on G is a bisimulation iff the following holds:

whenever xRy,

(zig) If x → x′, then there is some y → y′ such that x′Ry′.

(zag) If y → y′, then there is some x → x′ such that x′Ry′.

42/51


Let (G,→) be a graph.

A relation R on G is a bisimulation iff the following holds:

whenever xRy,

(zig) If x → x′, then there is some y → y′ such that x′Ry′.

(zag) If y → y′, then there is some x → x′ such that x′Ry′.

For an example, let’s look at the following graph G:

3b

3a 2aoo

@A

//

2boo

=={{{{{{{{1oo

OO

��

// 3c 3d

2c

=={{{{{{{{

aaCCCCCCCC

The largest bisimulation on our graph G is the relation that

relates 1 to itself,

all 2-points to all 2-points,

and all 3-points to all 3-points.42/51


The final coalgebra of Pfin

is the set of finitely braching graphs with the following features:

◮ There is a “top” point p

◮ Every point is reachable from the top.

◮ Every bisimulation on the graph is a subset of the diagonal.

43/51

Induction and bisimulation

Induction

Every subalgebra of an initial algebra is invertible.

Coinduction

Every bisimulation relation on a final coalgebra

is a subset of the diagonal.

44/51

arithmetic : induction ::

analysis : co-inductionThe proof here is due to Jan Rutten

For example, let’s prove that

sin(x + y) = sin x cos y + cos x sin y.

Fix a real a. Write b for sin a and c for cos a.

Consider the set of pairs of real functions, where

(sin(x + a), c sin x + b cos x)(cos(x + a),−b sin x + c cos x)(− sin(x + a),−c sin x − b cos x)(− cos(x + a), b sin x − c cos x)

45/51



For example, let’s prove that

sin(x + y) = sin x cos y + cos x sin y.

Fix a real a. Write b for sin a and c for cos a.

We get a relation on RA .

45/51




It’s a set of pairs of real functions.

In each pair, the value of the two functions at 0 is the same.

And for (f , g) in the list, (f ′, g′) is also in the list.

So we have a bisimulation.

And thus, by coinduction, the entries in each pair are the same.

So for all a,

sin(x + a) = cos a sin x + sin a cos x

46/51




It’s a set of pairs of real functions.

In each pair, the value of the two functions at 0 is the same.

And for (f , g) in the list, (f ′, g′) is also in the list.

So we have a bisimulation.

And thus, by coinduction, the entries in each pair are the same.

So for all a,

sin(x + a) = cos a sin x + sin a cos x

Other applications: enumerative combinatorics, power series,

continued fractions.46/51

Bisimulation in real-world settings

Let’s watch a video!

Click on this link.

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http://dl.acm.org/citation.cfm?id=2728770.2713167

Another success: coalgebraic generalizations of

modal logic

Modal logic

???=

the functor K (a) = P(a) × P(AtSen)

an arbitrary (?) functor F

The logic ??? should be interpreted on all coalgebras of F .

It should characterize points in (roughly) the sense that

points in a coalgebra have the same L theory

iff they are bisimilar

iff they are mapped to the same point in the final coalgebra

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What about ∆ : Meas→ Meas?

LM and Viglizzo 2006, building on Heifetz and Samet 1998

Every functor F : Meas→ Meas built from the usual stuff and

∆ : Meas→ Meas

has a final coalgebra.

The proof used a version of the probabilistic modal logic,

using the set of all theories of all points in all spaces,

and also using the π-λ Theorem of measure theory.

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The universe of sets

Consider the categoryA of classes.

P : A→ A gives the class of subsets of a given class.

Note that PV = V .

Work in ZF − Foundation

The Foundation Axiom is equivalent to the assertion that

(V , id : PV → V)

is an initial algebra of P.

The Anti-Foundation Axiom is equivalent to the assertion that

(V , id : V → PV)

is a final coalgebra of P.

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The conceptual comparison chartFilling out the details is my goal for coalgebra





least fixed point greatest fixed point

congruence relation bisimulation equivalence rel’n

equational logic modal logic

recursion: map out of corecursion: map into

an initial algebra a final coalgebra

Foundation Axiom Anti-Foundation Axiom

iterative conception of set coiterative conception of set


bottom-up top-down

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thirty yearsof coalgebra what havewe learned · 2015-12-01 · i had to engage with areas of...

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