states of convex sets - bram.westerbaan.name

States of Convex Sets

Bart Jacobsbart@cs.ru.nl

Bas Westerbaanbwesterb@cs.ru.nl

Bram Westerbaanawesterb@cs.ru.nl

Radboud University Nijmegen

June 29, 2015

States of Convex Sets

Bart Jacobsbart@cs.ru.nl

Bas Westerbaanbwesterb@cs.ru.nl

Bram Westerbaanawesterb@cs.ru.nl

Radboud University Nijmegen

June 29, 2015

The categorical quantum logic group in Nijmegen

The categorical quantum logic group in Nijmegen

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

In contrast to the friendlycompetition at Oxford: they emphasizeto axiomatize what is unique andnon-classical about quantum mechanics.

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ...

some advances on state spaces,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ... some advances on state spaces

,but we’ll come to that!

What we do in Nijmegen

1. The semantics and logic of quantum computation.

2. Focus on the common ground between the classical,probabilistic and quantum setting (States, predicates, ...)

3. Identify relevant structure (Effect algebras, ...)

4. Organise it with category theory and formal logic.

5. Ambition: to make quantum computation more accessible toexisting methods and techniques (of categorical logic, ...)

6. On the horizon: a categorical toolkit including a type theoryto formally verify quantum programs.

7. In this paper ... some advances on state spaces,but we’ll come to that!

Oxford & Nijmegen

Setting

Classical : Probabilistic : Quantum

Sets : K̀ (D) : vNop

sets with maps sets with von Neumann algebras

probabilistic maps with c.p. unital

normal linear maps

Setting

Classical : Probabilistic : Quantum

Sets : K̀ (D) : vNop

sets with maps sets with von Neumann algebras

probabilistic maps with c.p. unital

normal linear maps

Setting

Classical : Probabilistic : Quantum

Sets : K̀ (D) : vNop

sets with maps sets with von Neumann algebras

probabilistic maps with c.p. unital

normal linear maps

Logic?

Sets K̀ (D) vNop

classical probabilistic quantum

topos? X

7 7

CCC? X 7 7

effectus* X X X

* see next page

Logic?

Sets K̀ (D) vNop

classical probabilistic quantum

topos? X 7 7

CCC? X 7 7

effectus* X X X

* see next page

Logic?

Sets K̀ (D) vNop

classical probabilistic quantum

topos? X 7 7

CCC? X 7 7

effectus* X X X

* see next page

Logic?

Sets K̀ (D) vNop

classical probabilistic quantum

topos? X 7 7

CCC? X 7 7

effectus* X X X

* see next page

*Effectus

An effectus is a category with finite coproducts and 1 such that

I these diagrams are pullbacks:

A + Xid+g //

f +id��

A + Y

f +id��

B + Xid+g

// B + Y

Aid //

κ1

��

A

κ1

��A + X

id+f// A + Y

I these arrows are jointly monic:

X + X + X[κ1,κ2,κ2] //

[κ2,κ1,κ2]// X + X

(Rather weak assumptions!)

*Effectus

An effectus is a category with finite coproducts and 1 such that

I these diagrams are pullbacks:

A + Xid+g //

f +id��

A + Y

f +id��

B + Xid+g

// B + Y

Aid //

κ1

��

A

κ1

��A + X

id+f// A + Y

I these arrows are jointly monic:

X + X + X[κ1,κ2,κ2] //

[κ2,κ1,κ2]// X + X

(Rather weak assumptions!)

*Effectus

An effectus is a category with finite coproducts and 1 such that

I these diagrams are pullbacks:

A + Xid+g //

f +id��

A + Y

f +id��

B + Xid+g

// B + Y

Aid //

κ1

��

A

κ1

��A + X

id+f// A + Y

I these arrows are jointly monic:

X + X + X[κ1,κ2,κ2] //

[κ2,κ1,κ2]// X + X

(Rather weak assumptions!)

*Effectus

An effectus is a category with finite coproducts and 1 such that

I these diagrams are pullbacks:

A + Xid+g //

f +id��

A + Y

f +id��

B + Xid+g

// B + Y

Aid //

κ1

��

A

κ1

��A + X

id+f// A + Y

I these arrows are jointly monic:

X + X + X[κ1,κ2,κ2] //

[κ2,κ1,κ2]// X + X

(Rather weak assumptions!)

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X

state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1

predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Internal logic

effectus meaning

objects types

arrows programs

1 (final object) singleton/unit type

1ω // X state

Xp // 1 + 1 predicate

1ω //

ω�p22X

p // 1 + 1 validity

1λ// 1 + 1 scalar

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X

ω ∈ p {0, 1}probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉

fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I

ω(p) [0, 1]

Examples of states and predicates

State Predicate Validity Scalars

1ω→ X X

p→ 1 + 1 ω � p 1→ 1 + 1

classicalSets

elementω ∈ X

subsetp ⊆ X ω ∈ p {0, 1}

probabilistic

K̀ (D)distribution

ω ≡∑

i si |xi 〉fuzzy subset

Xp→ [0, 1]

∑i sip(xi ) [0, 1]

quantum

vNopnormal stateω : X → C

effect0 ≤ p ≤ I ω(p) [0, 1]

Structure on states and predicates

1. Predicates on X form an effect module

over M

(≈ an ordered vector space

over M

restricted to [0, 1])

2. States on X form an convex set

over M

(= algebra for the distribution monad

over M

)

3. The scalars form an effect monoid M.

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

Structure on states and predicates

1. Predicates on X form an effect module

over M

(≈ an ordered vector space

over M

restricted to [0, 1])

2. States on X form an convex set

over M

(= algebra for the distribution monad

over M

)

3. The scalars form an effect monoid M.

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

Structure on states and predicates

1. Predicates on X form an effect module

over M

(≈ an ordered vector space

over M

restricted to [0, 1])

2. States on X form an convex set

over M

(= algebra for the distribution monad

over M

)

3. The scalars form an effect monoid M.

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

Structure on states and predicates

1. Predicates on X form an effect module

over M

(≈ an ordered vector space

over M

restricted to [0, 1])

2. States on X form an convex set

over M

(= algebra for the distribution monad

over M

)

3. The scalars form an effect monoid M.

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

Structure on states and predicates

1. Predicates on X form an effect module over M(≈ an ordered vector space over M restricted to [0, 1])

2. States on X form an convex set over M(= algebra for the distribution monad over M)

3. The scalars form an effect monoid M.

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

Structure on states and predicates

1. Predicates on X form an effect module over M(≈ an ordered vector space over M restricted to [0, 1])

2. States on X form an convex set over M(= algebra for the distribution monad over M)

3. The scalars form an effect monoid M.

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

Examples of operatorions on states and predicates

I Negation of predicate: Xp //

¬p221 + 1

[κ2,κ1]// 1 + 1

I Convex combination of states 1λ //

λω+(1−λ)%

221 + 1[ω,%] // X

I Predicates p, q are summable whenever there is a b such that

Xp

yy

q

&&b��

1 + 1 1 + 1 + 1[κ1,κ2,κ2]oo

[κ2,κ1,κ2]// 1 + 1

and then their sum is given by p > q = [κ1, κ1, κ2] ◦ b.

Examples of operatorions on states and predicates

I Negation of predicate: Xp //

¬p221 + 1

[κ2,κ1]// 1 + 1

I Convex combination of states 1λ //

λω+(1−λ)%

221 + 1[ω,%] // X

I Predicates p, q are summable whenever there is a b such that

Xp

yy

q

&&b��

1 + 1 1 + 1 + 1[κ1,κ2,κ2]oo

[κ2,κ1,κ2]// 1 + 1

and then their sum is given by p > q = [κ1, κ1, κ2] ◦ b.

Examples of operatorions on states and predicates

I Negation of predicate: Xp //

¬p221 + 1

[κ2,κ1]// 1 + 1

I Convex combination of states 1λ //

λω+(1−λ)%

221 + 1[ω,%] // X

I Predicates p, q are summable whenever there is a b such that

Xp

yy

q

&&b��

1 + 1 1 + 1 + 1[κ1,κ2,κ2]oo

[κ2,κ1,κ2]// 1 + 1

and then their sum is given by p > q = [κ1, κ1, κ2] ◦ b.

Two problems?

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

1. EModopM is an effectus; Pred : C→ EModop

M preserves +.

2. ConvM is not an effectus; Stat : C→ ConvM does notalways preserve coproducts.

So what? They block treating conditional probability in an effectus.

Two problems?

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

1. EModopM is an effectus; Pred : C→ EModop

M preserves +.

2. ConvM is not an effectus; Stat : C→ ConvM does notalways preserve coproducts.

So what? They block treating conditional probability in an effectus.

Two problems?

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

1. EModopM is an effectus; Pred : C→ EModop

M preserves +.

2. ConvM is not an effectus; Stat : C→ ConvM does notalways preserve coproducts.

So what? They block treating conditional probability in an effectus.

Two problems?

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

1. EModopM is an effectus; Pred : C→ EModop

M preserves +.

2. ConvM is not an effectus; Stat : C→ ConvM does notalways preserve coproducts.

So what?

They block treating conditional probability in an effectus.

Two problems?

EModopM

Stat ,,> ConvM

Predll

CPred

bb

Stat

==

1. EModopM is an effectus; Pred : C→ EModop

M preserves +.

2. ConvM is not an effectus; Stat : C→ ConvM does notalways preserve coproducts.

So what? They block treating conditional probability in an effectus.

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;

3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + Aare jointly injective;

3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;

3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + Aare jointly injective;

3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;

3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + Aare jointly injective;

3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;

3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + Aare jointly injective;

3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + A

are jointly injective;

3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + A

are jointly injective;3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Cancellative Convex Sets

1.0

10

11

This is a convex set over [0, 1](that is, algebra for the distrubu-tion monad over [0, 1]):

2. A convex set A is cancellative if for λ 6= 1,λx + (1− λ)y1 = λx + (1− λ)y2 =⇒ y1 = y2.

3. Theorem For a convex set A over [0, 1] t.f.a.e.

3.1 A is cancellative;3.2 [κ1, κ2, κ2], [κ2, κ1, κ2] : A + A + A −→ A + A

are jointly injective;3.3 A is isomorphic to a convex subset of a real vector space.

4. The full subcategory CConv[0,1] of Conv[0,1] of cancellativeconvex sets over [0, 1] is an effectus!

Normalisation

Stat : C −→ CConv[0,1] preserves coproducts if ...

C has normalisation:For every 1

σ→ X + 1 with σ 6= κ2 there is a unique 1ω→ X such

that the following diagram commutes.

1σ //

σ��

X + 1

X + 1!+id

// 1 + 1

ω+id

OO

Normalisation

Stat : C −→ CConv[0,1] preserves coproducts if ...C has normalisation:

For every 1σ→ X + 1 with σ 6= κ2 there is a unique 1

ω→ X suchthat the following diagram commutes.

1σ //

σ��

X + 1

X + 1!+id

// 1 + 1

ω+id

OO

Normalisation

Stat : C −→ CConv[0,1] preserves coproducts if ...C has normalisation:For every 1

σ→ X + 1 with σ 6= κ2 there is a unique 1ω→ X such

that the following diagram commutes.

1σ //

σ��

X + 1

X + 1!+id

// 1 + 1

ω+id

OO

Conclusion and references

EModop[0,1]

Stat --> CConv[0,1]

Predmm

CPred

cc

Stat

;;

1. Every category above is an effectus;every functor above preserves coproducts.

2. For the relation with conditional probability,see Section 6 of the paper.

3. For more about effectuses:Bart Jacobs, New Directions in Categorical Logic, [...],arXiv:1205.3940v3.

Conclusion and references

EModop[0,1]

Stat --> CConv[0,1]

Predmm

CPred

cc

Stat

;;

1. Every category above is an effectus;every functor above preserves coproducts.

2. For the relation with conditional probability,see Section 6 of the paper.

3. For more about effectuses:Bart Jacobs, New Directions in Categorical Logic, [...],arXiv:1205.3940v3.

Conclusion and references

EModop[0,1]

Stat --> CConv[0,1]

Predmm

CPred

cc

Stat

;;

1. Every category above is an effectus;every functor above preserves coproducts.

2. For the relation with conditional probability,see Section 6 of the paper.

3. For more about effectuses:Bart Jacobs, New Directions in Categorical Logic, [...],arXiv:1205.3940v3.