States of Convex Sets - bram.westerbaan.name
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States of Convex Sets Bart Jacobs [email protected] Bas Westerbaan [email protected] Bram Westerbaan [email protected] Radboud University Nijmegen June 29, 2015
Transcript of States of Convex Sets - bram.westerbaan.name
States of Convex Sets
Bart [email protected]
Bram [email protected]
Radboud University Nijmegen
June 29, 2015
States of Convex Sets
Bart [email protected]
Bram [email protected]
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.
