Download - States of Convex Sets - bram.westerbaan.name
![Page 1: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/1.jpg)
States of Convex Sets
Bart [email protected]
Bram [email protected]
Radboud University Nijmegen
June 29, 2015
![Page 2: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/2.jpg)
States of Convex Sets
Bart [email protected]
Bram [email protected]
Radboud University Nijmegen
June 29, 2015
![Page 3: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/3.jpg)
The categorical quantum logic group in Nijmegen
![Page 4: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/4.jpg)
The categorical quantum logic group in Nijmegen
![Page 5: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/5.jpg)
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!
![Page 6: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/6.jpg)
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!
![Page 7: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/7.jpg)
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.
![Page 8: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/8.jpg)
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!
![Page 9: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/9.jpg)
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!
![Page 10: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/10.jpg)
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!
![Page 11: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/11.jpg)
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!
![Page 12: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/12.jpg)
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!
![Page 13: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/13.jpg)
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!
![Page 14: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/14.jpg)
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!
![Page 15: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/15.jpg)
Oxford & Nijmegen
![Page 16: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/16.jpg)
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
![Page 17: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/17.jpg)
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
![Page 18: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/18.jpg)
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
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Logic?
Sets K̀ (D) vNop
classical probabilistic quantum
topos? X
7 7
CCC? X 7 7
effectus* X X X
* see next page
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Logic?
Sets K̀ (D) vNop
classical probabilistic quantum
topos? X 7 7
CCC? X 7 7
effectus* X X X
* see next page
![Page 21: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/21.jpg)
Logic?
Sets K̀ (D) vNop
classical probabilistic quantum
topos? X 7 7
CCC? X 7 7
effectus* X X X
* see next page
![Page 22: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/22.jpg)
Logic?
Sets K̀ (D) vNop
classical probabilistic quantum
topos? X 7 7
CCC? X 7 7
effectus* X X X
* see next page
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*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!)
![Page 24: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/24.jpg)
*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!)
![Page 25: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/25.jpg)
*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!)
![Page 26: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/26.jpg)
*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!)
![Page 27: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/27.jpg)
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
![Page 28: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/28.jpg)
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
![Page 29: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/29.jpg)
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
![Page 30: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/30.jpg)
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
![Page 31: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/31.jpg)
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
![Page 32: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/32.jpg)
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
![Page 33: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/33.jpg)
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
![Page 34: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/34.jpg)
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
![Page 35: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/35.jpg)
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]
![Page 36: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/36.jpg)
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]
![Page 37: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/37.jpg)
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]
![Page 38: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/38.jpg)
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]
![Page 39: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/39.jpg)
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]
![Page 40: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/40.jpg)
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]
![Page 41: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/41.jpg)
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]
![Page 42: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/42.jpg)
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]
![Page 43: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/43.jpg)
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]
![Page 44: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/44.jpg)
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]
![Page 45: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/45.jpg)
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
==
![Page 46: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/46.jpg)
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
==
![Page 47: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/47.jpg)
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
==
![Page 48: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/48.jpg)
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
==
![Page 49: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/49.jpg)
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
==
![Page 50: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/50.jpg)
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
==
![Page 51: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/51.jpg)
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.
![Page 52: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/52.jpg)
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.
![Page 53: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/53.jpg)
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.
![Page 54: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/54.jpg)
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.
![Page 55: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/55.jpg)
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.
![Page 56: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/56.jpg)
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.
![Page 57: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/57.jpg)
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.
![Page 58: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/58.jpg)
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.
![Page 59: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/59.jpg)
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!
![Page 60: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/60.jpg)
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!
![Page 61: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/61.jpg)
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!
![Page 62: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/62.jpg)
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!
![Page 63: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/63.jpg)
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!
![Page 64: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/64.jpg)
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!
![Page 65: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/65.jpg)
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!
![Page 66: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/66.jpg)
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
![Page 67: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/67.jpg)
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
![Page 68: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/68.jpg)
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
![Page 69: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/69.jpg)
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.
![Page 70: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/70.jpg)
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.
![Page 71: States of Convex Sets - bram.westerbaan.name](https://reader030.vdocuments.site/reader030/viewer/2022020623/61f1f9a4f2f6cf6a2a767d68/html5/thumbnails/71.jpg)
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.