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Big Toy Models Workshop on Informatic Penomena 2009 – 1
Big Toy Models:
Representing Physical Systems As Chu Spaces
Samson Abramsky
Oxford University Computing Laboratory
Introduction
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models
Themes
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 3
Themes
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 3
• (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.
Exemplifies one of the main thrusts of our group in Oxford:
methods and concepts which have been developed in TheoreticalComputer Science are ripe for use in Physics.
Themes
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 3
• (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.
Exemplifies one of the main thrusts of our group in Oxford:
methods and concepts which have been developed in TheoreticalComputer Science are ripe for use in Physics.
• Models vs. Axioms. Examples: sheaves and toposes,domain-theoretic models of the λ-calculus.
Themes
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 3
• (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.
Exemplifies one of the main thrusts of our group in Oxford:
methods and concepts which have been developed in TheoreticalComputer Science are ripe for use in Physics.
• Models vs. Axioms. Examples: sheaves and toposes,domain-theoretic models of the λ-calculus.
• Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of
quantum states: A toy theory’.
Themes
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 3
• (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.
Exemplifies one of the main thrusts of our group in Oxford:
methods and concepts which have been developed in TheoreticalComputer Science are ripe for use in Physics.
• Models vs. Axioms. Examples: sheaves and toposes,domain-theoretic models of the λ-calculus.
• Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of
quantum states: A toy theory’.
• Big toy models.
Chu Spaces
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 4
Chu Spaces
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 4
We should understand Chu spaces as providing a very general (and, we
might reasonably say, rather simple) ‘logic of systems or structures’.
Indeed, they have been proposed by Barwise and Seligman as the
vehicle for a general logic of ‘distributed systems’ and information flow.
Chu Spaces
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 4
We should understand Chu spaces as providing a very general (and, we
might reasonably say, rather simple) ‘logic of systems or structures’.
Indeed, they have been proposed by Barwise and Seligman as the
vehicle for a general logic of ‘distributed systems’ and information flow.
This logic of Chu spaces was in no way biassed in its conception towardsthe description of quantum mechanics or any other kind of physical
system.
Chu Spaces
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 4
We should understand Chu spaces as providing a very general (and, we
might reasonably say, rather simple) ‘logic of systems or structures’.
Indeed, they have been proposed by Barwise and Seligman as the
vehicle for a general logic of ‘distributed systems’ and information flow.
This logic of Chu spaces was in no way biassed in its conception towardsthe description of quantum mechanics or any other kind of physical
system.
Just for this reason, it is interesting to see how much of
quantum-mechanical structure and concepts can be absorbed and
essentially determined by this more general systems logic.
Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 5
Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 5
• Chu spaces as a setting. We can find natural representations of
quantum (and other) systems as Chu spaces.
Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 5
• Chu spaces as a setting. We can find natural representations of
quantum (and other) systems as Chu spaces.
• The general ‘logic’ of Chu spaces and morphisms allow us to
‘rationally reconstruct’ many key quantum notions:
Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 5
• Chu spaces as a setting. We can find natural representations of
quantum (and other) systems as Chu spaces.
• The general ‘logic’ of Chu spaces and morphisms allow us to
‘rationally reconstruct’ many key quantum notions:
• States as rays of Hilbert spaces fall out as the biextensional
collapse of the Chu spaces.
Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 5
• Chu spaces as a setting. We can find natural representations of
quantum (and other) systems as Chu spaces.
• The general ‘logic’ of Chu spaces and morphisms allow us to
‘rationally reconstruct’ many key quantum notions:
• States as rays of Hilbert spaces fall out as the biextensional
collapse of the Chu spaces.
• Chu morphisms are automatically the unitaries and
antiunitaries — the physical symmetries of quantum systems.
Outline I
Introduction
• Themes
• Chu Spaces
• Outline I
• Outline II
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 5
• Chu spaces as a setting. We can find natural representations of
quantum (and other) systems as Chu spaces.
• The general ‘logic’ of Chu spaces and morphisms allow us to
‘rationally reconstruct’ many key quantum notions:
• States as rays of Hilbert spaces fall out as the biextensional
collapse of the Chu spaces.
• Chu morphisms are automatically the unitaries and
antiunitaries — the physical symmetries of quantum systems.
• This leads to a full and faithful representation of the
groupoid of Hilbert spaces and their physical symmetries in
Chu spaces over the unit interval.
Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6
Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6
• This leads to a further question of conceptual interest: is this representation
preserved by collapsing the unit interval to finitely many values?
Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6
• This leads to a further question of conceptual interest: is this representation
preserved by collapsing the unit interval to finitely many values?
• For the two canonical possibilistic collapses to two values, we showthat this fails .
Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6
• This leads to a further question of conceptual interest: is this representation
preserved by collapsing the unit interval to finitely many values?
• For the two canonical possibilistic collapses to two values, we showthat this fails .
• However, the natural collapse to three values works! — A possible rolefor 3-valued logic in quantum foundations?
Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6
• This leads to a further question of conceptual interest: is this representation
preserved by collapsing the unit interval to finitely many values?
• For the two canonical possibilistic collapses to two values, we showthat this fails .
• However, the natural collapse to three values works! — A possible rolefor 3-valued logic in quantum foundations?
• We also look at coalgebras as a possible alternative setting to Chu spaces.
Some interesting and novel points arise in comparing and relating these two
well-studied systems models.
Outline II
Big Toy Models Workshop on Informatic Penomena 2009 – 6
• This leads to a further question of conceptual interest: is this representation
preserved by collapsing the unit interval to finitely many values?
• For the two canonical possibilistic collapses to two values, we showthat this fails .
• However, the natural collapse to three values works! — A possible rolefor 3-valued logic in quantum foundations?
• We also look at coalgebras as a possible alternative setting to Chu spaces.
Some interesting and novel points arise in comparing and relating these two
well-studied systems models.
There is a paper available as an Oxford University Computing Laboratory Research
Report: RR–09–08 at
http://www.comlab.ox.ac.uk/techreports/cs/2009.html
Chu Spaces
Introduction
Chu Spaces
• Chu Spaces
• Definitions• Extensionality andSeparability
• BiextensionalCollapse
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTryBig Toy Models
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by
Po-Hsiang Chu. A generalization of constructions of dual pairings of topological
vector spaces from G. W. Mackey’s thesis.
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by
Po-Hsiang Chu. A generalization of constructions of dual pairings of topological
vector spaces from G. W. Mackey’s thesis.
Chu spaces have several interesting aspects:
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by
Po-Hsiang Chu. A generalization of constructions of dual pairings of topological
vector spaces from G. W. Mackey’s thesis.
Chu spaces have several interesting aspects:
• They have a rich type structure, and in particular form models of Linear Logic
(Seely).
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by
Po-Hsiang Chu. A generalization of constructions of dual pairings of topological
vector spaces from G. W. Mackey’s thesis.
Chu spaces have several interesting aspects:
• They have a rich type structure, and in particular form models of Linear Logic
(Seely).
• They have a rich representation theory; many concrete categories of interest
can be fully embedded into Chu spaces (Lafont and Streicher, Pratt).
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by
Po-Hsiang Chu. A generalization of constructions of dual pairings of topological
vector spaces from G. W. Mackey’s thesis.
Chu spaces have several interesting aspects:
• They have a rich type structure, and in particular form models of Linear Logic
(Seely).
• They have a rich representation theory; many concrete categories of interest
can be fully embedded into Chu spaces (Lafont and Streicher, Pratt).
• There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an
interesting characterization of information transfer across Chu morphisms
(van Benthem).
Chu Spaces
Big Toy Models Workshop on Informatic Penomena 2009 – 8
History: Michael Barr’s 1979 monograph on ∗-Autonomous categories, appendix by
Po-Hsiang Chu. A generalization of constructions of dual pairings of topological
vector spaces from G. W. Mackey’s thesis.
Chu spaces have several interesting aspects:
• They have a rich type structure, and in particular form models of Linear Logic
(Seely).
• They have a rich representation theory; many concrete categories of interest
can be fully embedded into Chu spaces (Lafont and Streicher, Pratt).
• There is a natural notion of ‘local logic’ on Chu spaces (Barwise), and an
interesting characterization of information transfer across Chu morphisms
(van Benthem).
Applications of Chu spaces have been proposed in a number of areas, including
concurrency, hardware verification, game theory and fuzzy systems.
Definitions
Big Toy Models Workshop on Informatic Penomena 2009 – 9
Definitions
Big Toy Models Workshop on Informatic Penomena 2009 – 9
Fix a set K. A Chu space over K is a structure (X,A, e), where X is a set of
‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X ×A→ K is an evaluation
function.
Definitions
Big Toy Models Workshop on Informatic Penomena 2009 – 9
Fix a set K. A Chu space over K is a structure (X,A, e), where X is a set of
‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X ×A→ K is an evaluation
function.
A morphism of Chu spaces f : (X,A, e) → (X ′, A′, e′) is a pair of functions
f = (f∗ : X → X ′, f∗ : A′ → A)
such that, for all x ∈ X and a′ ∈ A′:
e(x, f∗(a′)) = e′(f∗(x), a′).
Definitions
Big Toy Models Workshop on Informatic Penomena 2009 – 9
Fix a set K. A Chu space over K is a structure (X,A, e), where X is a set of
‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X ×A→ K is an evaluation
function.
A morphism of Chu spaces f : (X,A, e) → (X ′, A′, e′) is a pair of functions
f = (f∗ : X → X ′, f∗ : A′ → A)
such that, for all x ∈ X and a′ ∈ A′:
e(x, f∗(a′)) = e′(f∗(x), a′).
Chu morphisms compose componentwise: if f : (X1, A1, e1) → (X2, A2, e2) and
g : (X2, A2, e2) → (X3, A3, e3), then
(g ◦ f)∗ = g∗ ◦ f∗, (g ◦ f)∗ = f∗ ◦ g∗.
Definitions
Big Toy Models Workshop on Informatic Penomena 2009 – 9
Fix a set K. A Chu space over K is a structure (X,A, e), where X is a set of
‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X ×A→ K is an evaluation
function.
A morphism of Chu spaces f : (X,A, e) → (X ′, A′, e′) is a pair of functions
f = (f∗ : X → X ′, f∗ : A′ → A)
such that, for all x ∈ X and a′ ∈ A′:
e(x, f∗(a′)) = e′(f∗(x), a′).
Chu morphisms compose componentwise: if f : (X1, A1, e1) → (X2, A2, e2) and
g : (X2, A2, e2) → (X3, A3, e3), then
(g ◦ f)∗ = g∗ ◦ f∗, (g ◦ f)∗ = f∗ ◦ g∗.
Chu spaces over K and their morphisms form a category ChuK .
Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10
Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10
Given a Chu space C = (X,A, e), we say that C is:
Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10
Given a Chu space C = (X,A, e), we say that C is:
• extensional if for all a1, a2 ∈ A:
[∀x ∈ X. e(x, a1) = e(x, a2)] ⇒ a1 = a2
Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10
Given a Chu space C = (X,A, e), we say that C is:
• extensional if for all a1, a2 ∈ A:
[∀x ∈ X. e(x, a1) = e(x, a2)] ⇒ a1 = a2
• separable if for all x1, x2 ∈ X :
[∀a ∈ A. e(x1, a) = e(x2, a)] ⇒ x1 = x2
Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10
Given a Chu space C = (X,A, e), we say that C is:
• extensional if for all a1, a2 ∈ A:
[∀x ∈ X. e(x, a1) = e(x, a2)] ⇒ a1 = a2
• separable if for all x1, x2 ∈ X :
[∀a ∈ A. e(x1, a) = e(x2, a)] ⇒ x1 = x2
• biextensional if it is extensional and separable.
We define an equivalence relation on X by:
x1 ∼ x2 ⇐⇒ ∀a ∈ A. e(x1, a) = e(x2, a).
Extensionality and Separability
Big Toy Models Workshop on Informatic Penomena 2009 – 10
Given a Chu space C = (X,A, e), we say that C is:
• extensional if for all a1, a2 ∈ A:
[∀x ∈ X. e(x, a1) = e(x, a2)] ⇒ a1 = a2
• separable if for all x1, x2 ∈ X :
[∀a ∈ A. e(x1, a) = e(x2, a)] ⇒ x1 = x2
• biextensional if it is extensional and separable.
We define an equivalence relation on X by:
x1 ∼ x2 ⇐⇒ ∀a ∈ A. e(x1, a) = e(x2, a).
C is separable exactly when this relation is the identity. There is a Chu morphism
(q, idA) : (X,A, e) → (X/∼, A, e′)
where e′([x], a) = e(x, a) and q : X → X/∼ is the quotient map.
Biextensional Collapse
Introduction
Chu Spaces
• Chu Spaces
• Definitions• Extensionality andSeparability
• BiextensionalCollapse
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTryBig Toy Models Workshop on Informatic Penomena 2009 – 11
Proposition 1 If f : (X,A, e) → (X ′, A′, e′) is a Chu morphism, then
f∗ preserves ∼. That is, for all x1, x2 ∈ X ,
x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).
Biextensional Collapse
Introduction
Chu Spaces
• Chu Spaces
• Definitions• Extensionality andSeparability
• BiextensionalCollapse
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTryBig Toy Models Workshop on Informatic Penomena 2009 – 11
Proposition 1 If f : (X,A, e) → (X ′, A′, e′) is a Chu morphism, then
f∗ preserves ∼. That is, for all x1, x2 ∈ X ,
x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).
Proof For any a′ ∈ A′:
e′(f∗(x1), a′) = e(x1, f
∗(a′)) = e(x2, f∗(a′)) = e′(f∗(x2), a
′).
�
Biextensional Collapse
Introduction
Chu Spaces
• Chu Spaces
• Definitions• Extensionality andSeparability
• BiextensionalCollapse
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTryBig Toy Models Workshop on Informatic Penomena 2009 – 11
Proposition 1 If f : (X,A, e) → (X ′, A′, e′) is a Chu morphism, then
f∗ preserves ∼. That is, for all x1, x2 ∈ X ,
x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).
Proof For any a′ ∈ A′:
e′(f∗(x1), a′) = e(x1, f
∗(a′)) = e(x2, f∗(a′)) = e′(f∗(x2), a
′).
�
We shall write eChuK , sChuK and bChuK for the full subcategoriesof ChuK determined by the extensional, separated and biextensional
Chu spaces.
Biextensional Collapse
Introduction
Chu Spaces
• Chu Spaces
• Definitions• Extensionality andSeparability
• BiextensionalCollapse
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTryBig Toy Models Workshop on Informatic Penomena 2009 – 11
Proposition 1 If f : (X,A, e) → (X ′, A′, e′) is a Chu morphism, then
f∗ preserves ∼. That is, for all x1, x2 ∈ X ,
x1 ∼ x2 ⇒ f∗(x1) ∼ f∗(x2).
Proof For any a′ ∈ A′:
e′(f∗(x1), a′) = e(x1, f
∗(a′)) = e(x2, f∗(a′)) = e′(f∗(x2), a
′).
�
We shall write eChuK , sChuK and bChuK for the full subcategoriesof ChuK determined by the extensional, separated and biextensional
Chu spaces.
We shall mainly work with extensional and biextensional Chu spaces.
Obviously bChuK is a full sub-category of eChuK .
Proposition 2 The inclusion bChuK⊂ - eChuK has a left adjoint
Q, the biextensional collapse ..
Representing Physical Systems
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models
The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13
The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13
We take a system to be specified by its set of states S, and the set of questions Qwhich can be ‘asked’ of the system.
The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13
We take a system to be specified by its set of states S, and the set of questions Qwhich can be ‘asked’ of the system.
We shall consider only ‘yes/no’ questions; however, the result of asking a question in
a given state will in general be probabilistic . This will be represented by an
evaluation function
e : S ×Q→ [0, 1]
where e(s, q) is the probability that the question q will receive the answer ‘yes’ whenthe system is in state s.
The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13
We take a system to be specified by its set of states S, and the set of questions Qwhich can be ‘asked’ of the system.
We shall consider only ‘yes/no’ questions; however, the result of asking a question in
a given state will in general be probabilistic . This will be represented by an
evaluation function
e : S ×Q→ [0, 1]
where e(s, q) is the probability that the question q will receive the answer ‘yes’ whenthe system is in state s.
This is a Chu space!
The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13
We take a system to be specified by its set of states S, and the set of questions Qwhich can be ‘asked’ of the system.
We shall consider only ‘yes/no’ questions; however, the result of asking a question in
a given state will in general be probabilistic . This will be represented by an
evaluation function
e : S ×Q→ [0, 1]
where e(s, q) is the probability that the question q will receive the answer ‘yes’ whenthe system is in state s.
This is a Chu space!
N.B. This is essentially the point of view taken by Mackey in his classic
‘Mathematical foundations of Quantum Mechanics’. Note that we prefer ‘question’ to
‘property’, since QM we cannot think in terms of static properties which are
determinately possessed by a given state; questions imply a dynamic act of asking.
The General Paradigm
Big Toy Models Workshop on Informatic Penomena 2009 – 13
We take a system to be specified by its set of states S, and the set of questions Qwhich can be ‘asked’ of the system.
We shall consider only ‘yes/no’ questions; however, the result of asking a question in
a given state will in general be probabilistic . This will be represented by an
evaluation function
e : S ×Q→ [0, 1]
where e(s, q) is the probability that the question q will receive the answer ‘yes’ whenthe system is in state s.
This is a Chu space!
N.B. This is essentially the point of view taken by Mackey in his classic
‘Mathematical foundations of Quantum Mechanics’. Note that we prefer ‘question’ to
‘property’, since QM we cannot think in terms of static properties which are
determinately possessed by a given state; questions imply a dynamic act of asking.
It is standard in the foundational literature on QM to focus on yes/no questions.
However, the usual approaches to quantum logic avoid the direct introduction of
probabilities. More on this later!
Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 14
Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 14
A quantum system with a Hilbert space H as its state space will be
represented as
(H◦, L(H), eH)
where
Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 14
A quantum system with a Hilbert space H as its state space will be
represented as
(H◦, L(H), eH)
where
• H◦ is the set of non-zero vectors of H
Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 14
A quantum system with a Hilbert space H as its state space will be
represented as
(H◦, L(H), eH)
where
• H◦ is the set of non-zero vectors of H
• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions
of QM
Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 14
A quantum system with a Hilbert space H as its state space will be
represented as
(H◦, L(H), eH)
where
• H◦ is the set of non-zero vectors of H
• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions
of QM
• The evaluation function eH is the ‘statistical algorithm’ giving the
basic predictive content of Quantum Mechanics:
eH(ψ, S) =〈ψ | PSψ〉
〈ψ | ψ〉=
〈PSψ | PSψ〉
〈ψ | ψ〉=
‖PSψ‖2
‖ψ‖2.
Representing Quantum Systems As Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
• The GeneralParadigm
• RepresentingQuantum Systems AsChu Spaces
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountryBig Toy Models Workshop on Informatic Penomena 2009 – 14
A quantum system with a Hilbert space H as its state space will be
represented as
(H◦, L(H), eH)
where
• H◦ is the set of non-zero vectors of H
• L(H) is the set of closed subspaces of H — the ‘yes/no’ questions
of QM
• The evaluation function eH is the ‘statistical algorithm’ giving the
basic predictive content of Quantum Mechanics:
eH(ψ, S) =〈ψ | PSψ〉
〈ψ | ψ〉=
〈PSψ | PSψ〉
〈ψ | ψ〉=
‖PSψ‖2
‖ψ‖2.
We have thus directly transcribed the basic ingredients of the Dirac/von
Neumann-style formulation of Quantum Mechanics into the definition of
this Chu space.
Characterizing Chu Morphismson Quantum Chu Spaces
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models
Overview
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 16
Overview
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 16
We shall now see how the simple, discrete notions of Chu spaces suffice
to determine the appropriate notions of state equivalence, and to pick out
the physically significant symmetries on Hilbert space in a very striking
fashion. This leads to a full and faithful representation of the category of
quantum systems, with the groupoid structure of their physicalsymmetries, in the category of Chu spaces valued in the unit interval.
Overview
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 16
We shall now see how the simple, discrete notions of Chu spaces suffice
to determine the appropriate notions of state equivalence, and to pick out
the physically significant symmetries on Hilbert space in a very striking
fashion. This leads to a full and faithful representation of the category of
quantum systems, with the groupoid structure of their physicalsymmetries, in the category of Chu spaces valued in the unit interval.
The arguments here make use of Wigner’s theorem and the dualities of
projective geometry, in the modern form developed by Faure and
Frolicher, Modern Projective Geometry (2000).
Overview
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 16
We shall now see how the simple, discrete notions of Chu spaces suffice
to determine the appropriate notions of state equivalence, and to pick out
the physically significant symmetries on Hilbert space in a very striking
fashion. This leads to a full and faithful representation of the category of
quantum systems, with the groupoid structure of their physicalsymmetries, in the category of Chu spaces valued in the unit interval.
The arguments here make use of Wigner’s theorem and the dualities of
projective geometry, in the modern form developed by Faure and
Frolicher, Modern Projective Geometry (2000).
The surprising point is that unitarity/anitunitarity is essentially forced bythe mere requirement of being a Chu morphism. This even extends to
surjectivity, which here is derived rather than assumed.
Biextensionaity
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 17
Biextensionaity
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 17
Given a Hilbert space H, consider the Chu space (H◦, L(H), eH).
Biextensionaity
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 17
Given a Hilbert space H, consider the Chu space (H◦, L(H), eH).
A basic property of the evaluation.
Lemma 3 For ψ ∈ H◦ and S ∈ L(H):
ψ ∈ S ⇐⇒ eH(ψ, S) = 1.
Biextensionaity
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 17
Given a Hilbert space H, consider the Chu space (H◦, L(H), eH).
A basic property of the evaluation.
Lemma 3 For ψ ∈ H◦ and S ∈ L(H):
ψ ∈ S ⇐⇒ eH(ψ, S) = 1.
From this, we can prove:
Proposition 4 The Chu space (H◦, L(H), eH) is extensional but notseparable. The equivalence classes of the relation ∼ on states are
exactly the rays of H. That is:
φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.
Biextensionaity
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 17
Given a Hilbert space H, consider the Chu space (H◦, L(H), eH).
A basic property of the evaluation.
Lemma 3 For ψ ∈ H◦ and S ∈ L(H):
ψ ∈ S ⇐⇒ eH(ψ, S) = 1.
From this, we can prove:
Proposition 4 The Chu space (H◦, L(H), eH) is extensional but notseparable. The equivalence classes of the relation ∼ on states are
exactly the rays of H. That is:
φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.
Thus we have recovered the standard notion of pure states as the rays of
the Hilbert space from the general notion of state equivalence in Chu
spaces.
Projectivity = Biextensionality
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 18
Projectivity = Biextensionality
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 18
We shall now use some notions and results from projective geometry.
Projectivity = Biextensionality
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 18
We shall now use some notions and results from projective geometry.
Given a vector ψ ∈ H◦, we write ψ = {λψ | λ ∈ C} for the ray which it
generates. The rays are the atoms in the lattice L(H).
Projectivity = Biextensionality
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 18
We shall now use some notions and results from projective geometry.
Given a vector ψ ∈ H◦, we write ψ = {λψ | λ ∈ C} for the ray which it
generates. The rays are the atoms in the lattice L(H).
We write P(H) for the set of rays of H. By virtue of Proposition 4, we can
write the biextensional collapse of (H◦, L(H), eH) given by Proposition 2as
(P(H), L(H), eH)
where eH(ψ, S) = eH(ψ, S).
Projectivity = Biextensionality
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 18
We shall now use some notions and results from projective geometry.
Given a vector ψ ∈ H◦, we write ψ = {λψ | λ ∈ C} for the ray which it
generates. The rays are the atoms in the lattice L(H).
We write P(H) for the set of rays of H. By virtue of Proposition 4, we can
write the biextensional collapse of (H◦, L(H), eH) given by Proposition 2as
(P(H), L(H), eH)
where eH(ψ, S) = eH(ψ, S).
We restate Lemma 3 for the biextensional case.
Lemma 5 For ψ ∈ H◦ and S ∈ L(H):
eH(ψ, S) = 1 ⇐⇒ ψ ⊆ S.
Characterizing Chu Morphisms
Big Toy Models Workshop on Informatic Penomena 2009 – 19
Characterizing Chu Morphisms
Big Toy Models Workshop on Informatic Penomena 2009 – 19
To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism
(f∗, f∗) : (P(H), L(H), eH) → (P(K), L(K), eK).
Characterizing Chu Morphisms
Big Toy Models Workshop on Informatic Penomena 2009 – 19
To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism
(f∗, f∗) : (P(H), L(H), eH) → (P(K), L(K), eK).
Proposition 6 For ψ ∈ H◦ and S ∈ L(K):
ψ ⊆ f∗(S) ⇐⇒ f∗(ψ) ⊆ S.
Proof By Lemma 5:
ψ ⊆ f∗(S) ⇔ eH(ψ, f∗(S)) = 1 ⇔ eK(f∗(ψ), S) = 1 ⇔ f∗(ψ) ⊆ S.
�
Characterizing Chu Morphisms
Big Toy Models Workshop on Informatic Penomena 2009 – 19
To fix notation, suppose we have Hilbert spaces H and K, and a Chu morphism
(f∗, f∗) : (P(H), L(H), eH) → (P(K), L(K), eK).
Proposition 6 For ψ ∈ H◦ and S ∈ L(K):
ψ ⊆ f∗(S) ⇐⇒ f∗(ψ) ⊆ S.
Proof By Lemma 5:
ψ ⊆ f∗(S) ⇔ eH(ψ, f∗(S)) = 1 ⇔ eK(f∗(ψ), S) = 1 ⇔ f∗(ψ) ⊆ S.
�
Note that P(H) ⊆ L(H).
Injectivity Assumption
Big Toy Models Workshop on Informatic Penomena 2009 – 20
Injectivity Assumption
Big Toy Models Workshop on Informatic Penomena 2009 – 20
Proposition 7 If f∗ is injective, then the following diagram commutes:
P(H)f∗
- P(K)
L(H)?
∩
�
f∗L(K)
?
∩
(1)
That is, for all ψ ∈ H◦:ψ = f∗(f∗(ψ)).
Injectivity Assumption
Big Toy Models Workshop on Informatic Penomena 2009 – 20
Proposition 7 If f∗ is injective, then the following diagram commutes:
P(H)f∗
- P(K)
L(H)?
∩
�
f∗L(K)
?
∩
(1)
That is, for all ψ ∈ H◦:ψ = f∗(f∗(ψ)).
Proof Proposition 6 implies that ψ ⊆ f∗(f∗(ψ)). For the converse, suppose that
φ ⊆ f∗(f∗(ψ)). Applying Proposition 6 again, this implies that f∗(φ) ⊆ f∗(ψ).
Since f∗(φ) and f∗(ψ) are atoms, this implies that f∗(φ) = f∗(ψ), which since f∗is injective implies that φ = ψ. Thus the only atom below f∗(f∗(ψ)) is ψ. Since
L(H) is atomistic , this implies that f∗(f∗(ψ)) ⊆ ψ. �
Orthogonality is Preserved
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 21
Another basic property of the evaluation.
Lemma 8 For any φ, ψ ∈ H◦:
eH(φ, ψ) = 0 ⇐⇒ φ⊥ψ.
Orthogonality is Preserved
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 21
Another basic property of the evaluation.
Lemma 8 For any φ, ψ ∈ H◦:
eH(φ, ψ) = 0 ⇐⇒ φ⊥ψ.
Proposition 9 If f∗ is injective, it preserves and reflectsorthogonality . That is, for all φ, ψ ∈ H◦:
φ⊥ψ ⇐⇒ f∗(φ)⊥ f∗(ψ).
Orthogonality is Preserved
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 21
Another basic property of the evaluation.
Lemma 8 For any φ, ψ ∈ H◦:
eH(φ, ψ) = 0 ⇐⇒ φ⊥ψ.
Proposition 9 If f∗ is injective, it preserves and reflectsorthogonality . That is, for all φ, ψ ∈ H◦:
φ⊥ψ ⇐⇒ f∗(φ)⊥ f∗(ψ).
Proof
φ⊥ψ ⇐⇒ eH(φ, ψ) = 0 Lemma 8
⇐⇒ eH(φ, f∗(f∗(ψ))) = 0 Proposition 7
⇐⇒ eK(f∗(φ), f∗(ψ)) = 0
⇐⇒ f∗(φ)⊥ f∗(ψ) Lemma 8.
Constructing the Left Adjoint
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 22
We define a map f→ : L(H) → L(K):
f→(S) =∨
{f∗(ψ) | ψ ∈ S◦}
where S◦ = S \ {0}.
Constructing the Left Adjoint
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 22
We define a map f→ : L(H) → L(K):
f→(S) =∨
{f∗(ψ) | ψ ∈ S◦}
where S◦ = S \ {0}.
Lemma 10 The map f→ is left adjoint to f∗:
f→(S) ⊆ T ⇐⇒ S ⊆ f∗(T ).
Constructing the Left Adjoint
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 22
We define a map f→ : L(H) → L(K):
f→(S) =∨
{f∗(ψ) | ψ ∈ S◦}
where S◦ = S \ {0}.
Lemma 10 The map f→ is left adjoint to f∗:
f→(S) ⊆ T ⇐⇒ S ⊆ f∗(T ).
We can now extend the diagram (1):
P(H)f∗
- P(K)
L(H)?
∩
f→-
⊥�
f∗L(K)
?
∩
(2)
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
By construction, f→ extends f∗: this says that f→ preserves atoms. Since f→ is a
left adjoint, it preserves sups. Hence f→ and f∗ are paired under the duality ofprojective lattices and projective geometries .
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
By construction, f→ extends f∗: this says that f→ preserves atoms. Since f→ is a
left adjoint, it preserves sups. Hence f→ and f∗ are paired under the duality ofprojective lattices and projective geometries .
Proposition 11 f∗ is a total map of projective geometries .
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
By construction, f→ extends f∗: this says that f→ preserves atoms. Since f→ is a
left adjoint, it preserves sups. Hence f→ and f∗ are paired under the duality ofprojective lattices and projective geometries .
Proposition 11 f∗ is a total map of projective geometries .
We can now apply Wigner’s Theorem, in the modernized form given by Faure(2002).
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
By construction, f→ extends f∗: this says that f→ preserves atoms. Since f→ is a
left adjoint, it preserves sups. Hence f→ and f∗ are paired under the duality ofprojective lattices and projective geometries .
Proposition 11 f∗ is a total map of projective geometries .
We can now apply Wigner’s Theorem, in the modernized form given by Faure(2002).
Let V1 be a vector space over the field F and V2 a vector space over the field G. A
semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a field
homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and
v ∈ V1:f(λv) = α(λ)f(v).
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
By construction, f→ extends f∗: this says that f→ preserves atoms. Since f→ is a
left adjoint, it preserves sups. Hence f→ and f∗ are paired under the duality ofprojective lattices and projective geometries .
Proposition 11 f∗ is a total map of projective geometries .
We can now apply Wigner’s Theorem, in the modernized form given by Faure(2002).
Let V1 be a vector space over the field F and V2 a vector space over the field G. A
semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a field
homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and
v ∈ V1:f(λv) = α(λ)f(v).
Note that semilinear maps compose: if (f, α) : V1 → V2 and (g, β) : V2 → V3,
then (g ◦ f, β ◦ α) : V1 → V2 is a semilinear map.
Using Projective Duality
Big Toy Models Workshop on Informatic Penomena 2009 – 23
By construction, f→ extends f∗: this says that f→ preserves atoms. Since f→ is a
left adjoint, it preserves sups. Hence f→ and f∗ are paired under the duality ofprojective lattices and projective geometries .
Proposition 11 f∗ is a total map of projective geometries .
We can now apply Wigner’s Theorem, in the modernized form given by Faure(2002).
Let V1 be a vector space over the field F and V2 a vector space over the field G. A
semilinear map from V1 to V2 is a pair (f, α) where α : F → G is a field
homomorphism, and f : V1 → V2 is an additive map such that, for all λ ∈ F and
v ∈ V1:f(λv) = α(λ)f(v).
Note that semilinear maps compose: if (f, α) : V1 → V2 and (g, β) : V2 → V3,
then (g ◦ f, β ◦ α) : V1 → V2 is a semilinear map.
N.B. There are lots of (horrible) automorphisms, and non-surjective
endomorphisms, of the complex field!
Wigner’s Theorem
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 24
Wigner’s Theorem
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 24
Given a semilinear map g : V1 → V2, we define Pg : PV1 → PV2 by
P(g)(ψ) = g(ψ).
Wigner’s Theorem
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 24
Given a semilinear map g : V1 → V2, we define Pg : PV1 → PV2 by
P(g)(ψ) = g(ψ).
We can now state Wigner’s Theorem in the form we shall use it.
Theorem 12 Let f : P(H) → P(K) be a total map of projective
geometries, where dimH > 2. If f preserves orthogonality, meaning
that
φ⊥ ψ ⇒ f(φ)⊥ f(ψ)
then there is a semilinear map g : H → K such that P(g) = f , and
〈g(φ) | g(ψ)〉 = σ(〈φ | ψ〉),
where σ is the homomorphism associated with g. Moreover, thishomomorphism is either the identity or complex conjugation, so g is either
linear or antilinear . The map g is unique up to a phase , i.e. a scalar of
modulus 1.
Remarks
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 25
Remarks
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 25
• Note that in our case, taking f∗ = f , Pg is just the action of the
biextensional collapse functor on Chu morphisms.
Remarks
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 25
• Note that in our case, taking f∗ = f , Pg is just the action of the
biextensional collapse functor on Chu morphisms.
• Note that a total map of projective geometries must necessarily
come from an injective map g on the underlying vector spaces,
since P(g) maps rays to rays, and hence g must have trivial kernel.
Remarks
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 25
• Note that in our case, taking f∗ = f , Pg is just the action of the
biextensional collapse functor on Chu morphisms.
• Note that a total map of projective geometries must necessarily
come from an injective map g on the underlying vector spaces,
since P(g) maps rays to rays, and hence g must have trivial kernel.
• For this reason, partial maps of projective geometries are
considered in the Faure-Frolicher approach. However, we are
simply following the ‘logic’ of Chu space morphisms here.
A Surprise: Surjectivity Comes for Free!
Big Toy Models Workshop on Informatic Penomena 2009 – 26
A Surprise: Surjectivity Comes for Free!
Big Toy Models Workshop on Informatic Penomena 2009 – 26
Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗where f is a Chu space morphism, and dim(H) > 0. Suppose that the
endomorphism σ : C → C associated with g is surjective, and hence anautomorphism. Then g is surjective.
A Surprise: Surjectivity Comes for Free!
Big Toy Models Workshop on Informatic Penomena 2009 – 26
Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗where f is a Chu space morphism, and dim(H) > 0. Suppose that the
endomorphism σ : C → C associated with g is surjective, and hence anautomorphism. Then g is surjective.
Proof We write Im g for the set-theoretic direct image of g, which is a linear
subspace of K, since σ is an automorphism. In particular, g carries rays to rays,
since λg(φ) = g(σ−1(λ)φ).
A Surprise: Surjectivity Comes for Free!
Big Toy Models Workshop on Informatic Penomena 2009 – 26
Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗where f is a Chu space morphism, and dim(H) > 0. Suppose that the
endomorphism σ : C → C associated with g is surjective, and hence anautomorphism. Then g is surjective.
Proof We write Im g for the set-theoretic direct image of g, which is a linear
subspace of K, since σ is an automorphism. In particular, g carries rays to rays,
since λg(φ) = g(σ−1(λ)φ).
We claim that for any vector ψ ∈ K◦ which is not in the image of g, ψ⊥ Im g.Given such a ψ, for any φ ∈ H◦ it is not the case that f∗(φ) ⊆ ψ; for otherwise, for
some λ, g(φ) = λψ, and hence g(σ−1(λ−1)φ) = ψ. Then by Proposition 6,
f∗(ψ) = {0}. It follows that for all φ ∈ H◦,
eK(f∗(φ), ψ) = eH(φ, {0}) = 0,
and hence by Lemma 8 that ψ⊥ Im g.
A Surprise: Surjectivity Comes for Free!
Big Toy Models Workshop on Informatic Penomena 2009 – 26
Proposition 13 Let g : H → K be a semilinear morphism such that P(g) = f∗where f is a Chu space morphism, and dim(H) > 0. Suppose that the
endomorphism σ : C → C associated with g is surjective, and hence anautomorphism. Then g is surjective.
Proof We write Im g for the set-theoretic direct image of g, which is a linear
subspace of K, since σ is an automorphism. In particular, g carries rays to rays,
since λg(φ) = g(σ−1(λ)φ).
We claim that for any vector ψ ∈ K◦ which is not in the image of g, ψ⊥ Im g.Given such a ψ, for any φ ∈ H◦ it is not the case that f∗(φ) ⊆ ψ; for otherwise, for
some λ, g(φ) = λψ, and hence g(σ−1(λ−1)φ) = ψ. Then by Proposition 6,
f∗(ψ) = {0}. It follows that for all φ ∈ H◦,
eK(f∗(φ), ψ) = eH(φ, {0}) = 0,
and hence by Lemma 8 that ψ⊥ Im g.
Now suppose for a contradiction that such a ψ exists. Consider the vector ψ + χwhere χ is a non-zero vector in Im g, which must exist since g is injective and Hhas positive dimension. This vector is not in Im g, nor is it orthogonal to Im g, sincee.g. 〈ψ + χ | χ〉 = 〈χ | χ〉 6= 0. This yields the required contradiction. �
Putting The Pieces Together
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 27
Putting The Pieces Together
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 27
We say that a map U : H → K is semiunitary if it is either unitary or
antiunitary; that is, if it is a bijective map satisfying
U(φ+ψ) = Uφ+Uψ, U(λφ) = σ(λ)Uφ, 〈Uφ | Uψ〉 = σ(〈φ | ψ〉)
where σ is the identity if U is unitary, and complex conjugation if U is
antiunitary. Note that semiunitaries preserve norm, so if U and V are
semiunitaries and U = λV , then |λ| = 1.
Putting The Pieces Together
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
• Overview
• Biextensionaity
• Projectivity =Biextensionality
• Characterizing ChuMorphisms• InjectivityAssumption
• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality
• Wigner’s Theorem
• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogether
The RepresentationTheorem
Reducing The ValueSet
Big Toy Models Workshop on Informatic Penomena 2009 – 27
We say that a map U : H → K is semiunitary if it is either unitary or
antiunitary; that is, if it is a bijective map satisfying
U(φ+ψ) = Uφ+Uψ, U(λφ) = σ(λ)Uφ, 〈Uφ | Uψ〉 = σ(〈φ | ψ〉)
where σ is the identity if U is unitary, and complex conjugation if U is
antiunitary. Note that semiunitaries preserve norm, so if U and V are
semiunitaries and U = λV , then |λ| = 1.
Theorem 14 Let H, K be Hilbert spaces of dimension greater than 2.
Consider a Chu morphism
(f∗, f∗) : (P(H), L(H), eH) → (P(K), L(K), eK).
where f∗ is injective. Then there is a semiunitary U : H → K such that
f∗ = P(U). U is unique up to a phase.
The Representation Theorem
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
We define a category SymmH as follows:
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
We define a category SymmH as follows:
• The objects are Hilbert spaces of dimension > 2.
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
We define a category SymmH as follows:
• The objects are Hilbert spaces of dimension > 2.
• Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps.
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
We define a category SymmH as follows:
• The objects are Hilbert spaces of dimension > 2.
• Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps.
• Semiunitaries compose as explained more generally for semilinear maps in
the previous subsection. Since complex conjugation is an involution,
semiunitaries compose according to the rule of signs: two antiunitaries or twounitaries compose to form a unitary, while a unitary and an antiunitary
compose to form an antiunitary.
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
We define a category SymmH as follows:
• The objects are Hilbert spaces of dimension > 2.
• Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps.
• Semiunitaries compose as explained more generally for semilinear maps in
the previous subsection. Since complex conjugation is an involution,
semiunitaries compose according to the rule of signs: two antiunitaries or twounitaries compose to form a unitary, while a unitary and an antiunitary
compose to form an antiunitary.
This category is a groupoid, i.e. every arrow is an isomorphism.
The Big Picture
Big Toy Models Workshop on Informatic Penomena 2009 – 29
We define a category SymmH as follows:
• The objects are Hilbert spaces of dimension > 2.
• Morphisms U : H → K are semiunitary (i.e. unitary or antiunitary) maps.
• Semiunitaries compose as explained more generally for semilinear maps in
the previous subsection. Since complex conjugation is an involution,
semiunitaries compose according to the rule of signs: two antiunitaries or twounitaries compose to form a unitary, while a unitary and an antiunitary
compose to form an antiunitary.
This category is a groupoid, i.e. every arrow is an isomorphism.
The seminunitaries are the physically significant symmetries of Hilbert spacefrom the point of view of Quantum Mechanics. The usual dynamics according to the
Schrodinger equation is given by a continuous one-parameter group {U(t)} of
these symmetries; the requirement of continuity forces the U(t) to be unitaries.
However, some important physical symmetries are represented by antiunitaries, e.g.
time reversal and charge conjugation .
Remarks
Big Toy Models Workshop on Informatic Penomena 2009 – 30
Remarks
Big Toy Models Workshop on Informatic Penomena 2009 – 30
• By the results of the previous subsection, Chu morphisms essentially force us
to consider the symmetries on Hilbert space. As pointed out there, linear
maps which can be represented as Chu morphisms in the biextensional
category must be injective; and if L : H → K is an injective linear or
antilinear map, then P(L) is injective.
Remarks
Big Toy Models Workshop on Informatic Penomena 2009 – 30
• By the results of the previous subsection, Chu morphisms essentially force us
to consider the symmetries on Hilbert space. As pointed out there, linear
maps which can be represented as Chu morphisms in the biextensional
category must be injective; and if L : H → K is an injective linear or
antilinear map, then P(L) is injective.
• Our results then show that if L can be represented as a Chu morphism, it
must in fact be semiunitary.
Remarks
Big Toy Models Workshop on Informatic Penomena 2009 – 30
• By the results of the previous subsection, Chu morphisms essentially force us
to consider the symmetries on Hilbert space. As pointed out there, linear
maps which can be represented as Chu morphisms in the biextensional
category must be injective; and if L : H → K is an injective linear or
antilinear map, then P(L) is injective.
• Our results then show that if L can be represented as a Chu morphism, it
must in fact be semiunitary.
• This delineation of the physically significant symmetries by the logic of Chu
morphisms should be seen as a strong point in favour of this representation by
Chu spaces.
Functors
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 31
We define a functor R : SymmH → eChu[0,1]:
R : U : H → K 7−→ (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK)
where U◦ is the restriction of U to H◦.
As noted in Proposition 2, the inclusion bChu[0,1]⊂ - eChu[0,1] has
a left adjoint, which we name Q. By Proposition 4, this can be defined onthe image of R as follows:
Q : (H◦, L(H), eH) 7→ (PH, L(H), eH)
and for (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK),
Q : (U◦, U−1) 7−→ (PU,U−1).
Not Quite Right Yet
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 32
Not Quite Right Yet
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 32
We write emChu, bmChu for the subcategories of eChu[0,1] and
bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is
injective. The functors R and Q factor through these subcategories.
Not Quite Right Yet
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 32
We write emChu, bmChu for the subcategories of eChu[0,1] and
bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is
injective. The functors R and Q factor through these subcategories.
Proposition 15 Both
R : SymmH → emChu
and
Q : emChu → bmChu
are well-defined functors. R is faithful but not full; Q is full but not faithful.
Not Quite Right Yet
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 32
We write emChu, bmChu for the subcategories of eChu[0,1] and
bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is
injective. The functors R and Q factor through these subcategories.
Proposition 15 Both
R : SymmH → emChu
and
Q : emChu → bmChu
are well-defined functors. R is faithful but not full; Q is full but not faithful.
This involves verifying that unitaries and antiunitaries U : H → K doindeed yield Chu morphisms!
Not Quite Right Yet
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 32
We write emChu, bmChu for the subcategories of eChu[0,1] and
bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is
injective. The functors R and Q factor through these subcategories.
Proposition 15 Both
R : SymmH → emChu
and
Q : emChu → bmChu
are well-defined functors. R is faithful but not full; Q is full but not faithful.
This involves verifying that unitaries and antiunitaries U : H → K doindeed yield Chu morphisms!
The key property, for ψ ∈ H◦ and S ∈ L(H), is:
PS(Uψ) = U(PU−1(S)ψ).
Biextensionality and Scalar Factors
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 33
Biextensionality and Scalar Factors
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 33
We can analyze the non-fullness of R more precisely as follows.
Proposition 16 Let (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK)
be a Chu morphism in the image of R. Given an arbitrary function
f : H → C \ {0}, define fU : H◦ → K◦ by:
fU(ψ) = f(ψ)U(ψ).
Then (fU, U−1) ∼ (U◦, U−1). Moreover, the ∼-equivalence class of U
is exactly the set of functions of this form.
Biextensionality and Scalar Factors
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
• The Big Picture
• Remarks
• Functors
• Not Quite Right Yet
• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid
• Jes’ Right
• PR is anembedding up to aphase
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic ConceptsBig Toy Models Workshop on Informatic Penomena 2009 – 33
We can analyze the non-fullness of R more precisely as follows.
Proposition 16 Let (U◦, U−1) : (H◦, L(H), eH) → (K◦, L(K), eK)
be a Chu morphism in the image of R. Given an arbitrary function
f : H → C \ {0}, define fU : H◦ → K◦ by:
fU(ψ) = f(ψ)U(ψ).
Then (fU, U−1) ∼ (U◦, U−1). Moreover, the ∼-equivalence class of U
is exactly the set of functions of this form.
Thus before biextensional collapse, Chu morphisms can introduce
arbitrary scalar factors pointwise. Once we move to the biextensional
category, we know by Theorem 14 that our representation will be full, and
essentially faithful — up to a global phase. This points to the need for aprojective version of the symmetry groupoid.
Projectivising The Symmetry Groupoid
Big Toy Models Workshop on Informatic Penomena 2009 – 34
Projectivising The Symmetry Groupoid
Big Toy Models Workshop on Informatic Penomena 2009 – 34
The mathematical object underlying phases is the circle group U(1):
U(1) = {λ ∈ C | |λ| = 1} = {eiθ | θ ∈ R}
Since λ has modulus 1 if and only if λλ = 1, U(1) is the unitary group on the
one-dimensional Hilbert space.
Projectivising The Symmetry Groupoid
Big Toy Models Workshop on Informatic Penomena 2009 – 34
The mathematical object underlying phases is the circle group U(1):
U(1) = {λ ∈ C | |λ| = 1} = {eiθ | θ ∈ R}
Since λ has modulus 1 if and only if λλ = 1, U(1) is the unitary group on the
one-dimensional Hilbert space.The circle group acts on the symmetry groupoid SymmH by scalar multiplication.
For Hilbert spaces H, K we can define
U(1) × SymmH(H,K) → SymmH(H,K) :: (λ, U) 7→ λU.
Moreover, this is a category action, meaning that
(λU) ◦ V = U ◦ (λV ) = λ(U ◦ V ).
Projectivising The Symmetry Groupoid
Big Toy Models Workshop on Informatic Penomena 2009 – 34
The mathematical object underlying phases is the circle group U(1):
U(1) = {λ ∈ C | |λ| = 1} = {eiθ | θ ∈ R}
Since λ has modulus 1 if and only if λλ = 1, U(1) is the unitary group on the
one-dimensional Hilbert space.The circle group acts on the symmetry groupoid SymmH by scalar multiplication.
For Hilbert spaces H, K we can define
U(1) × SymmH(H,K) → SymmH(H,K) :: (λ, U) 7→ λU.
Moreover, this is a category action, meaning that
(λU) ◦ V = U ◦ (λV ) = λ(U ◦ V ).
It follows that we can form a quotient category (in fact again a groupoid) with the
same objects as SymmH, and in which the morphisms will be the orbits of this
group action:
U ∼ V ⇔ ∃λ ∈ U(1). U = λV.
Jes’ Right
Big Toy Models Workshop on Informatic Penomena 2009 – 35
Jes’ Right
Big Toy Models Workshop on Informatic Penomena 2009 – 35
We call the resulting category PSymmH, the projective quantum symmetrygroupoid . It is a natural generalization of the standard notion of the projectiveunitary group on Hilbert space.
Jes’ Right
Big Toy Models Workshop on Informatic Penomena 2009 – 35
We call the resulting category PSymmH, the projective quantum symmetrygroupoid . It is a natural generalization of the standard notion of the projectiveunitary group on Hilbert space.
There is a quotient functor P : SymmH → PSymmH, and by virtue of
Theorem 14, we can factor Q ◦R through this quotient to obtain a functor
PR : PSymmH → bmChu.
Jes’ Right
Big Toy Models Workshop on Informatic Penomena 2009 – 35
We call the resulting category PSymmH, the projective quantum symmetrygroupoid . It is a natural generalization of the standard notion of the projectiveunitary group on Hilbert space.
There is a quotient functor P : SymmH → PSymmH, and by virtue of
Theorem 14, we can factor Q ◦R through this quotient to obtain a functor
PR : PSymmH → bmChu.
The situation can be summarized by the following diagram:
SymmH >R
> emChu
PSymmH
P∨∨
>PR
>> bmChu
Q∨∨
Jes’ Right
Big Toy Models Workshop on Informatic Penomena 2009 – 35
We call the resulting category PSymmH, the projective quantum symmetrygroupoid . It is a natural generalization of the standard notion of the projectiveunitary group on Hilbert space.
There is a quotient functor P : SymmH → PSymmH, and by virtue of
Theorem 14, we can factor Q ◦R through this quotient to obtain a functor
PR : PSymmH → bmChu.
The situation can be summarized by the following diagram:
SymmH >R
> emChu
PSymmH
P∨∨
>PR
>> bmChu
Q∨∨
Theorem 17 The functor PR : PSymmH → bmChu is full and faithful.
PR is an embedding up to a phase
Big Toy Models Workshop on Informatic Penomena 2009 – 36
PR is an embedding up to a phase
Big Toy Models Workshop on Informatic Penomena 2009 – 36
• To see that PR is essentially injective on objects, we can use therepresentation theorems of Piron and Soler, which tell us that we can
reconstruct H as a Hilbert space from L(H).
PR is an embedding up to a phase
Big Toy Models Workshop on Informatic Penomena 2009 – 36
• To see that PR is essentially injective on objects, we can use therepresentation theorems of Piron and Soler, which tell us that we can
reconstruct H as a Hilbert space from L(H).
• This reconstruction will give us a Hilbert space H′ such that L(H) ∼= L(H′),
and P(H) ∼= P(H′).
PR is an embedding up to a phase
Big Toy Models Workshop on Informatic Penomena 2009 – 36
• To see that PR is essentially injective on objects, we can use therepresentation theorems of Piron and Soler, which tell us that we can
reconstruct H as a Hilbert space from L(H).
• This reconstruction will give us a Hilbert space H′ such that L(H) ∼= L(H′),
and P(H) ∼= P(H′).
• We can apply Wigner’s theorem to this isomorphism to obtain a semiunitary
U : H ∼= H′ from which we can recover the Hilbert space structure on H.
PR is an embedding up to a phase
Big Toy Models Workshop on Informatic Penomena 2009 – 36
• To see that PR is essentially injective on objects, we can use therepresentation theorems of Piron and Soler, which tell us that we can
reconstruct H as a Hilbert space from L(H).
• This reconstruction will give us a Hilbert space H′ such that L(H) ∼= L(H′),
and P(H) ∼= P(H′).
• We can apply Wigner’s theorem to this isomorphism to obtain a semiunitary
U : H ∼= H′ from which we can recover the Hilbert space structure on H.
• This means that we have recovered H uniquely to within the coset of idH inPSymmH.
Reducing The Value Set
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models
Generalities
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 38
Generalities
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 38
We now return to the issue of whether it is necessary to use the full unit
interval as the value set for our Chu spaces.
Generalities
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 38
We now return to the issue of whether it is necessary to use the full unit
interval as the value set for our Chu spaces.
We begin with some generalities. Given a function v : K → L, we define
a functor Fv : ChuK → ChuL:
Fv : (X,A, e) 7→ (X,A, v ◦ e)
and Fvf = f for Chu morphisms f .
Generalities
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 38
We now return to the issue of whether it is necessary to use the full unit
interval as the value set for our Chu spaces.
We begin with some generalities. Given a function v : K → L, we define
a functor Fv : ChuK → ChuL:
Fv : (X,A, e) 7→ (X,A, v ◦ e)
and Fvf = f for Chu morphisms f .
Proposition 18 Fv is a faithful functor. If v is injective, it is full.
The Question
Big Toy Models Workshop on Informatic Penomena 2009 – 39
The Question
Big Toy Models Workshop on Informatic Penomena 2009 – 39
We can now state the question we wish to pose more precisely:
Is there a mapping v : [0, 1] → K from the unit interval to some
finite set K such that the restriction of the functor Fv to the image ofPR is full, and thus the composition
Fv ◦ PR : PSymmH → bmChuK
is a representation?
The Question
Big Toy Models Workshop on Informatic Penomena 2009 – 39
We can now state the question we wish to pose more precisely:
Is there a mapping v : [0, 1] → K from the unit interval to some
finite set K such that the restriction of the functor Fv to the image ofPR is full, and thus the composition
Fv ◦ PR : PSymmH → bmChuK
is a representation?
There is no general reason to suppose that this is possible; in fact, we shall showthat it is, although not quite in the obvious fashion.
Two Values?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 40
Two Values?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 40
We shall write n = {0, . . . , n− 1} for the finite ordinals. The most
popular choice of value set for Chu spaces, by far, has been 2, and
indeed many interesting categories can be strictly (and even concretely)
represented in Chu2.
Two Values?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 40
We shall write n = {0, . . . , n− 1} for the finite ordinals. The most
popular choice of value set for Chu spaces, by far, has been 2, and
indeed many interesting categories can be strictly (and even concretely)
represented in Chu2.
This makes the following question natural:
Question 19 Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full
and faithful?
Two Values?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 40
We shall write n = {0, . . . , n− 1} for the finite ordinals. The most
popular choice of value set for Chu spaces, by far, has been 2, and
indeed many interesting categories can be strictly (and even concretely)
represented in Chu2.
This makes the following question natural:
Question 19 Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full
and faithful?
What we can show is that the most plausible candidates for suchfunctions, yielding the two canonical forms of possibilistic semantics as
a coarse-graining of probabilistic semantics, both in fact fail .
The Canonical Possibilistic Reductions
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 41
The Canonical Possibilistic Reductions
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 41
Note that any function v : [0, 1] → {0, 1} must lose information either on
0 or on 1 — or both.
The Canonical Possibilistic Reductions
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 41
Note that any function v : [0, 1] → {0, 1} must lose information either on
0 or on 1 — or both.
In this sense, the two ‘sharpest’ mappings will be:
v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1.
The Canonical Possibilistic Reductions
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 41
Note that any function v : [0, 1] → {0, 1} must lose information either on
0 or on 1 — or both.
In this sense, the two ‘sharpest’ mappings will be:
v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1.
These are the two canonical reductions of probabilistic to possibilisticinformation: the first maps ‘definitely not’ to ‘no’, and anything else to
‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitelyyes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’.
The Canonical Possibilistic Reductions
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
• Generalities
• The Question
• Two Values?• The CanonicalPossibilistic Reductions
• Two is Too Few
• Other Case
• Analysis
• Three ValuesSuffice!
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsBig Toy Models Workshop on Informatic Penomena 2009 – 41
Note that any function v : [0, 1] → {0, 1} must lose information either on
0 or on 1 — or both.
In this sense, the two ‘sharpest’ mappings will be:
v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1.
These are the two canonical reductions of probabilistic to possibilisticinformation: the first maps ‘definitely not’ to ‘no’, and anything else to
‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitelyyes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’.
Note that, under the first of these, we no longer have
eH(ψ, S) = 1 ⇐⇒ ψ ∈ S
while under the second, we no longer have
eH(ψ, S) = 0 ⇐⇒ ψ⊥S.
Two is Too Few
Big Toy Models Workshop on Informatic Penomena 2009 – 42
Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full.
Two is Too Few
Big Toy Models Workshop on Informatic Penomena 2009 – 42
Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full.
Let H be a Hilbert space with 2 < dimH <∞, and let (g, σ) be any semilinear
automorphism of H, where σ can be any automorphism of the complex field. (We
can extend the argument to infinite-dimensional Hilbert space by requiring g to becontinuous.) For each of the above two mappings of the unit interval to 2, we shall
construct a Chu2 endomorphism f : Fv ◦ PR(H) → Fv ◦ PR(H) with
f∗ = P(g). This will show the non-fullness of Fv .
Two is Too Few
Big Toy Models Workshop on Informatic Penomena 2009 – 42
Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full.
Let H be a Hilbert space with 2 < dimH <∞, and let (g, σ) be any semilinear
automorphism of H, where σ can be any automorphism of the complex field. (We
can extend the argument to infinite-dimensional Hilbert space by requiring g to becontinuous.) For each of the above two mappings of the unit interval to 2, we shall
construct a Chu2 endomorphism f : Fv ◦ PR(H) → Fv ◦ PR(H) with
f∗ = P(g). This will show the non-fullness of Fv .
Case 1 Here we consider the mapping v1 which sends [0, 1) to 0 and fixes 1. In this
case:eH(ψ, S) = 1 ⇐⇒ ψ ∈ S
and hence the Chu morphism condition on (f∗, f∗), where f∗ = P(g), is:
ψ ∈ f∗(S) ⇐⇒ g(ψ) ∈ S.
Taking f∗ = g−1 obviously fulfills this condition. Note that, since g is a semilinear
automorphism, and H is finite-dimensional, g−1 : L(H) → L(H) is well-defined.
Other Case
Big Toy Models Workshop on Informatic Penomena 2009 – 43
Case 2 Now consider the mapping v0 keeping 0 fixed and sending (0, 1] to 1. In
this case:
eH(ψ, S) = 0 ⇐⇒ ψ⊥S
and hence the Chu morphism condition on (f∗, f∗), where f∗ = P(g), is:
ψ⊥ f∗(S) ⇐⇒ g(ψ)⊥S.
Other Case
Big Toy Models Workshop on Informatic Penomena 2009 – 43
Case 2 Now consider the mapping v0 keeping 0 fixed and sending (0, 1] to 1. In
this case:
eH(ψ, S) = 0 ⇐⇒ ψ⊥S
and hence the Chu morphism condition on (f∗, f∗), where f∗ = P(g), is:
ψ⊥ f∗(S) ⇐⇒ g(ψ)⊥S.
We define f∗(S) = g−1(S⊥)⊥. Note that f∗ : L(H) → L(H) is well defined, and
also that g−1(S⊥) is a subspace of H; hence g−1(S⊥)⊥⊥ = g−1(S⊥).
Other Case
Big Toy Models Workshop on Informatic Penomena 2009 – 43
Case 2 Now consider the mapping v0 keeping 0 fixed and sending (0, 1] to 1. In
this case:
eH(ψ, S) = 0 ⇐⇒ ψ⊥S
and hence the Chu morphism condition on (f∗, f∗), where f∗ = P(g), is:
ψ⊥ f∗(S) ⇐⇒ g(ψ)⊥S.
We define f∗(S) = g−1(S⊥)⊥. Note that f∗ : L(H) → L(H) is well defined, and
also that g−1(S⊥) is a subspace of H; hence g−1(S⊥)⊥⊥ = g−1(S⊥).
ψ⊥ f∗S ⇐⇒ ψ ∈ g−1(S⊥)⊥⊥ = g−1(S⊥)
⇐⇒ g(ψ) ∈ S⊥
⇐⇒ g(ψ)⊥S.
and hence (f∗, f∗) is a Chu morphism as required.
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:
v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1
Thus we lose information only on the probabilities strictly between 0 and 1, which
are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:
v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1
Thus we lose information only on the probabilities strictly between 0 and 1, which
are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.
Why is this adequate? Given a vector ψ and a subspace S, we can write ψ uniquely
as θ + χ, where θ ∈ S and χ ∈ S⊥. For non-zero ψ, there are only three
possibilities:
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:
v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1
Thus we lose information only on the probabilities strictly between 0 and 1, which
are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.
Why is this adequate? Given a vector ψ and a subspace S, we can write ψ uniquely
as θ + χ, where θ ∈ S and χ ∈ S⊥. For non-zero ψ, there are only three
possibilities:
• θ = 0 and χ 6= 0, so eH(φ, S) = 0
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:
v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1
Thus we lose information only on the probabilities strictly between 0 and 1, which
are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.
Why is this adequate? Given a vector ψ and a subspace S, we can write ψ uniquely
as θ + χ, where θ ∈ S and χ ∈ S⊥. For non-zero ψ, there are only three
possibilities:
• θ = 0 and χ 6= 0, so eH(φ, S) = 0
• θ 6= 0 and χ = 0, so eH(φ, S) = 1
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:
v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1
Thus we lose information only on the probabilities strictly between 0 and 1, which
are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.
Why is this adequate? Given a vector ψ and a subspace S, we can write ψ uniquely
as θ + χ, where θ ∈ S and χ ∈ S⊥. For non-zero ψ, there are only three
possibilities:
• θ = 0 and χ 6= 0, so eH(φ, S) = 0
• θ 6= 0 and χ = 0, so eH(φ, S) = 1
• θ 6= 0 and χ 6= 0, so eH(ψ, S) ∈ (0, 1), and hence v ◦ eH(ψ, S) = 2.
Analysis
Big Toy Models Workshop on Informatic Penomena 2009 – 44
However, this negative result immediately suggests a remedy: to keep theinterpretations of both 0 and 1 sharp . We can do this with three values! Namely:
v : 0 7→ 0, (0, 1) 7→ 2, 1 7→ 1
Thus we lose information only on the probabilities strictly between 0 and 1, which
are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2.
Why is this adequate? Given a vector ψ and a subspace S, we can write ψ uniquely
as θ + χ, where θ ∈ S and χ ∈ S⊥. For non-zero ψ, there are only three
possibilities:
• θ = 0 and χ 6= 0, so eH(φ, S) = 0
• θ 6= 0 and χ = 0, so eH(φ, S) = 1
• θ 6= 0 and χ 6= 0, so eH(ψ, S) ∈ (0, 1), and hence v ◦ eH(ψ, S) = 2.
These are the only case discriminations which are used in pro ving theRepresentation Theorem .
Three Values Suffice!
Big Toy Models Workshop on Informatic Penomena 2009 – 45
Three Values Suffice!
Big Toy Models Workshop on Informatic Penomena 2009 – 45
Hence we have:
Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is arepresentation.
Three Values Suffice!
Big Toy Models Workshop on Informatic Penomena 2009 – 45
Hence we have:
Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is arepresentation.
We note that Chu3 has found some uses in concurrency and verification (Pratt03,
Ivanov08), under a temporal interpretation: the three values are read as ‘before’,
‘during’ and ‘after’, whereas in our setting the three values represent ‘definitely yes’,
‘definitely no’ and ‘maybe’.
Three Values Suffice!
Big Toy Models Workshop on Informatic Penomena 2009 – 45
Hence we have:
Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is arepresentation.
We note that Chu3 has found some uses in concurrency and verification (Pratt03,
Ivanov08), under a temporal interpretation: the three values are read as ‘before’,
‘during’ and ‘after’, whereas in our setting the three values represent ‘definitely yes’,
‘definitely no’ and ‘maybe’.
Theorem 21 may suggest some interesting uses for 3-valued ‘local logics’ in the
sense of Jon Barwise.
Discussion
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models
Where Next?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47
Where Next?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47
• Connections and contrasts between Chu spaces and coalgebras .
Where Next?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47
• Connections and contrasts between Chu spaces and coalgebras .
• Mixed states — handled generally at the level of Chu spaces.
Where Next?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47
• Connections and contrasts between Chu spaces and coalgebras .
• Mixed states — handled generally at the level of Chu spaces.
• Universal Chu spaces.
Where Next?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47
• Connections and contrasts between Chu spaces and coalgebras .
• Mixed states — handled generally at the level of Chu spaces.
• Universal Chu spaces.
• Linear and other type theories.
Where Next?
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
• Where Next?
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 47
• Connections and contrasts between Chu spaces and coalgebras .
• Mixed states — handled generally at the level of Chu spaces.
• Universal Chu spaces.
• Linear and other type theories.
• Local logics.
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49
• Coalgebras over Set; ‘universal coalgebra’.
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49
• Coalgebras over Set; ‘universal coalgebra’.
• Each of these general systems models has been studied
extensively.
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49
• Coalgebras over Set; ‘universal coalgebra’.
• Each of these general systems models has been studied
extensively.Their connections have not been studied at all.
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49
• Coalgebras over Set; ‘universal coalgebra’.
• Each of these general systems models has been studied
extensively.Their connections have not been studied at all.
• They have complementary merits and deficiencies.
• Chu spaces have , coalgebras lack : contravariance.
• Coalgebras have , Chu spaces lack : extension in time.
• Symmetry vs. rigidity.
Chu Spaces and Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
• Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsBig Toy Models Workshop on Informatic Penomena 2009 – 49
• Coalgebras over Set; ‘universal coalgebra’.
• Each of these general systems models has been studied
extensively.Their connections have not been studied at all.
• They have complementary merits and deficiencies.
• Chu spaces have , coalgebras lack : contravariance.
• Coalgebras have , Chu spaces lack : extension in time.
• Symmetry vs. rigidity.
• Interesting formal consequences:
• Indexed structure (‘externalising contravariance’)
• Grothendieck construction: new description of Chu spaces.
• Truncation functors.
Primer on coalgebra
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models
Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 51
Category theory allows us to dualize algebras to obtain a notion of
coalgebras of an endofunctor . However, while algebras abstract a
familiar set of notions, coalgebras open up a new and rather unexpected
territory, and provides an effective abstraction and mathematical theory
for a central class of computational phenomena:
Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 51
Category theory allows us to dualize algebras to obtain a notion of
coalgebras of an endofunctor . However, while algebras abstract a
familiar set of notions, coalgebras open up a new and rather unexpected
territory, and provides an effective abstraction and mathematical theory
for a central class of computational phenomena:
• Programming over infinite data structures : streams, lazy lists,
infinite trees . . .
Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 51
Category theory allows us to dualize algebras to obtain a notion of
coalgebras of an endofunctor . However, while algebras abstract a
familiar set of notions, coalgebras open up a new and rather unexpected
territory, and provides an effective abstraction and mathematical theory
for a central class of computational phenomena:
• Programming over infinite data structures : streams, lazy lists,
infinite trees . . .
• A novel notion of coinduction
Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 51
Category theory allows us to dualize algebras to obtain a notion of
coalgebras of an endofunctor . However, while algebras abstract a
familiar set of notions, coalgebras open up a new and rather unexpected
territory, and provides an effective abstraction and mathematical theory
for a central class of computational phenomena:
• Programming over infinite data structures : streams, lazy lists,
infinite trees . . .
• A novel notion of coinduction
• Modelling state-based computations of all kinds
Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 51
Category theory allows us to dualize algebras to obtain a notion of
coalgebras of an endofunctor . However, while algebras abstract a
familiar set of notions, coalgebras open up a new and rather unexpected
territory, and provides an effective abstraction and mathematical theory
for a central class of computational phenomena:
• Programming over infinite data structures : streams, lazy lists,
infinite trees . . .
• A novel notion of coinduction
• Modelling state-based computations of all kinds
• The key notion of bisimulation equivalence between processes.
Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
• Coalgebras
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexingBig Toy Models Workshop on Informatic Penomena 2009 – 51
Category theory allows us to dualize algebras to obtain a notion of
coalgebras of an endofunctor . However, while algebras abstract a
familiar set of notions, coalgebras open up a new and rather unexpected
territory, and provides an effective abstraction and mathematical theory
for a central class of computational phenomena:
• Programming over infinite data structures : streams, lazy lists,
infinite trees . . .
• A novel notion of coinduction
• Modelling state-based computations of all kinds
• The key notion of bisimulation equivalence between processes.
• A general coalgebraic logic can be read off from the functor, andused to specify and reason about properties of systems.
Basic Concepts
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models
F -Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 53
Let F : C −→ C be a functor.An F -coalgebra is a pair (A, γ : A −→ FA) for some object A of C.
We say that A is the carrier of the coalgebra, while γ is the behaviourmap .
An F -coalgebra homomorphism from (A, γ : A −→ FA) to
(B, δ : B −→ FB) is an arrow h : A −→ B such that
Aγ
- FA
B
h
?
δ- FB
Fh
?
F -coalgebras and their homomorphisms form a category F−Coalg.
Final F -coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 54
An F -coalgebra (C, γ) is final if for every F -coalgebra (A,α) there is a
unique homomorphism from (A,α) to (C, γ).
Proposition 22 If a final F -coalgebra exists, it is unique up to
isomorphism.
Proposition 23 (Lambek Lemma) If γ : A −→ FA is final, it is an
isomorphism
Labelled Transition Systems
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 55
Let A be a set of actions . A labelled transition system over A is a
coalgebra for the functor
LTA : Set −→ Set :: X 7→ Pf(A×X).
Such a coalgebra
γ : S −→ Pf(A× S)
can be understood operationally as follows:
• S is a set of states
• For each state s ∈ S, γ(s) specifies the possible transitions from
that state. We write sa
−→ s′ if (a, s′) ∈ γ(s). We think of such a
transition as consisting of the system performing the action a, and
changing state from s to s′. Note that we regard actions as directlyobservable , while states are not.
Transition Graphs as Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 56
Note that any labelled transition graph (or “state machine”) with labels in
A is a coalgebra for LTA.
Examples 1.
1 2b ca
This corresponds to the coalgebra ({1, 2}, γ)
γ : 1 7→ {(a, 1), (b, 2)}, 2 7→ {(c, 2)}
2.
1 2 3b
a
ac
1 7→ {(b, 2), (c, 1)}, 2 7→ {(a, 1), (a, 3)}, 3 7→ ∅
The Final Coalgebra
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 57
The key point is this.
Proposition 24 For any set A of actions, there is a final LTA-coalgebra
(ProcA, out).
The Final Coalgebra
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 57
The key point is this.
Proposition 24 For any set A of actions, there is a final LTA-coalgebra
(ProcA, out).
We think of elements of the final coalgebra as processes . The final
coalgebra provides a “universal semantics” for transition systems, which
is “fully abstract” with respect to observable behaviour.
The Final Coalgebra
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
• F -Coalgebras
• Final F -coalgebras
• Labelled TransitionSystems
• Transition Graphs asCoalgebras
• The Final Coalgebra
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Big Toy Models Workshop on Informatic Penomena 2009 – 57
The key point is this.
Proposition 24 For any set A of actions, there is a final LTA-coalgebra
(ProcA, out).
We think of elements of the final coalgebra as processes . The final
coalgebra provides a “universal semantics” for transition systems, which
is “fully abstract” with respect to observable behaviour.
All of this generalizes to a large class of endofunctors.
Representing Physical SystemsAs Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
• Coalgebras asModels of PhysicalSystems
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingBig Toy Models
Coalgebras as Models of Physical Systems
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
• Coalgebras asModels of PhysicalSystems
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingBig Toy Models Workshop on Informatic Penomena 2009 – 59
Coalgebras as Models of Physical Systems
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
• Coalgebras asModels of PhysicalSystems
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingBig Toy Models Workshop on Informatic Penomena 2009 – 59
Recall our basic setup: systems are modelled by a set of states S, of
questions Q, and an evaluation
e : S ×Q→ [0, 1].
Coalgebras as Models of Physical Systems
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
• Coalgebras asModels of PhysicalSystems
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingBig Toy Models Workshop on Informatic Penomena 2009 – 59
Recall our basic setup: systems are modelled by a set of states S, of
questions Q, and an evaluation
e : S ×Q→ [0, 1].
Problems:
Coalgebras as Models of Physical Systems
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
• Coalgebras asModels of PhysicalSystems
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingBig Toy Models Workshop on Informatic Penomena 2009 – 59
Recall our basic setup: systems are modelled by a set of states S, of
questions Q, and an evaluation
e : S ×Q→ [0, 1].
Problems:
• In Chu spaces, we get to specify Q as well as S. How do we do
this with coalgebras?
Coalgebras as Models of Physical Systems
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
• Coalgebras asModels of PhysicalSystems
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingBig Toy Models Workshop on Informatic Penomena 2009 – 59
Recall our basic setup: systems are modelled by a set of states S, of
questions Q, and an evaluation
e : S ×Q→ [0, 1].
Problems:
• In Chu spaces, we get to specify Q as well as S. How do we do
this with coalgebras?
• Q is contravariant (the maps f∗ go backwards.). Coalgebras are
based on covariant functors. (We could work with domains, but
there are drawbacks).
Comparison: A First Try
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models
First Approximation
Big Toy Models Workshop on Informatic Penomena 2009 – 61
Fix a set K. We can define a functor on Set:
FK : X 7→ KPX .
First Approximation
Big Toy Models Workshop on Informatic Penomena 2009 – 61
Fix a set K. We can define a functor on Set:
FK : X 7→ KPX .
If we use the contravariant powerset functor, F will be covariant. Explicitly, for
f : X → Y :
FKf(g)(S) = g(f−1(S)),
where g ∈ KPX and S ∈ PY .
First Approximation
Big Toy Models Workshop on Informatic Penomena 2009 – 61
Fix a set K. We can define a functor on Set:
FK : X 7→ KPX .
If we use the contravariant powerset functor, F will be covariant. Explicitly, for
f : X → Y :
FKf(g)(S) = g(f−1(S)),
where g ∈ KPX and S ∈ PY .
A coalgebra for this functor will be a map of the form
α : X → KPX .
First Approximation
Big Toy Models Workshop on Informatic Penomena 2009 – 61
Fix a set K. We can define a functor on Set:
FK : X 7→ KPX .
If we use the contravariant powerset functor, F will be covariant. Explicitly, for
f : X → Y :
FKf(g)(S) = g(f−1(S)),
where g ∈ KPX and S ∈ PY .
A coalgebra for this functor will be a map of the form
α : X → KPX .
Consider a Chu space C = (X,A, e) over K. We suppose that this Chu space is
normal , meaning that A = PX . We can define an FK -coalgebra on X by
α(x)(S) = e(x, S).
We write GC = (X,α).
Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 62
Proposition 25 Suppose we are given a Chu morphism f : C → C ′, where Cand C ′ are normal Chu spaces, such that f∗ = f−1
∗ . Then f∗ : GC → GC ′ is an
FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism
f : GC → GC ′, then (f, f−1) : C → C ′ is a Chu morphism.
Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 62
Proposition 25 Suppose we are given a Chu morphism f : C → C ′, where Cand C ′ are normal Chu spaces, such that f∗ = f−1
∗ . Then f∗ : GC → GC ′ is an
FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism
f : GC → GC ′, then (f, f−1) : C → C ′ is a Chu morphism.
Let NChuK be the category of normal Chu spaces and Chu morphisms of the
form (f, f−1). Then by the Proposition, G extends to a functorG : NChuK → FK−Coalg, with G(f, f−1) = f .
Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 62
Proposition 25 Suppose we are given a Chu morphism f : C → C ′, where Cand C ′ are normal Chu spaces, such that f∗ = f−1
∗ . Then f∗ : GC → GC ′ is an
FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism
f : GC → GC ′, then (f, f−1) : C → C ′ is a Chu morphism.
Let NChuK be the category of normal Chu spaces and Chu morphisms of the
form (f, f−1). Then by the Proposition, G extends to a functorG : NChuK → FK−Coalg, with G(f, f−1) = f .
There is an evident inverse to this functor.
Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 62
Proposition 25 Suppose we are given a Chu morphism f : C → C ′, where Cand C ′ are normal Chu spaces, such that f∗ = f−1
∗ . Then f∗ : GC → GC ′ is an
FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism
f : GC → GC ′, then (f, f−1) : C → C ′ is a Chu morphism.
Let NChuK be the category of normal Chu spaces and Chu morphisms of the
form (f, f−1). Then by the Proposition, G extends to a functorG : NChuK → FK−Coalg, with G(f, f−1) = f .
There is an evident inverse to this functor.
Proposition 26 NChuK and FK−Coalg are isomorphic categories.
Discussion: Critique of Coalgebras
Big Toy Models Workshop on Informatic Penomena 2009 – 63
Discussion: Critique of Coalgebras
Big Toy Models Workshop on Informatic Penomena 2009 – 63
• Assuming Chu spaces are normal is overly restrictive.
Discussion: Critique of Coalgebras
Big Toy Models Workshop on Informatic Penomena 2009 – 63
• Assuming Chu spaces are normal is overly restrictive.
The use of powersets, full or not, to represent ‘questions’ is fairly crude and ad
hoc. The degree of freedom afforded by Chu spaces to choose both the
states and the questions appropriately is a major benefit to conceptually
natural and formally adequate modelling of a wide range of situations.
Discussion: Critique of Coalgebras
Big Toy Models Workshop on Informatic Penomena 2009 – 63
• Assuming Chu spaces are normal is overly restrictive.
The use of powersets, full or not, to represent ‘questions’ is fairly crude and ad
hoc. The degree of freedom afforded by Chu spaces to choose both the
states and the questions appropriately is a major benefit to conceptually
natural and formally adequate modelling of a wide range of situations.
• The functors FK are somewhat problematic from the point of view of
coalgebra — they fail to preserve weak pullbacks.
Discussion: Critique of Coalgebras
Big Toy Models Workshop on Informatic Penomena 2009 – 63
• Assuming Chu spaces are normal is overly restrictive.
The use of powersets, full or not, to represent ‘questions’ is fairly crude and ad
hoc. The degree of freedom afforded by Chu spaces to choose both the
states and the questions appropriately is a major benefit to conceptually
natural and formally adequate modelling of a wide range of situations.
• The functors FK are somewhat problematic from the point of view of
coalgebra — they fail to preserve weak pullbacks.
• They will also fail to have final coalgebras. However, this can be fixed easily
enough by using bounded powerset and bounded partial functions.
Discussion: In Praise of Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 64
Discussion: In Praise of Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 64
• The coalgebraic point of view can be described as state-based , but
in a way that emphasizes that the meaning of states lies in their
observable behaviour . Indeed, in the “universal model” we shallconstruct, the states are determined exactly as the possible
observable behaviours — we actually find a canonical solution for
what the state space should be in these terms. States are
identified exactly if they have the same observable behaviour.
Discussion: In Praise of Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 64
• The coalgebraic point of view can be described as state-based , but
in a way that emphasizes that the meaning of states lies in their
observable behaviour . Indeed, in the “universal model” we shallconstruct, the states are determined exactly as the possible
observable behaviours — we actually find a canonical solution for
what the state space should be in these terms. States are
identified exactly if they have the same observable behaviour.
We can see this as a kind of reconciliation between the ontic and
epistemic standpoints.
Discussion: In Praise of Coalgebras
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 64
• The coalgebraic point of view can be described as state-based , but
in a way that emphasizes that the meaning of states lies in their
observable behaviour . Indeed, in the “universal model” we shallconstruct, the states are determined exactly as the possible
observable behaviours — we actually find a canonical solution for
what the state space should be in these terms. States are
identified exactly if they have the same observable behaviour.
We can see this as a kind of reconciliation between the ontic and
epistemic standpoints.
• Coalgebras allow us to capture the ‘dynamics of measurement’ —
what happens after a measurement — in a way that Chu spaces
don’t. They have ‘extension in time’.
Extension in Time
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 65
Extension in Time
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 65
Consider a coalgebraic representation of stochastic transducers :
F : X 7→ Prob(O ×X)I
where I is a fixed set of inputs , O a fixed set of outputs , and Prob(S) is
the set of probability distributions on S.
Extension in Time
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 65
Consider a coalgebraic representation of stochastic transducers :
F : X 7→ Prob(O ×X)I
where I is a fixed set of inputs , O a fixed set of outputs , and Prob(S) is
the set of probability distributions on S.
We can think of I as a set of questions , and O as a set of answers(which we could standardize by only considering yes/no questions).
Extension in Time
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
• First Approximation
• Comparison
• Discussion: Critiqueof Coalgebras
• Discussion: In Praiseof Coalgebras
• Extension in Time
Big Toy Models Workshop on Informatic Penomena 2009 – 65
Consider a coalgebraic representation of stochastic transducers :
F : X 7→ Prob(O ×X)I
where I is a fixed set of inputs , O a fixed set of outputs , and Prob(S) is
the set of probability distributions on S.
We can think of I as a set of questions , and O as a set of answers(which we could standardize by only considering yes/no questions).
What we learn from this is that
QM is less nondeterministic/probabilistic than stochastic transducers
since in QM if we know the preparation and the outcome of the
measurement, we know (by the projection postulate) exactly what theresulting quantum state will be.
Semantics in One Country
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
• CoalgebraicSemantics For OneSystem
• Well BehavedFunctorsBig Toy Models
Coalgebraic Semantics For One System
Big Toy Models Workshop on Informatic Penomena 2009 – 67
Coalgebraic Semantics For One System
Big Toy Models Workshop on Informatic Penomena 2009 – 67
We fix attention on a single Hilbert space H. This determines a set of question
Q = L(H).
Coalgebraic Semantics For One System
Big Toy Models Workshop on Informatic Penomena 2009 – 67
We fix attention on a single Hilbert space H. This determines a set of question
Q = L(H).
We now define an endofunctor on Set:
FQ : X 7→ ({0} + (0, 1] ×X)Q.
Coalgebraic Semantics For One System
Big Toy Models Workshop on Informatic Penomena 2009 – 67
We fix attention on a single Hilbert space H. This determines a set of question
Q = L(H).
We now define an endofunctor on Set:
FQ : X 7→ ({0} + (0, 1] ×X)Q.
A coalgebra for this functor is then a map
α : X → ({0} + (0, 1] ×X)Q
The interpretation is that X is a set of states; the coalgebra map sends its state to
its behaviour, which is a function from questions in Q to the probability that the
answer is ‘yes’; and, if the probability is not 0 , to the successor state following a
‘yes’ answer.
Well Behaved Functors
Big Toy Models Workshop on Informatic Penomena 2009 – 68
Unlike the functors FK , the functors FQ are very well-behaved from the point of
view of coalgebra (they are in fact polynomial functors ). They preserve weak
pull-backs, which guarantees a number of nice properties, and they are bounded
and admit final coalgebras
γQ : UQ → ({0} + (0, 1] × UQ)Q.
Well Behaved Functors
Big Toy Models Workshop on Informatic Penomena 2009 – 68
Unlike the functors FK , the functors FQ are very well-behaved from the point of
view of coalgebra (they are in fact polynomial functors ). They preserve weak
pull-backs, which guarantees a number of nice properties, and they are bounded
and admit final coalgebras
γQ : UQ → ({0} + (0, 1] × UQ)Q.
The elements of UQ can be visualized as ‘Q-branching trees’ with the arcs labelledby probabilities.
Representing One Quantum System As A Coalgebra
Big Toy Models Workshop on Informatic Penomena 2009 – 69
Representing One Quantum System As A Coalgebra
Big Toy Models Workshop on Informatic Penomena 2009 – 69
The FQ-coalgebra which is of primary interest to us is the map
aH : H◦ → ({0} + (0, 1] ×H◦)Q
defined by:
aH(ψ)(S) =
0, eH(ψ, S) = 0
(r, PSψ), eH(ψ, S) = r > 0
Representing One Quantum System As A Coalgebra
Big Toy Models Workshop on Informatic Penomena 2009 – 69
The FQ-coalgebra which is of primary interest to us is the map
aH : H◦ → ({0} + (0, 1] ×H◦)Q
defined by:
aH(ψ)(S) =
0, eH(ψ, S) = 0
(r, PSψ), eH(ψ, S) = r > 0
The new ingredient compared with the Chu space representation of H is the statewhich results in the case of a ‘yes’ answer to the question, which is computed
according to the Luders rule .
Externalising Contravariance AsIndexing
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models
The Indexed Category
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 71
We define a functor
F : Setop → CAT
with
Q 7→ FQ−Coalg
and for f : Q′ → Q:
tf : FQ → FQ′
:: Θ 7→ Θ ◦ f
is a natural transformation, and
F(f) = f∗ : Coalg−FQ → Coalg−FQ′
f∗ : (A,α) 7→ (A, tfA ◦ α)
is a functor.
The Indexed Category
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 71
We define a functor
F : Setop → CAT
with
Q 7→ FQ−Coalg
and for f : Q′ → Q:
tf : FQ → FQ′
:: Θ 7→ Θ ◦ f
is a natural transformation, and
F(f) = f∗ : Coalg−FQ → Coalg−FQ′
f∗ : (A,α) 7→ (A, tfA ◦ α)
is a functor.
Thus we get a strict indexed category of coalgebra categories, with
contravariant indexing.
The Grothendieck Construction
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 72
Where we have an indexed category, we can apply the Grothendieckconstruction to glue all the fibres together (and get a fibration).
The Grothendieck Construction
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 72
Where we have an indexed category, we can apply the Grothendieckconstruction to glue all the fibres together (and get a fibration).
Given a functorI : C
op → CAT
define∫
I with objects (A, a), where A is an object of C and a is an
object of I(A).
The Grothendieck Construction
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 72
Where we have an indexed category, we can apply the Grothendieckconstruction to glue all the fibres together (and get a fibration).
Given a functorI : C
op → CAT
define∫
I with objects (A, a), where A is an object of C and a is an
object of I(A).
Arrows are (G, g) : (A, a) → (B, b), where G : B → A and
g : I(G)(a) → b.
The Grothendieck Construction
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
• The IndexedBig Toy Models Workshop on Informatic Penomena 2009 – 72
Where we have an indexed category, we can apply the Grothendieckconstruction to glue all the fibres together (and get a fibration).
Given a functorI : C
op → CAT
define∫
I with objects (A, a), where A is an object of C and a is an
object of I(A).
Arrows are (G, g) : (A, a) → (B, b), where G : B → A and
g : I(G)(a) → b.
Composition of (G, g) : (A, a) → (B, b) and (H,h) : (B, b) → (C, c)is given by
(G ◦H,h ◦ I(G)(g)) : (A, a) → (C, c).
Indexed Comparison With ChuSpaces
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models
Slicing and Dicing Chu
Big Toy Models Workshop on Informatic Penomena 2009 – 74
For each Q, we define ChuQK to be the subcategory of ChuK of Chu spaces
(X,Q, e) and morphisms of the form (f∗, idQ).
Slicing and Dicing Chu
Big Toy Models Workshop on Informatic Penomena 2009 – 74
For each Q, we define ChuQK to be the subcategory of ChuK of Chu spaces
(X,Q, e) and morphisms of the form (f∗, idQ).
This doesn’t look too exciting. In fact, it is just the comma category
(−×Q, K)
where K : 1 → Set picks out the object K.
Slicing and Dicing Chu
Big Toy Models Workshop on Informatic Penomena 2009 – 74
For each Q, we define ChuQK to be the subcategory of ChuK of Chu spaces
(X,Q, e) and morphisms of the form (f∗, idQ).
This doesn’t look too exciting. In fact, it is just the comma category
(−×Q, K)
where K : 1 → Set picks out the object K.
Given f : Q′ → Q, we define a functor
f∗ : ChuQK → Chu
Q′
K :: (X,Q, e) 7→ (X,Q′, e ◦ (1 × f))
and which is the identity on morphisms.
Slicing and Dicing Chu
Big Toy Models Workshop on Informatic Penomena 2009 – 74
For each Q, we define ChuQK to be the subcategory of ChuK of Chu spaces
(X,Q, e) and morphisms of the form (f∗, idQ).
This doesn’t look too exciting. In fact, it is just the comma category
(−×Q, K)
where K : 1 → Set picks out the object K.
Given f : Q′ → Q, we define a functor
f∗ : ChuQK → Chu
Q′
K :: (X,Q, e) 7→ (X,Q′, e ◦ (1 × f))
and which is the identity on morphisms.
This gives an indexed category
Chu : Setop → CAT
Grothendieck puts Chu back together again
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 75
Grothendieck puts Chu back together again
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 75
Proposition 27∫
Chu ∼= ChuK .
The Truncation Functor
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 76
The Truncation Functor
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 76
The relationship between coalgebras and Chu spaces is further clarified
by an indexed truncation functor T : F → Chu.
The Truncation Functor
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 76
The relationship between coalgebras and Chu spaces is further clarified
by an indexed truncation functor T : F → Chu.
For each set Q there is a functor
TQ : FQ−Coalg → Chu
QK
TQ(X,α) = (X,Q, e)
where
e(x, q) =
0, α(x)(q) = 0
r, α(x)(q) = (r, x′)
The Truncation Functor
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models Workshop on Informatic Penomena 2009 – 76
The relationship between coalgebras and Chu spaces is further clarified
by an indexed truncation functor T : F → Chu.
For each set Q there is a functor
TQ : FQ−Coalg → Chu
QK
TQ(X,α) = (X,Q, e)
where
e(x, q) =
0, α(x)(q) = 0
r, α(x)(q) = (r, x′)
For f : Q′ → Q there is a natural transformation
τ f : TQ → TQ′
τ f
(X,α)= (idX , f) : TQ(X,α) → TQ′
(X,α).
A Universal Model
Introduction
Chu Spaces
Representing PhysicalSystems
Characterizing ChuMorphisms onQuantum Chu Spaces
The RepresentationTheorem
Reducing The ValueSet
Discussion
Chu Spaces andCoalgebras
Primer on coalgebra
Basic Concepts
Representing PhysicalSystems AsCoalgebras
Comparison: A FirstTry
Semantics in OneCountry
ExternalisingContravariance AsIndexing
Indexed Comparison
Big Toy Models
A Universal Model
Big Toy Models Workshop on Informatic Penomena 2009 – 78
A Universal Model
Big Toy Models Workshop on Informatic Penomena 2009 – 78
We can now define a single coalgebra which is universal for quantum systems in
the following sense:
A Universal Model
Big Toy Models Workshop on Informatic Penomena 2009 – 78
We can now define a single coalgebra which is universal for quantum systems in
the following sense:
• Fix a countably-infinite dimensional Hilbert space, e.g. H = ℓ2(N). Take
Q = L(H). Let (U, γ) be the final coalgebra for FQ.
A Universal Model
Big Toy Models Workshop on Informatic Penomena 2009 – 78
We can now define a single coalgebra which is universal for quantum systems in
the following sense:
• Fix a countably-infinite dimensional Hilbert space, e.g. H = ℓ2(N). Take
Q = L(H). Let (U, γ) be the final coalgebra for FQ.
• Any quantum system is described by a separable Hilbert space K, say with a
preferred basis. This basis will induce an isometric embedding i : K- - H.
Taking Q′ = L(K), this induces a map f = i−1 : Q→ Q′. This in turninduces a functor f∗ : FQ′
−Coalg → FQ−Coalg.
A Universal Model
Big Toy Models Workshop on Informatic Penomena 2009 – 78
We can now define a single coalgebra which is universal for quantum systems in
the following sense:
• Fix a countably-infinite dimensional Hilbert space, e.g. H = ℓ2(N). Take
Q = L(H). Let (U, γ) be the final coalgebra for FQ.
• Any quantum system is described by a separable Hilbert space K, say with a
preferred basis. This basis will induce an isometric embedding i : K- - H.
Taking Q′ = L(K), this induces a map f = i−1 : Q→ Q′. This in turninduces a functor f∗ : FQ′
−Coalg → FQ−Coalg.
• This functor can be applied to the coalgebra (K◦, α) corresponding to theHilbert space K to yield a coalgebra in FQ−Coalg.
Universality
Big Toy Models Workshop on Informatic Penomena 2009 – 79
Universality
Big Toy Models Workshop on Informatic Penomena 2009 – 79
• Since (U, γ) is the final coalgebra in FQ−Coalg, there is a unique
coalgebra homomorphism h : f∗(K◦, α) → (U, γ).
Universality
Big Toy Models Workshop on Informatic Penomena 2009 – 79
• Since (U, γ) is the final coalgebra in FQ−Coalg, there is a unique
coalgebra homomorphism h : f∗(K◦, α) → (U, γ).
• This homomorphism maps the quantum system (K◦, α) into (U, γ) in a fullyabstract fashion , i.e. identifying states precisely according to observational
equivalence.
Universality
Big Toy Models Workshop on Informatic Penomena 2009 – 79
• Since (U, γ) is the final coalgebra in FQ−Coalg, there is a unique
coalgebra homomorphism h : f∗(K◦, α) → (U, γ).
• This homomorphism maps the quantum system (K◦, α) into (U, γ) in a fullyabstract fashion , i.e. identifying states precisely according to observational
equivalence.
• This homomorphism is an arrow in the Grothendieck category.
Universality
Big Toy Models Workshop on Informatic Penomena 2009 – 79
• Since (U, γ) is the final coalgebra in FQ−Coalg, there is a unique
coalgebra homomorphism h : f∗(K◦, α) → (U, γ).
• This homomorphism maps the quantum system (K◦, α) into (U, γ) in a fullyabstract fashion , i.e. identifying states precisely according to observational
equivalence.
• This homomorphism is an arrow in the Grothendieck category.
• This works for all quantum systems, with respect to a single coalgebra.
Universality
Big Toy Models Workshop on Informatic Penomena 2009 – 79
• Since (U, γ) is the final coalgebra in FQ−Coalg, there is a unique
coalgebra homomorphism h : f∗(K◦, α) → (U, γ).
• This homomorphism maps the quantum system (K◦, α) into (U, γ) in a fullyabstract fashion , i.e. identifying states precisely according to observational
equivalence.
• This homomorphism is an arrow in the Grothendieck category.
• This works for all quantum systems, with respect to a single coalgebra.
This is truly a Big Toy Model!