Big Toy Models: Representing Physical Systems as Chu Spaces
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WORKSHOP ON INFORMATIC PHENOMENA (2009):
http://dauns.math.tulane.edu/~mwm/WIP2009/Titles_and_Abstracts.html

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  • 'Samson Abramsky, Information Flow in Physics, Geometry, Logic and Computation

    In my lectures, I will describe a broad program that has been initiated to model information flow in these and related areas. This includes a high-level reformulation of quantum information and quantum computing using category theory that have been shown to capture all of the fundamental components of the theory. The approach supports reasoning about classical and quantum communication in the same model. The approach also has provided what are arguably the first completely formal descriptions and proofs of correctness of several key quantum informatic protocols, e.g. (logic-gate) teleportation, superdense coding, and one-way computational schemes. It also provides a description of the quantum state, as well as the flow of information from the quantum state to the classical world (measurements), and from the classical world to the quantum state (control), all of which are important for reasoning about security in a quantum setting.'
    http://129.81.170.14/~mwm/clifford/Site/Abstracts.html

    Speaker: Samson Abramsky (University of Oxford)
    Title: Information flow in physics, geometry, logic and computation V
    Event: Clifford Lectures 2008 (March 2008, Tulane University)
    http://www.youtube.com/watch?v=pmwbcW4E35c
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Big Toy Models: Representing Physical Systems as Chu Spaces Big Toy Models: Representing Physical Systems as Chu Spaces Presentation Transcript

  • Big Toy Models: Representing Physical Systems As Chu Spaces Samson Abramsky Oxford University Computing LaboratoryBig Toy Models Workshop on Informatic Penomena 2009 – 1
  • Introduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces IntroductionThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One
  • ThemesIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  • ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  • ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  • ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view ofReducing The Value quantum states: A toy theory’.SetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  • ThemesIntroduction• Themes • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’.• Chu Spaces• Outline I Exemplifies one of the main thrusts of our group in Oxford:• Outline II methods and concepts which have been developed in TheoreticalChu Spaces Computer Science are ripe for use in Physics.Representing PhysicalSystemsCharacterizing Chu • Models vs. Axioms. Examples: sheaves and toposes,Morphisms on domain-theoretic models of the λ-calculus.Quantum Chu SpacesThe RepresentationTheorem • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view ofReducing The Value quantum states: A toy theory’.SetDiscussionChu Spaces and • Big toy models.CoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 3
  • Chu SpacesIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  • Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information flow.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  • Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information flow.Representing PhysicalSystems This logic of Chu spaces was in no way biassed in its conception towardsCharacterizing Chu the description of quantum mechanics or any other kind of physicalMorphisms onQuantum Chu Spaces system.The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  • Chu SpacesIntroduction We should understand Chu spaces as providing a very general (and, we• Themes• Chu Spaces might reasonably say, rather simple) ‘logic of systems or structures’.• Outline I• Outline II Indeed, they have been proposed by Barwise and Seligman as theChu Spaces vehicle for a general logic of ‘distributed systems’ and information flow.Representing PhysicalSystems This logic of Chu spaces was in no way biassed in its conception towardsCharacterizing Chu the description of quantum mechanics or any other kind of physicalMorphisms onQuantum Chu Spaces system.The RepresentationTheorem Just for this reason, it is interesting to see how much ofReducing The ValueSet quantum-mechanical structure and concepts can be absorbed andDiscussion essentially determined by this more general systems logic.Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 4
  • Outline IIntroduction• Themes• Chu Spaces• Outline I• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  • Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  • Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  • Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms on • States as rays of Hilbert spaces fall out as the biextensionalQuantum Chu Spaces collapse of the Chu spaces.The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  • Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms on • States as rays of Hilbert spaces fall out as the biextensionalQuantum Chu Spaces collapse of the Chu spaces.The RepresentationTheorem • Chu morphisms are automatically the unitaries andReducing The ValueSet antiunitaries — the physical symmetries of quantum systems.DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  • Outline IIntroduction• Themes • Chu spaces as a setting. We can find natural representations of• Chu Spaces• Outline I quantum (and other) systems as Chu spaces.• Outline IIChu Spaces • The general ‘logic’ of Chu spaces and morphisms allow us toRepresenting Physical ‘rationally reconstruct’ many key quantum notions:SystemsCharacterizing ChuMorphisms on • States as rays of Hilbert spaces fall out as the biextensionalQuantum Chu Spaces collapse of the Chu spaces.The RepresentationTheorem • Chu morphisms are automatically the unitaries andReducing The ValueSet antiunitaries — the physical symmetries of quantum systems.Discussion • This leads to a full and faithful representation of theChu Spaces andCoalgebras groupoid of Hilbert spaces and their physical symmetries inPrimer on coalgebra Chu spaces over the unit interval.Basic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 5
  • Outline IIBig Toy Models Workshop on Informatic Penomena 2009 – 6
  • Outline II • This leads to a further question of conceptual interest: is this representation preserved by collapsing the unit interval to finitely many values?Big Toy Models Workshop on Informatic Penomena 2009 – 6
  • Outline II • 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 show that this fails.Big Toy Models Workshop on Informatic Penomena 2009 – 6
  • Outline II • 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 show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 3-valued logic in quantum foundations?Big Toy Models Workshop on Informatic Penomena 2009 – 6
  • Outline II • 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 show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 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.Big Toy Models Workshop on Informatic Penomena 2009 – 6
  • Outline II • 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 show that this fails. • However, the natural collapse to three values works! — A possible role ˆ for 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.htmlBig Toy Models Workshop on Informatic Penomena 2009 – 6
  • IntroductionChu Spaces• Chu Spaces• Definitions• Extensionality andSeparability• BiextensionalCollapseRepresenting PhysicalSystemsCharacterizing Chu Chu SpacesMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big
  • Chu SpacesBig Toy Models Workshop on Informatic Penomena 2009 – 8
  • Chu Spaces 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.Big Toy Models Workshop on Informatic Penomena 2009 – 8
  • Chu Spaces 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:Big Toy Models Workshop on Informatic Penomena 2009 – 8
  • Chu Spaces 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).Big Toy Models Workshop on Informatic Penomena 2009 – 8
  • Chu Spaces 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).Big Toy Models Workshop on Informatic Penomena 2009 – 8
  • Chu Spaces 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).Big Toy Models Workshop on Informatic Penomena 2009 – 8
  • Chu Spaces 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.Big Toy Models Workshop on Informatic Penomena 2009 – 8
  • DefinitionsBig Toy Models Workshop on Informatic Penomena 2009 – 9
  • Definitions 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.Big Toy Models Workshop on Informatic Penomena 2009 – 9
  • Definitions 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′ ).Big Toy Models Workshop on Informatic Penomena 2009 – 9
  • Definitions 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 ∗ .Big Toy Models Workshop on Informatic Penomena 2009 – 9
  • Definitions 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 .Big Toy Models Workshop on Informatic Penomena 2009 – 9
  • Extensionality and SeparabilityBig Toy Models Workshop on Informatic Penomena 2009 – 10
  • Extensionality and Separability Given a Chu space C = (X, A, e), we say that C is:Big Toy Models Workshop on Informatic Penomena 2009 – 10
  • Extensionality and Separability 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 = a2Big Toy Models Workshop on Informatic Penomena 2009 – 10
  • Extensionality and Separability 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 = x2Big Toy Models Workshop on Informatic Penomena 2009 – 10
  • Extensionality and Separability 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).Big Toy Models Workshop on Informatic Penomena 2009 – 10
  • Extensionality and Separability 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.Big Toy Models Workshop on Informatic Penomena 2009 – 10
  • Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big Workshop on Informatic Penomena 2009 – 11
  • Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystems Proof For any a′ ∈ A′ :Characterizing ChuMorphisms onQuantum Chu Spaces e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big Workshop on Informatic Penomena 2009 – 11
  • Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystems Proof For any a′ ∈ A′ :Characterizing ChuMorphisms onQuantum Chu Spaces e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).The RepresentationTheoremReducing The ValueSet We shall write eChuK , sChuK and bChuK for the full subcategoriesDiscussion of ChuK determined by the extensional, separated and biextensionalChu Spaces andCoalgebras Chu spaces.Primer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Toy Models Big Workshop on Informatic Penomena 2009 – 11
  • Biextensional CollapseIntroduction Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, thenChu Spaces• Chu Spaces f∗ preserves ∼. That is, for all x1 , x2 ∈ X ,• Definitions• Extensionality andSeparability x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ).• BiextensionalCollapseRepresenting PhysicalSystems Proof For any a′ ∈ A′ :Characterizing ChuMorphisms onQuantum Chu Spaces e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ).The RepresentationTheoremReducing The ValueSet We shall write eChuK , sChuK and bChuK for the full subcategoriesDiscussion of ChuK determined by the extensional, separated and biextensionalChu Spaces andCoalgebras Chu spaces.Primer on coalgebra We shall mainly work with extensional and biextensional Chu spaces.Basic Concepts Obviously bChuK is a full sub-category of eChuK .Representing PhysicalSystems AsCoalgebras Proposition 2 The inclusion bChuK ⊂ - eChuK has a left adjointComparison: A First Q, the biextensional collapse..Try Toy Models Big Workshop on Informatic Penomena 2009 – 11
  • IntroductionChu SpacesRepresenting PhysicalSystems• The GeneralParadigm• RepresentingQuantum Systems AsChu SpacesCharacterizing ChuMorphisms onQuantum Chu Spaces Representing Physical SystemsThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One
  • The General ParadigmBig Toy Models Workshop on Informatic Penomena 2009 – 13
  • The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system.Big Toy Models Workshop on Informatic Penomena 2009 – 13
  • The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which 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’ when the system is in state s.Big Toy Models Workshop on Informatic Penomena 2009 – 13
  • The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which 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’ when the system is in state s. This is a Chu space!Big Toy Models Workshop on Informatic Penomena 2009 – 13
  • The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which 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’ when the 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.Big Toy Models Workshop on Informatic Penomena 2009 – 13
  • The General Paradigm We take a system to be specified by its set of states S , and the set of questions Q which 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’ when the 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!Big Toy Models Workshop on Informatic Penomena 2009 – 13
  • Representing Quantum Systems As Chu SpacesIntroductionChu SpacesRepresenting PhysicalSystems• The GeneralParadigm• RepresentingQuantum Systems AsChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  • Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems AsChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  • Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  • Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  • Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The Value • The evaluation function eH is the ‘statistical algorithm’ giving theSet basic predictive content of Quantum Mechanics:DiscussionChu Spaces andCoalgebras ψ | PS ψ PS ψ | PS ψ PS ψ 2 eH (ψ, S) = = = 2 .Primer on coalgebra ψ|ψ ψ|ψ ψBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  • Representing Quantum Systems As Chu Spaces A quantum system with a Hilbert space H as its state space will beIntroduction represented asChu Spaces (H◦ , L(H), eH )Representing PhysicalSystems• The General whereParadigm• RepresentingQuantum Systems As • H◦ is the set of non-zero vectors of HChu SpacesCharacterizing ChuMorphisms on • L(H) is the set of closed subspaces of H — the ‘yes/no’ questionsQuantum Chu Spaces of QMThe RepresentationTheoremReducing The Value • The evaluation function eH is the ‘statistical algorithm’ giving theSet basic predictive content of Quantum Mechanics:DiscussionChu Spaces andCoalgebras ψ | PS ψ PS ψ | PS ψ PS ψ 2 eH (ψ, S) = = = 2 .Primer on coalgebra ψ|ψ ψ|ψ ψBasic ConceptsRepresenting PhysicalSystems As We have thus directly transcribed the basic ingredients of the Dirac/vonCoalgebras Neumann-style formulation of Quantum Mechanics into the definition ofComparison: A FirstTry this Chu space. Big Toy ModelsSemantics in One Workshop on Informatic Penomena 2009 – 14
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity• Projectivity = Characterizing Chu MorphismsBiextensionality• Characterizing ChuMorphisms on Quantum Chu Spaces• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value
  • OverviewIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  • OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufficeChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically significant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  • OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufficeChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically significant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing Chu The arguments here make use of Wigner’s theorem and the dualities ofMorphisms• Injectivity projective geometry, in the modern form developed by Faure andAssumption ¨ Frolicher, Modern Projective Geometry (2000).• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  • OverviewIntroduction We shall now see how the simple, discrete notions of Chu spaces sufficeChu Spaces to determine the appropriate notions of state equivalence, and to pick outRepresenting PhysicalSystems the physically significant symmetries on Hilbert space in a very strikingCharacterizing Chu fashion. This leads to a full and faithful representation of the category ofMorphisms onQuantum Chu Spaces quantum systems, with the groupoid structure of their physical• Overview• Biextensionaity symmetries, in the category of Chu spaces valued in the unit interval.• Projectivity =Biextensionality• Characterizing Chu The arguments here make use of Wigner’s theorem and the dualities ofMorphisms• Injectivity projective geometry, in the modern form developed by Faure andAssumption ¨ Frolicher, Modern Projective Geometry (2000).• Orthogonality isPreserved• Constructing the LeftAdjoint The surprising point is that unitarity/anitunitarity is essentially forced by• Using Projective the mere requirement of being a Chu morphism. This even extends toDuality• Wigner’s Theorem surjectivity, which here is derived rather than assumed.• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 16
  • BiextensionaityIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  • BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  • BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting Physical A basic property of the evaluation.SystemsCharacterizing ChuMorphisms on Lemma 3 For ψ ∈ H◦ and S ∈ L(H):Quantum Chu Spaces• Overview• Biextensionaity ψ ∈ S ⇐⇒ eH (ψ, S) = 1.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  • BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting Physical A basic property of the evaluation.SystemsCharacterizing ChuMorphisms on Lemma 3 For ψ ∈ H◦ and S ∈ L(H):Quantum Chu Spaces• Overview• Biextensionaity ψ ∈ S ⇐⇒ eH (ψ, S) = 1.• Projectivity =Biextensionality• Characterizing Chu From this, we can prove:Morphisms• InjectivityAssumption• Orthogonality is Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but notPreserved• Constructing the Left separable. The equivalence classes of the relation ∼ on states areAdjoint exactly the rays of H. That is:• Using ProjectiveDuality• Wigner’s Theorem• Remarks φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  • BiextensionaityIntroduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ).Chu SpacesRepresenting Physical A basic property of the evaluation.SystemsCharacterizing ChuMorphisms on Lemma 3 For ψ ∈ H◦ and S ∈ L(H):Quantum Chu Spaces• Overview• Biextensionaity ψ ∈ S ⇐⇒ eH (ψ, S) = 1.• Projectivity =Biextensionality• Characterizing Chu From this, we can prove:Morphisms• InjectivityAssumption• Orthogonality is Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but notPreserved• Constructing the Left separable. The equivalence classes of the relation ∼ on states areAdjoint exactly the rays of H. That is:• Using ProjectiveDuality• Wigner’s Theorem• Remarks φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ.• A Surprise:Surjectivity Comes forFree! Thus we have recovered the standard notion of pure states as the rays of• Putting The PiecesTogether the Hilbert space from the general notion of state equivalence in ChuThe Representation spaces.Theorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 17
  • Projectivity = BiextensionalityIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  • Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  • Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting Physical ¯ Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which itSystemsCharacterizing Chu generates. The rays are the atoms in the lattice L(H).Morphisms 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  • Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting Physical ¯ Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which itSystemsCharacterizing Chu generates. The rays are the atoms in the lattice L(H).Morphisms onQuantum Chu Spaces We write P(H) for the set of rays of H. By virtue of Proposition 4, we can• Overview• Biextensionaity write the biextensional collapse of (H◦ , L(H), eH ) given by Proposition 2• Projectivity =Biextensionality as• Characterizing ChuMorphisms (P(H), L(H), eH) ¯• InjectivityAssumption• Orthogonality is ¯ ¯ where eH (ψ, S) = eH (ψ, S).Preserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  • Projectivity = BiextensionalityIntroduction We shall now use some notions and results from projective geometry.Chu SpacesRepresenting Physical ¯ Given a vector ψ ∈ H◦ , we write ψ = {λψ | λ ∈ C} for the ray which itSystemsCharacterizing Chu generates. The rays are the atoms in the lattice L(H).Morphisms onQuantum Chu Spaces We write P(H) for the set of rays of H. By virtue of Proposition 4, we can• Overview• Biextensionaity write the biextensional collapse of (H◦ , L(H), eH ) given by Proposition 2• Projectivity =Biextensionality as• Characterizing ChuMorphisms (P(H), L(H), eH) ¯• InjectivityAssumption• Orthogonality is ¯ ¯ where eH (ψ, S) = eH (ψ, S).Preserved• Constructing the LeftAdjoint We restate Lemma 3 for the biextensional case.• Using ProjectiveDuality• Wigner’s Theorem Lemma 5 For ψ ∈ H◦ and S ∈ L(H):• Remarks• A Surprise:Surjectivity Comes for ¯ ¯ ¯ eH (ψ, S) = 1 ⇐⇒ ψ ⊆ S.Free!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 18
  • Characterizing Chu MorphismsBig Toy Models Workshop on Informatic Penomena 2009 – 19
  • Characterizing Chu Morphisms 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 ). ¯ ¯Big Toy Models Workshop on Informatic Penomena 2009 – 19
  • Characterizing Chu Morphisms 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. ¯Big Toy Models Workshop on Informatic Penomena 2009 – 19
  • Characterizing Chu Morphisms 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).Big Toy Models Workshop on Informatic Penomena 2009 – 19
  • Injectivity AssumptionBig Toy Models Workshop on Informatic Penomena 2009 – 20
  • Injectivity Assumption Proposition 7 If f∗ is injective, then the following diagram commutes: f∗ P(H) - P(K) ∩ ∩ (1) ? ? L(H)  ∗ L(K) f That is, for all ψ ∈ H◦ : ¯ ¯ ψ = f ∗ (f∗ (ψ)).Big Toy Models Workshop on Informatic Penomena 2009 – 20
  • Injectivity Assumption Proposition 7 If f∗ is injective, then the following diagram commutes: f∗ P(H) - P(K) ∩ ∩ (1) ? ? L(H)  ∗ L(K) f 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 .Big Toy Models Workshop on Informatic Penomena 2009 – 20
  • Orthogonality is Preserved Another basic property of the evaluation.IntroductionChu Spaces Lemma 8 For any φ, ψ ∈ H◦ :Representing PhysicalSystems ¯ ¯ ¯ eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 21
  • Orthogonality is Preserved Another basic property of the evaluation.IntroductionChu Spaces Lemma 8 For any φ, ψ ∈ H◦ :Representing PhysicalSystems ¯ ¯ ¯ eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.Characterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity Proposition 9 If f∗ is injective, it preserves and reflects• Projectivity =Biextensionality orthogonality. That is, for all φ, ψ ∈ H◦ :• Characterizing ChuMorphisms• Injectivity ¯ ¯ φ ⊥ ψ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ).Assumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 21
  • Orthogonality is Preserved Another basic property of the evaluation.IntroductionChu Spaces Lemma 8 For any φ, ψ ∈ H◦ :Representing PhysicalSystems ¯ ¯ ¯ eH (φ, ψ) = 0 ⇐⇒ φ ⊥ ψ.Characterizing ChuMorphisms onQuantum Chu Spaces• Overview• Biextensionaity Proposition 9 If f∗ is injective, it preserves and reflects• Projectivity =Biextensionality orthogonality. That is, for all φ, ψ ∈ H◦ :• Characterizing ChuMorphisms• Injectivity ¯ ¯ φ ⊥ ψ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ).Assumption• Orthogonality isPreserved• Constructing the Left ProofAdjoint• Using Projective ¯ ¯ ¯Duality• Wigner’s Theorem φ ⊥ ψ ⇐⇒ eH (φ, ψ) = 0 Lemma 8• Remarks• A Surprise:Surjectivity Comes for ¯ ¯ ¯ ⇐⇒ eH (φ, f ∗ (f∗ (ψ))) = 0 Proposition 7Free!• Putting The Pieces ¯ ¯ ⇐⇒ eK (f∗ (φ), f∗ (ψ)) = 0 ¯TogetherThe RepresentationTheorem ¯ ¯ ⇐⇒ f∗ (φ) ⊥ f∗ (ψ) Lemma 8. Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 21
  • Constructing the Left AdjointIntroduction We define a map f → : L(H) → L(K):Chu SpacesRepresenting PhysicalSystems f → (S) = ¯ {f∗ (ψ) | ψ ∈ S◦ }Characterizing ChuMorphisms onQuantum Chu Spaces where S◦ = S {0}.• 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 22
  • Constructing the Left AdjointIntroduction We define a map f → : L(H) → L(K):Chu SpacesRepresenting PhysicalSystems f → (S) = ¯ {f∗ (ψ) | ψ ∈ S◦ }Characterizing ChuMorphisms onQuantum Chu Spaces where S◦ = S {0}.• Overview• Biextensionaity• Projectivity = Lemma 10 The map f → is left adjoint to f ∗ :Biextensionality• Characterizing ChuMorphisms• Injectivity f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ).Assumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 22
  • Constructing the Left AdjointIntroduction We define a map f → : L(H) → L(K):Chu SpacesRepresenting PhysicalSystems f → (S) = ¯ {f∗ (ψ) | ψ ∈ S◦ }Characterizing ChuMorphisms onQuantum Chu Spaces where S◦ = S {0}.• Overview• Biextensionaity• Projectivity = Lemma 10 The map f → is left adjoint to f ∗ :Biextensionality• Characterizing ChuMorphisms• Injectivity f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ).Assumption• Orthogonality isPreserved We can now extend the diagram (1):• Constructing the LeftAdjoint• Using Projective f∗Duality• Wigner’s Theorem P(H) - P(K) ∩ ∩• Remarks• A Surprise:Surjectivity Comes for (2)Free!• Putting The PiecesTogether ? f→ - ?The Representation L(H) ⊥ L(K)Theorem Big Toy ModelsReducing The Value f∗ Workshop on Informatic Penomena 2009 – 22
  • Using Projective DualityBig Toy Models Workshop on Informatic Penomena 2009 – 23
  • Using Projective Duality 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 of projective lattices and projective geometries.Big Toy Models Workshop on Informatic Penomena 2009 – 23
  • Using Projective Duality 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 of projective lattices and projective geometries. Proposition 11 f∗ is a total map of projective geometries.Big Toy Models Workshop on Informatic Penomena 2009 – 23
  • Using Projective Duality 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 of projective 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).Big Toy Models Workshop on Informatic Penomena 2009 – 23
  • Using Projective Duality 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 of projective 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).Big Toy Models Workshop on Informatic Penomena 2009 – 23
  • Using Projective Duality 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 of projective 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.Big Toy Models Workshop on Informatic Penomena 2009 – 23
  • Using Projective Duality 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 of projective 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!Big Toy Models Workshop on Informatic Penomena 2009 – 23
  • Wigner’s TheoremIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 24
  • Wigner’s TheoremIntroduction Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 byChu SpacesRepresenting PhysicalSystems ¯ P(g)(ψ) = g(ψ).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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 24
  • Wigner’s TheoremIntroduction Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 byChu SpacesRepresenting PhysicalSystems ¯ P(g)(ψ) = g(ψ).Characterizing ChuMorphisms onQuantum Chu Spaces We can now state Wigner’s Theorem in the form we shall use it.• Overview• Biextensionaity Theorem 12 Let f : P(H) → P(K) be a total map of projective• Projectivity =Biextensionality geometries, where dim H > 2. If f preserves orthogonality, meaning• Characterizing ChuMorphisms that• InjectivityAssumption ¯ ¯ ¯ ¯ φ ⊥ ψ ⇒ f (φ) ⊥ f (ψ)• Orthogonality isPreserved• Constructing the Left then there is a semilinear map g : H → K such that P(g) = f , andAdjoint• Using ProjectiveDuality g(φ) | g(ψ) = σ( φ | ψ ),• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes for where σ is the homomorphism associated with g . Moreover, thisFree!• Putting The Pieces homomorphism is either the identity or complex conjugation, so g is eitherTogether linear or antilinear. The map g is unique up to a phase, i.e. a scalar ofThe RepresentationTheorem modulus 1. Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 24
  • RemarksIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  • RemarksIntroductionChu Spaces • Note that in our case, taking f∗ = f , Pg is just the action of theRepresenting Physical biextensional collapse functor on Chu morphisms.SystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  • RemarksIntroductionChu Spaces • Note that in our case, taking f∗ = f , Pg is just the action of theRepresenting Physical biextensional collapse functor on Chu morphisms.SystemsCharacterizing ChuMorphisms on • Note that a total map of projective geometries must necessarilyQuantum Chu Spaces come from an injective map g on the underlying vector spaces,• Overview• Biextensionaity since P(g) maps rays to rays, and hence g must have trivial kernel.• Projectivity =Biextensionality• Characterizing ChuMorphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  • RemarksIntroductionChu Spaces • Note that in our case, taking f∗ = f , Pg is just the action of theRepresenting Physical biextensional collapse functor on Chu morphisms.SystemsCharacterizing ChuMorphisms on • Note that a total map of projective geometries must necessarilyQuantum Chu Spaces come from an injective map g on the underlying vector spaces,• Overview• Biextensionaity since P(g) maps rays to rays, and hence g must have trivial kernel.• Projectivity =Biextensionality• Characterizing Chu • For this reason, partial maps of projective geometries areMorphisms• Injectivity ¨ considered in the Faure-Frolicher approach. However, we areAssumption• Orthogonality is simply following the ‘logic’ of Chu space morphisms here.Preserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 25
  • A Surprise: Surjectivity Comes for Free!Big Toy Models Workshop on Informatic Penomena 2009 – 26
  • A Surprise: Surjectivity Comes for Free! 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 an automorphism. Then g is surjective.Big Toy Models Workshop on Informatic Penomena 2009 – 26
  • A Surprise: Surjectivity Comes for Free! 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 an automorphism. 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).Big Toy Models Workshop on Informatic Penomena 2009 – 26
  • A Surprise: Surjectivity Comes for Free! 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 an automorphism. 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 .Big Toy Models Workshop on Informatic Penomena 2009 – 26
  • A Surprise: Surjectivity Comes for Free! 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 an automorphism. 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 + χ where χ is a non-zero vector in Im g , which must exist since g is injective and H has positive dimension. This vector is not in Im g , nor is it orthogonal to Im g , since e.g. ψ + χ | χ = χ | χ = 0. This yields the required contradiction.Big Toy Models Workshop on Informatic Penomena 2009 – 26
  • Putting The Pieces TogetherIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing 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 PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 27
  • Putting The Pieces TogetherIntroduction We say that a map U : H → K is semiunitary if it is either unitary orChu Spaces antiunitary; that is, if it is a bijective map satisfyingRepresenting PhysicalSystemsCharacterizing Chu U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, U φ | U ψ = σ( φ | ψ )Morphisms onQuantum Chu Spaces• Overview where σ is the identity if U is unitary, and complex conjugation if U is• Biextensionaity• Projectivity = antiunitary. Note that semiunitaries preserve norm, so if U and V areBiextensionality• Characterizing Chu semiunitaries and U = λV , then |λ| = 1.Morphisms• InjectivityAssumption• Orthogonality isPreserved• Constructing the LeftAdjoint• Using ProjectiveDuality• Wigner’s Theorem• Remarks• A Surprise:Surjectivity Comes forFree!• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 27
  • Putting The Pieces TogetherIntroduction We say that a map U : H → K is semiunitary if it is either unitary orChu Spaces antiunitary; that is, if it is a bijective map satisfyingRepresenting PhysicalSystemsCharacterizing Chu U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, U φ | U ψ = σ( φ | ψ )Morphisms onQuantum Chu Spaces• Overview where σ is the identity if U is unitary, and complex conjugation if U is• Biextensionaity• Projectivity = antiunitary. Note that semiunitaries preserve norm, so if U and V areBiextensionality• Characterizing Chu semiunitaries and U = λV , then |λ| = 1.Morphisms• InjectivityAssumption• Orthogonality is Theorem 14 Let H, K be Hilbert spaces of dimension greater than 2.Preserved• Constructing the Left Consider a Chu morphismAdjoint (f∗ , f ∗ ) : (P(H), L(H), eH) → (P(K), L(K), eK ).• Using ProjectiveDuality ¯ ¯• Wigner’s Theorem• Remarks• A Surprise: where f∗ is injective. Then there is a semiunitary U : H → K such thatSurjectivity Comes forFree! f∗ = P(U ). U is unique up to a phase.• Putting The PiecesTogetherThe RepresentationTheorem Big Toy ModelsReducing The Value Workshop on Informatic Penomena 2009 – 27
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheorem• The Big Picture• Remarks The Representation Theorem• Functors• Not Quite Right Yet• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy ModelsBasic Concepts
  • The Big PictureBig Toy Models Workshop on Informatic Penomena 2009 – 29
  • The Big Picture We define a category SymmH as follows:Big Toy Models Workshop on Informatic Penomena 2009 – 29
  • The Big Picture We define a category SymmH as follows: • The objects are Hilbert spaces of dimension > 2.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  • The Big Picture 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.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  • The Big Picture 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 two unitaries compose to form a unitary, while a unitary and an antiunitary compose to form an antiunitary.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  • The Big Picture 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 two unitaries 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.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  • The Big Picture 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 two unitaries 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 space from 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.Big Toy Models Workshop on Informatic Penomena 2009 – 29
  • RemarksBig Toy Models Workshop on Informatic Penomena 2009 – 30
  • Remarks • 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.Big Toy Models Workshop on Informatic Penomena 2009 – 30
  • Remarks • 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.Big Toy Models Workshop on Informatic Penomena 2009 – 30
  • Remarks • 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.Big Toy Models Workshop on Informatic Penomena 2009 – 30
  • FunctorsIntroduction We define a functor R : SymmH → eChu[0,1] :Chu Spaces R : U : H → K −→ (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK )Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces where U◦ is the restriction of U to H◦ .The Representation As noted in Proposition 2, the inclusion bChu[0,1] ⊂ - eChu[0,1] hasTheorem• The Big Picture a left adjoint, which we name Q. By Proposition 4, this can be defined on• Remarks the image of R as follows:• Functors• Not Quite Right Yet• Biextensionality andScalar Factors Q : (H◦ , L(H), eH ) → (PH, L(H), eH ) ¯• Projectivising TheSymmetry Groupoid• Jes’ Right and for (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK ),• PR is anembedding up to aphase Q : (U◦ , U −1 ) −→ (PU, U −1 ).Reducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 31Basic Concepts
  • Not Quite Right YetIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheorem• The Big Picture• Remarks• Functors• Not Quite Right Yet• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 32Basic Concepts
  • Not Quite Right YetIntroduction We write emChu, bmChu for the subcategories of eChu[0,1] andChu Spaces bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ isRepresenting PhysicalSystems injective. The functors R and Q factor through these subcategories.Characterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheorem• The Big Picture• Remarks• Functors• Not Quite Right Yet• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 32Basic Concepts
  • Not Quite Right YetIntroduction We write emChu, bmChu for the subcategories of eChu[0,1] andChu Spaces bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ isRepresenting PhysicalSystems injective. The functors R and Q factor through these subcategories.Characterizing ChuMorphisms onQuantum Chu Spaces Proposition 15 BothThe RepresentationTheorem• The Big Picture R : SymmH → emChu• Remarks• Functors• Not Quite Right Yet and• Biextensionality andScalar Factors Q : emChu → bmChu• Projectivising TheSymmetry Groupoid are well-defined functors. R is faithful but not full; Q is full but not faithful.• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 32Basic Concepts
  • Not Quite Right YetIntroduction We write emChu, bmChu for the subcategories of eChu[0,1] andChu Spaces bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ isRepresenting PhysicalSystems injective. The functors R and Q factor through these subcategories.Characterizing ChuMorphisms onQuantum Chu Spaces Proposition 15 BothThe RepresentationTheorem• The Big Picture R : SymmH → emChu• Remarks• Functors• Not Quite Right Yet and• Biextensionality andScalar Factors Q : emChu → bmChu• Projectivising TheSymmetry Groupoid are well-defined functors. R is faithful but not full; Q is full but not faithful.• Jes’ Right• PR is anembedding up to aphase This involves verifying that unitaries and antiunitaries U : H → K doReducing The Value indeed yield Chu morphisms!SetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 32Basic Concepts
  • Not Quite Right YetIntroduction We write emChu, bmChu for the subcategories of eChu[0,1] andChu Spaces bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ isRepresenting PhysicalSystems injective. The functors R and Q factor through these subcategories.Characterizing ChuMorphisms onQuantum Chu Spaces Proposition 15 BothThe RepresentationTheorem• The Big Picture R : SymmH → emChu• Remarks• Functors• Not Quite Right Yet and• Biextensionality andScalar Factors Q : emChu → bmChu• Projectivising TheSymmetry Groupoid are well-defined functors. R is faithful but not full; Q is full but not faithful.• Jes’ Right• PR is anembedding up to aphase This involves verifying that unitaries and antiunitaries U : H → K doReducing The Value indeed yield Chu morphisms!Set The key property, for ψ ∈ H◦ and S ∈ L(H), is:DiscussionChu Spaces andCoalgebras PS (U ψ) = U (PU −1 (S) ψ).Primer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 32Basic Concepts
  • Biextensionality and Scalar FactorsIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheorem• The Big Picture• Remarks• Functors• Not Quite Right Yet• Biextensionality andScalar Factors• Projectivising TheSymmetry Groupoid• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 33Basic Concepts
  • Biextensionality and Scalar FactorsIntroduction We can analyze the non-fullness of R more precisely as follows.Chu SpacesRepresenting PhysicalSystems Proposition 16 Let (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK )Characterizing Chu be a Chu morphism in the image of R. Given an arbitrary functionMorphisms onQuantum Chu Spaces f : H → C {0}, define f U : H◦ → K◦ by:The RepresentationTheorem• The Big Picture f U (ψ) = f (ψ)U (ψ).• Remarks• Functors Then (f U, U −1 ) ∼ (U◦ , U −1 ). Moreover, the ∼-equivalence class of U• Not Quite Right Yet• Biextensionality and is exactly the set of functions of this form.Scalar Factors• Projectivising TheSymmetry Groupoid• Jes’ Right• PR is anembedding up to aphaseReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 33Basic Concepts
  • Biextensionality and Scalar FactorsIntroduction We can analyze the non-fullness of R more precisely as follows.Chu SpacesRepresenting PhysicalSystems Proposition 16 Let (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK )Characterizing Chu be a Chu morphism in the image of R. Given an arbitrary functionMorphisms onQuantum Chu Spaces f : H → C {0}, define f U : H◦ → K◦ by:The RepresentationTheorem• The Big Picture f U (ψ) = f (ψ)U (ψ).• Remarks• Functors Then (f U, U −1 ) ∼ (U◦ , U −1 ). Moreover, the ∼-equivalence class of U• Not Quite Right Yet• Biextensionality and is exactly the set of functions of this form.Scalar Factors• Projectivising TheSymmetry Groupoid Thus before biextensional collapse, Chu morphisms can introduce• Jes’ Right• PR is an arbitrary scalar factors pointwise. Once we move to the biextensionalembedding up to aphase category, we know by Theorem 14 that our representation will be full, andReducing The Value essentially faithful — up to a global phase. This points to the need for aSetDiscussion projective version of the symmetry groupoid.Chu Spaces andCoalgebrasPrimer on coalgebra Big Toy Models Workshop on Informatic Penomena 2009 – 33Basic Concepts
  • Projectivising The Symmetry GroupoidBig Toy Models Workshop on Informatic Penomena 2009 – 34
  • Projectivising The Symmetry Groupoid 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.Big Toy Models Workshop on Informatic Penomena 2009 – 34
  • Projectivising The Symmetry Groupoid 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 ) → λU. Moreover, this is a category action, meaning that (λU ) ◦ V = U ◦ (λV ) = λ(U ◦ V ).Big Toy Models Workshop on Informatic Penomena 2009 – 34
  • Projectivising The Symmetry Groupoid 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 ) → 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.Big Toy Models Workshop on Informatic Penomena 2009 – 34
  • Jes’ RightBig Toy Models Workshop on Informatic Penomena 2009 – 35
  • Jes’ Right We call the resulting category PSymmH, the projective quantum symmetry groupoid. It is a natural generalization of the standard notion of the projective unitary group on Hilbert space.Big Toy Models Workshop on Informatic Penomena 2009 – 35
  • Jes’ Right We call the resulting category PSymmH, the projective quantum symmetry groupoid. It is a natural generalization of the standard notion of the projective unitary 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.Big Toy Models Workshop on Informatic Penomena 2009 – 35
  • Jes’ Right We call the resulting category PSymmH, the projective quantum symmetry groupoid. It is a natural generalization of the standard notion of the projective unitary 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: R SymmH > > emChu P Q ∨ ∨ ∨ ∨ PSymmH > > bmChu > PRBig Toy Models Workshop on Informatic Penomena 2009 – 35
  • Jes’ Right We call the resulting category PSymmH, the projective quantum symmetry groupoid. It is a natural generalization of the standard notion of the projective unitary 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: R SymmH > > emChu P Q ∨ ∨ ∨ ∨ PSymmH > > bmChu > PR Theorem 17 The functor PR : PSymmH → bmChu is full and faithful.Big Toy Models Workshop on Informatic Penomena 2009 – 35
  • PR is an embedding up to a phaseBig Toy Models Workshop on Informatic Penomena 2009 – 36
  • PR is an embedding up to a phase • To see that PR is essentially injective on objects, we can use the ` representation theorems of Piron and Soler, which tell us that we can reconstruct H as a Hilbert space from L(H).Big Toy Models Workshop on Informatic Penomena 2009 – 36
  • PR is an embedding up to a phase • To see that PR is essentially injective on objects, we can use the ` representation 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′ ). =Big Toy Models Workshop on Informatic Penomena 2009 – 36
  • PR is an embedding up to a phase • To see that PR is essentially injective on objects, we can use the ` representation 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. =Big Toy Models Workshop on Informatic Penomena 2009 – 36
  • PR is an embedding up to a phase • To see that PR is essentially injective on objects, we can use the ` representation 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 in PSymmH.Big Toy Models Workshop on Informatic Penomena 2009 – 36
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet Reducing The Value Set• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical
  • GeneralitiesIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 38
  • GeneralitiesIntroduction We now return to the issue of whether it is necessary to use the full unitChu Spaces interval as the value set for our Chu spaces.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 38
  • GeneralitiesIntroduction We now return to the issue of whether it is necessary to use the full unitChu Spaces interval as the value set for our Chu spaces.Representing PhysicalSystemsCharacterizing Chu We begin with some generalities. Given a function v : K → L, we defineMorphisms onQuantum Chu Spaces a functor Fv : ChuK → ChuL :The RepresentationTheorem Fv : (X, A, e) → (X, A, v ◦ e)Reducing The ValueSet• Generalities and Fv f = f for Chu morphisms f .• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 38
  • GeneralitiesIntroduction We now return to the issue of whether it is necessary to use the full unitChu Spaces interval as the value set for our Chu spaces.Representing PhysicalSystemsCharacterizing Chu We begin with some generalities. Given a function v : K → L, we defineMorphisms onQuantum Chu Spaces a functor Fv : ChuK → ChuL :The RepresentationTheorem Fv : (X, A, e) → (X, A, v ◦ e)Reducing The ValueSet• Generalities and Fv f = f for Chu morphisms f .• The Question• Two Values?• The CanonicalPossibilistic Reductions Proposition 18 Fv is a faithful functor. If v is injective, it is full.• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 38
  • The QuestionBig Toy Models Workshop on Informatic Penomena 2009 – 39
  • The Question 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 of PR is full, and thus the composition Fv ◦ PR : PSymmH → bmChuK is a representation?Big Toy Models Workshop on Informatic Penomena 2009 – 39
  • The Question 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 of PR 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 show that it is, although not quite in the obvious fashion.Big Toy Models Workshop on Informatic Penomena 2009 – 39
  • Two Values?IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 40
  • Two Values?Introduction We shall write n = {0, . . . , n − 1} for the finite ordinals. The mostChu Spaces popular choice of value set for Chu spaces, by far, has been 2, andRepresenting PhysicalSystems indeed many interesting categories can be strictly (and even concretely)Characterizing ChuMorphisms on represented in Chu2 .Quantum Chu SpacesThe RepresentationTheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 40
  • Two Values?Introduction We shall write n = {0, . . . , n − 1} for the finite ordinals. The mostChu Spaces popular choice of value set for Chu spaces, by far, has been 2, andRepresenting PhysicalSystems indeed many interesting categories can be strictly (and even concretely)Characterizing ChuMorphisms on represented in Chu2 .Quantum Chu SpacesThe Representation This makes the following question natural:TheoremReducing The ValueSet Question 19 Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full• Generalities• The Question and faithful?• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 40
  • Two Values?Introduction We shall write n = {0, . . . , n − 1} for the finite ordinals. The mostChu Spaces popular choice of value set for Chu spaces, by far, has been 2, andRepresenting PhysicalSystems indeed many interesting categories can be strictly (and even concretely)Characterizing ChuMorphisms on represented in Chu2 .Quantum Chu SpacesThe Representation This makes the following question natural:TheoremReducing The ValueSet Question 19 Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full• Generalities• The Question and faithful?• Two Values?• The CanonicalPossibilistic Reductions What we can show is that the most plausible candidates for such• Two is Too Few• Other Case functions, yielding the two canonical forms of possibilistic semantics as• Analysis a coarse-graining of probabilistic semantics, both in fact fail.• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 40
  • The Canonical Possibilistic ReductionsIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 41
  • The Canonical Possibilistic ReductionsIntroduction Note that any function v : [0, 1] → {0, 1} must lose information either onChu Spaces 0 or on 1 — or both.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 41
  • The Canonical Possibilistic ReductionsIntroduction Note that any function v : [0, 1] → {0, 1} must lose information either onChu Spaces 0 or on 1 — or both.Representing PhysicalSystemsCharacterizing Chu In this sense, the two ‘sharpest’ mappings will be:Morphisms onQuantum Chu SpacesThe Representation v0 : 0 → 0, (0, 1] → 1 v1 : [0, 1) → 0, 1 → 1.TheoremReducing The ValueSet• Generalities• The Question• Two Values?• The CanonicalPossibilistic Reductions• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 41
  • The Canonical Possibilistic ReductionsIntroduction Note that any function v : [0, 1] → {0, 1} must lose information either onChu Spaces 0 or on 1 — or both.Representing PhysicalSystemsCharacterizing Chu In this sense, the two ‘sharpest’ mappings will be:Morphisms onQuantum Chu SpacesThe Representation v0 : 0 → 0, (0, 1] → 1 v1 : [0, 1) → 0, 1 → 1.TheoremReducing The ValueSet These are the two canonical reductions of probabilistic to possibilistic• Generalities information: the first maps ‘definitely not’ to ‘no’, and anything else to• The Question• Two Values? ‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitely• The CanonicalPossibilistic Reductions yes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’.• Two is Too Few• Other Case• Analysis• Three ValuesSuffice!DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 41
  • The Canonical Possibilistic ReductionsIntroduction Note that any function v : [0, 1] → {0, 1} must lose information either onChu Spaces 0 or on 1 — or both.Representing PhysicalSystemsCharacterizing Chu In this sense, the two ‘sharpest’ mappings will be:Morphisms onQuantum Chu SpacesThe Representation v0 : 0 → 0, (0, 1] → 1 v1 : [0, 1) → 0, 1 → 1.TheoremReducing The ValueSet These are the two canonical reductions of probabilistic to possibilistic• Generalities information: the first maps ‘definitely not’ to ‘no’, and anything else to• The Question• Two Values? ‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitely• The CanonicalPossibilistic Reductions yes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’.• Two is Too Few• Other Case Note that, under the first of these, we no longer have• Analysis• Three ValuesSuffice! eH (ψ, S) = 1 ⇐⇒ ψ ∈ SDiscussionChu Spaces andCoalgebras while under the second, we no longer havePrimer on coalgebraBasic Concepts eH (ψ, S) = 0 ⇐⇒ ψ ⊥ S. Big Toy ModelsRepresenting Physical Workshop on Informatic Penomena 2009 – 41
  • Two is Too Few Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full.Big Toy Models Workshop on Informatic Penomena 2009 – 42
  • Two is Too Few Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full. Let H be a Hilbert space with 2 < dim H < ∞, 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 be continuous.) 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 .Big Toy Models Workshop on Informatic Penomena 2009 – 42
  • Two is Too Few Proposition 20 For neither v = v0 nor v = v1 is Fv ◦ PR full. Let H be a Hilbert space with 2 < dim H < ∞, 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 be continuous.) 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.Big Toy Models Workshop on Informatic Penomena 2009 – 42
  • Other Case 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.Big Toy Models Workshop on Informatic Penomena 2009 – 43
  • Other Case 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 ⊥ ).Big Toy Models Workshop on Informatic Penomena 2009 – 43
  • Other Case 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.Big Toy Models Workshop on Informatic Penomena 2009 – 43
  • AnalysisBig Toy Models Workshop on Informatic Penomena 2009 – 44
  • Analysis However, this negative result immediately suggests a remedy: to keep the interpretations of both 0 and 1 sharp. We can do this with three values! Namely: v : 0 → 0, (0, 1) → 2, 1→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.Big Toy Models Workshop on Informatic Penomena 2009 – 44
  • Analysis However, this negative result immediately suggests a remedy: to keep the interpretations of both 0 and 1 sharp. We can do this with three values! Namely: v : 0 → 0, (0, 1) → 2, 1→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:Big Toy Models Workshop on Informatic Penomena 2009 – 44
  • Analysis However, this negative result immediately suggests a remedy: to keep the interpretations of both 0 and 1 sharp. We can do this with three values! Namely: v : 0 → 0, (0, 1) → 2, 1→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 = 0, so eH (φ, S) = 0Big Toy Models Workshop on Informatic Penomena 2009 – 44
  • Analysis However, this negative result immediately suggests a remedy: to keep the interpretations of both 0 and 1 sharp. We can do this with three values! Namely: v : 0 → 0, (0, 1) → 2, 1→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 = 0, so eH (φ, S) = 0 • θ = 0 and χ = 0, so eH (φ, S) = 1Big Toy Models Workshop on Informatic Penomena 2009 – 44
  • Analysis However, this negative result immediately suggests a remedy: to keep the interpretations of both 0 and 1 sharp. We can do this with three values! Namely: v : 0 → 0, (0, 1) → 2, 1→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 = 0, so eH (φ, S) = 0 • θ = 0 and χ = 0, so eH (φ, S) = 1 • θ = 0 and χ = 0, so eH (ψ, S) ∈ (0, 1), and hence v ◦ eH (ψ, S) = 2.Big Toy Models Workshop on Informatic Penomena 2009 – 44
  • Analysis However, this negative result immediately suggests a remedy: to keep the interpretations of both 0 and 1 sharp. We can do this with three values! Namely: v : 0 → 0, (0, 1) → 2, 1→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 = 0, so eH (φ, S) = 0 • θ = 0 and χ = 0, so eH (φ, S) = 1 • θ = 0 and χ = 0, so eH (ψ, S) ∈ (0, 1), and hence v ◦ eH (ψ, S) = 2. These are the only case discriminations which are used in proving the Representation Theorem.Big Toy Models Workshop on Informatic Penomena 2009 – 44
  • Three Values Suffice!Big Toy Models Workshop on Informatic Penomena 2009 – 45
  • Three Values Suffice! Hence we have: Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is a representation.Big Toy Models Workshop on Informatic Penomena 2009 – 45
  • Three Values Suffice! Hence we have: Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is a representation. 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’.Big Toy Models Workshop on Informatic Penomena 2009 – 45
  • Three Values Suffice! Hence we have: Theorem 21 The functor Fv ◦ PR : PSymmH → bmChu3 is a representation. 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.Big Toy Models Workshop on Informatic Penomena 2009 – 45
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet DiscussionDiscussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models
  • Where Next?IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47
  • Where Next?IntroductionChu Spaces • Connections and contrasts between Chu spaces and coalgebras.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47
  • Where Next?IntroductionChu Spaces • Connections and contrasts between Chu spaces and coalgebras.Representing PhysicalSystems • Mixed states — handled generally at the level of Chu spaces.Characterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47
  • Where Next?IntroductionChu Spaces • Connections and contrasts between Chu spaces and coalgebras.Representing PhysicalSystems • Mixed states — handled generally at the level of Chu spaces.Characterizing ChuMorphisms onQuantum Chu Spaces • Universal Chu spaces.The RepresentationTheoremReducing The ValueSetDiscussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47
  • Where Next?IntroductionChu Spaces • Connections and contrasts between Chu spaces and coalgebras.Representing PhysicalSystems • Mixed states — handled generally at the level of Chu spaces.Characterizing ChuMorphisms onQuantum Chu Spaces • Universal Chu spaces.The RepresentationTheorem • Linear and other type theories.Reducing The ValueSetDiscussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47
  • Where Next?IntroductionChu Spaces • Connections and contrasts between Chu spaces and coalgebras.Representing PhysicalSystems • Mixed states — handled generally at the level of Chu spaces.Characterizing ChuMorphisms onQuantum Chu Spaces • Universal Chu spaces.The RepresentationTheorem • Linear and other type theories.Reducing The ValueSet • Local logics.Discussion• Where Next?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet Chu Spaces and CoalgebrasDiscussionChu Spaces andCoalgebras• Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalising Big Toy ModelsContravariance As
  • Chu Spaces and CoalgebrasIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebras• Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalising Big Toy Models Workshop on Informatic Penomena 2009 – 49Contravariance As
  • Chu Spaces and CoalgebrasIntroductionChu Spaces • Coalgebras over Set; ‘universal coalgebra’.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebras• Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalising Big Toy Models Workshop on Informatic Penomena 2009 – 49Contravariance As
  • Chu Spaces and CoalgebrasIntroductionChu Spaces • Coalgebras over Set; ‘universal coalgebra’.Representing PhysicalSystems • Each of these general systems models has been studiedCharacterizing ChuMorphisms on extensively.Quantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebras• Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalising Big Toy Models Workshop on Informatic Penomena 2009 – 49Contravariance As
  • Chu Spaces and CoalgebrasIntroductionChu Spaces • Coalgebras over Set; ‘universal coalgebra’.Representing PhysicalSystems • Each of these general systems models has been studiedCharacterizing ChuMorphisms on extensively.Quantum Chu Spaces Their connections have not been studied at all.The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebras• Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalising Big Toy Models Workshop on Informatic Penomena 2009 – 49Contravariance As
  • Chu Spaces and CoalgebrasIntroductionChu Spaces • Coalgebras over Set; ‘universal coalgebra’.Representing PhysicalSystems • Each of these general systems models has been studiedCharacterizing ChuMorphisms on extensively.Quantum Chu Spaces Their connections have not been studied at all.The RepresentationTheoremReducing The Value • They have complementary merits and deficiencies.SetDiscussion • Chu spaces have, coalgebras lack: contravariance.Chu Spaces andCoalgebras • Coalgebras have, Chu spaces lack: extension in time.• Chu Spaces andCoalgebras • Symmetry vs. rigidity.Primer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalising Big Toy Models Workshop on Informatic Penomena 2009 – 49Contravariance As
  • Chu Spaces and CoalgebrasIntroductionChu Spaces • Coalgebras over Set; ‘universal coalgebra’.Representing PhysicalSystems • Each of these general systems models has been studiedCharacterizing ChuMorphisms on extensively.Quantum Chu Spaces Their connections have not been studied at all.The RepresentationTheoremReducing The Value • They have complementary merits and deficiencies.SetDiscussion • Chu spaces have, coalgebras lack: contravariance.Chu Spaces andCoalgebras • Coalgebras have, Chu spaces lack: extension in time.• Chu Spaces andCoalgebras • Symmetry vs. rigidity.Primer on coalgebraBasic Concepts • Interesting formal consequences:Representing PhysicalSystems AsCoalgebras • Indexed structure (‘externalising contravariance’)Comparison: A FirstTry • Grothendieck construction: new description of Chu spaces.Semantics in OneCountry • Truncation functors.Externalising Big Toy Models Workshop on Informatic Penomena 2009 – 49Contravariance As
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet Primer on coalgebraDiscussionChu Spaces andCoalgebrasPrimer on coalgebra• CoalgebrasBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models
  • CoalgebrasIntroduction Category theory allows us to dualize algebras to obtain a notion ofChu Spaces coalgebras of an endofunctor. However, while algebras abstract aRepresenting PhysicalSystems familiar set of notions, coalgebras open up a new and rather unexpectedCharacterizing Chu territory, and provides an effective abstraction and mathematical theoryMorphisms onQuantum Chu Spaces for a central class of computational phenomena:The RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebra• CoalgebrasBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51
  • CoalgebrasIntroduction Category theory allows us to dualize algebras to obtain a notion ofChu Spaces coalgebras of an endofunctor. However, while algebras abstract aRepresenting PhysicalSystems familiar set of notions, coalgebras open up a new and rather unexpectedCharacterizing Chu territory, and provides an effective abstraction and mathematical theoryMorphisms onQuantum Chu Spaces for a central class of computational phenomena:The RepresentationTheoremReducing The Value • Programming over infinite data structures: streams, lazy lists,Set infinite trees . . .DiscussionChu Spaces andCoalgebrasPrimer on coalgebra• CoalgebrasBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51
  • CoalgebrasIntroduction Category theory allows us to dualize algebras to obtain a notion ofChu Spaces coalgebras of an endofunctor. However, while algebras abstract aRepresenting PhysicalSystems familiar set of notions, coalgebras open up a new and rather unexpectedCharacterizing Chu territory, and provides an effective abstraction and mathematical theoryMorphisms onQuantum Chu Spaces for a central class of computational phenomena:The RepresentationTheoremReducing The Value • Programming over infinite data structures: streams, lazy lists,Set infinite trees . . .DiscussionChu Spaces andCoalgebras • A novel notion of coinductionPrimer on coalgebra• CoalgebrasBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51
  • CoalgebrasIntroduction Category theory allows us to dualize algebras to obtain a notion ofChu Spaces coalgebras of an endofunctor. However, while algebras abstract aRepresenting PhysicalSystems familiar set of notions, coalgebras open up a new and rather unexpectedCharacterizing Chu territory, and provides an effective abstraction and mathematical theoryMorphisms onQuantum Chu Spaces for a central class of computational phenomena:The RepresentationTheoremReducing The Value • Programming over infinite data structures: streams, lazy lists,Set infinite trees . . .DiscussionChu Spaces andCoalgebras • A novel notion of coinductionPrimer on coalgebra• Coalgebras • Modelling state-based computations of all kindsBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51
  • CoalgebrasIntroduction Category theory allows us to dualize algebras to obtain a notion ofChu Spaces coalgebras of an endofunctor. However, while algebras abstract aRepresenting PhysicalSystems familiar set of notions, coalgebras open up a new and rather unexpectedCharacterizing Chu territory, and provides an effective abstraction and mathematical theoryMorphisms onQuantum Chu Spaces for a central class of computational phenomena:The RepresentationTheoremReducing The Value • Programming over infinite data structures: streams, lazy lists,Set infinite trees . . .DiscussionChu Spaces andCoalgebras • A novel notion of coinductionPrimer on coalgebra• Coalgebras • Modelling state-based computations of all kindsBasic ConceptsRepresenting Physical • The key notion of bisimulation equivalence between processes.Systems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51
  • CoalgebrasIntroduction Category theory allows us to dualize algebras to obtain a notion ofChu Spaces coalgebras of an endofunctor. However, while algebras abstract aRepresenting PhysicalSystems familiar set of notions, coalgebras open up a new and rather unexpectedCharacterizing Chu territory, and provides an effective abstraction and mathematical theoryMorphisms onQuantum Chu Spaces for a central class of computational phenomena:The RepresentationTheoremReducing The Value • Programming over infinite data structures: streams, lazy lists,Set infinite trees . . .DiscussionChu Spaces andCoalgebras • A novel notion of coinductionPrimer on coalgebra• Coalgebras • Modelling state-based computations of all kindsBasic ConceptsRepresenting Physical • The key notion of bisimulation equivalence between processes.Systems AsCoalgebras • A general coalgebraic logic can be read off from the functor, andComparison: A FirstTry used to specify and reason about properties of systems.Semantics in OneCountryExternalisingContravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet Basic ConceptsDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts• F -Coalgebras• Final F -coalgebras• Labelled TransitionSystems• Transition Graphs asCoalgebras• The Final CoalgebraRepresenting PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First
  • F -CoalgebrasIntroduction Let F : C −→ C be a functor.Chu Spaces An F -coalgebra is a pair (A, γ : A −→ F A) for some object A of C.Representing PhysicalSystems We say that A is the carrier of the coalgebra, while γ is the behaviourCharacterizing ChuMorphisms on map.Quantum Chu Spaces An F -coalgebra homomorphism from (A, γ : A −→ F A) toThe RepresentationTheorem (B, δ : B −→ F B) is an arrow h : A −→ B such thatReducing The ValueSet γ-Discussion A FAChu Spaces andCoalgebrasPrimer on coalgebra h FhBasic Concepts ? ?• F -Coalgebras B - FB• Final F -coalgebras• Labelled Transition δSystems• Transition Graphs asCoalgebras F -coalgebras and their homomorphisms form a category F −Coalg.• The Final CoalgebraRepresenting PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 53
  • Final F -coalgebrasIntroduction An F -coalgebra (C, γ) is final if for every F -coalgebra (A, α) there is aChu Spaces unique homomorphism from (A, α) to (C, γ).Representing PhysicalSystemsCharacterizing Chu Proposition 22 If a final F -coalgebra exists, it is unique up toMorphisms onQuantum Chu Spaces isomorphism.The RepresentationTheoremReducing The ValueSet Proposition 23 (Lambek Lemma) If γ : A −→ F A is final, it is anDiscussion isomorphismChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts• F -Coalgebras• Final F -coalgebras• Labelled TransitionSystems• Transition Graphs asCoalgebras• The Final CoalgebraRepresenting PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 54
  • Labelled Transition SystemsIntroduction Let A be a set of actions. A labelled transition system over A is aChu Spaces coalgebra for the functorRepresenting PhysicalSystemsCharacterizing Chu LTA : Set −→ Set :: X → Pf (A × X).Morphisms onQuantum Chu SpacesThe Representation Such a coalgebraTheoremReducing The Value γ : S −→ Pf (A × S)SetDiscussion can be understood operationally as follows:Chu Spaces andCoalgebras • S is a set of statesPrimer on coalgebraBasic Concepts • For each state s ∈ S , γ(s) specifies the possible transitions from a• F -Coalgebras• Final F -coalgebras that state. We write s −→ s′ if (a, s′ ) ∈ γ(s). We think of such a• Labelled Transition transition as consisting of the system performing the action a, andSystems• Transition Graphs as changing state from s to s′ . Note that we regard actions as directlyCoalgebras• The Final Coalgebra observable, while states are not.Representing PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 55
  • Transition Graphs as CoalgebrasIntroduction Note that any labelled transition graph (or “state machine”) with labels inChu Spaces A is a coalgebra for LTA .Representing PhysicalSystems Examples 1.Characterizing ChuMorphisms on a b cQuantum Chu Spaces 1 2The RepresentationTheoremReducing The Value This corresponds to the coalgebra ({1, 2}, γ)SetDiscussionChu Spaces and γ : 1 → {(a, 1), (b, 2)}, 2 → {(c, 2)}CoalgebrasPrimer on coalgebra 2.Basic Concepts c b• F -Coalgebras a• Final F -coalgebras 1 2 3• Labelled TransitionSystems a• Transition Graphs asCoalgebras• The Final Coalgebra 1 → {(b, 2), (c, 1)}, 2 → {(a, 1), (a, 3)}, 3→∅Representing PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 56
  • The Final CoalgebraIntroduction The key point is this.Chu SpacesRepresenting Physical Proposition 24 For any set A of actions, there is a final LTA -coalgebraSystemsCharacterizing Chu (ProcA , out).Morphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts• F -Coalgebras• Final F -coalgebras• Labelled TransitionSystems• Transition Graphs asCoalgebras• The Final CoalgebraRepresenting PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 57
  • The Final CoalgebraIntroduction The key point is this.Chu SpacesRepresenting Physical Proposition 24 For any set A of actions, there is a final LTA -coalgebraSystemsCharacterizing Chu (ProcA , out).Morphisms onQuantum Chu Spaces We think of elements of the final coalgebra as processes. The finalThe RepresentationTheorem coalgebra provides a “universal semantics” for transition systems, whichReducing The ValueSet is “fully abstract” with respect to observable behaviour.DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic Concepts• F -Coalgebras• Final F -coalgebras• Labelled TransitionSystems• Transition Graphs asCoalgebras• The Final CoalgebraRepresenting PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 57
  • The Final CoalgebraIntroduction The key point is this.Chu SpacesRepresenting Physical Proposition 24 For any set A of actions, there is a final LTA -coalgebraSystemsCharacterizing Chu (ProcA , out).Morphisms onQuantum Chu Spaces We think of elements of the final coalgebra as processes. The finalThe RepresentationTheorem coalgebra provides a “universal semantics” for transition systems, whichReducing The ValueSet is “fully abstract” with respect to observable behaviour.DiscussionChu Spaces and All of this generalizes to a large class of endofunctors.CoalgebrasPrimer on coalgebraBasic Concepts• F -Coalgebras• Final F -coalgebras• Labelled TransitionSystems• Transition Graphs asCoalgebras• The Final CoalgebraRepresenting PhysicalSystems AsCoalgebras Big Toy ModelsComparison: A First Workshop on Informatic Penomena 2009 – 57
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The Value Representing Physical SystemsSetDiscussion As CoalgebrasChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebras• Coalgebras asModels of PhysicalSystemsComparison: A FirstTrySemantics in OneCountry Big Toy ModelsExternalising
  • Coalgebras as Models of Physical SystemsIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebras• Coalgebras asModels of PhysicalSystemsComparison: A FirstTrySemantics in OneCountry Big Toy Models Workshop on Informatic Penomena 2009 – 59Externalising
  • Coalgebras as Models of Physical SystemsIntroduction Recall our basic setup: systems are modelled by a set of states S , ofChu Spaces questions Q, and an evaluationRepresenting PhysicalSystemsCharacterizing Chu e : S × Q → [0, 1].Morphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebras• Coalgebras asModels of PhysicalSystemsComparison: A FirstTrySemantics in OneCountry Big Toy Models Workshop on Informatic Penomena 2009 – 59Externalising
  • Coalgebras as Models of Physical SystemsIntroduction Recall our basic setup: systems are modelled by a set of states S , ofChu Spaces questions Q, and an evaluationRepresenting PhysicalSystemsCharacterizing Chu e : S × Q → [0, 1].Morphisms onQuantum Chu SpacesThe Representation Problems:TheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebras• Coalgebras asModels of PhysicalSystemsComparison: A FirstTrySemantics in OneCountry Big Toy Models Workshop on Informatic Penomena 2009 – 59Externalising
  • Coalgebras as Models of Physical SystemsIntroduction Recall our basic setup: systems are modelled by a set of states S , ofChu Spaces questions Q, and an evaluationRepresenting PhysicalSystemsCharacterizing Chu e : S × Q → [0, 1].Morphisms onQuantum Chu SpacesThe Representation Problems:TheoremReducing The ValueSet • In Chu spaces, we get to specify Q as well as S . How do we doDiscussion this with coalgebras?Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebras• Coalgebras asModels of PhysicalSystemsComparison: A FirstTrySemantics in OneCountry Big Toy Models Workshop on Informatic Penomena 2009 – 59Externalising
  • Coalgebras as Models of Physical SystemsIntroduction Recall our basic setup: systems are modelled by a set of states S , ofChu Spaces questions Q, and an evaluationRepresenting PhysicalSystemsCharacterizing Chu e : S × Q → [0, 1].Morphisms onQuantum Chu SpacesThe Representation Problems:TheoremReducing The ValueSet • In Chu spaces, we get to specify Q as well as S . How do we doDiscussion this with coalgebras?Chu Spaces andCoalgebras • Q is contravariant (the maps f ∗ go backwards.). Coalgebras arePrimer on coalgebra based on covariant functors. (We could work with domains, butBasic ConceptsRepresenting Physical there are drawbacks).Systems AsCoalgebras• Coalgebras asModels of PhysicalSystemsComparison: A FirstTrySemantics in OneCountry Big Toy Models Workshop on Informatic Penomena 2009 – 59Externalising
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet Comparison: A First TryDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras
  • First Approximation Fix a set K . We can define a functor on Set: FK : X → K PX .Big Toy Models Workshop on Informatic Penomena 2009 – 61
  • First Approximation Fix a set K . We can define a functor on Set: FK : X → K PX . If we use the contravariant powerset functor, F will be covariant. Explicitly, for f :X →Y: FK f (g)(S) = g(f −1 (S)), where g ∈ K PX and S ∈ PY .Big Toy Models Workshop on Informatic Penomena 2009 – 61
  • First Approximation Fix a set K . We can define a functor on Set: FK : X → K PX . If we use the contravariant powerset functor, F will be covariant. Explicitly, for f :X →Y: FK f (g)(S) = g(f −1 (S)), where g ∈ K PX and S ∈ PY . A coalgebra for this functor will be a map of the form α : X → K PX .Big Toy Models Workshop on Informatic Penomena 2009 – 61
  • First Approximation Fix a set K . We can define a functor on Set: FK : X → K PX . If we use the contravariant powerset functor, F will be covariant. Explicitly, for f :X →Y: FK f (g)(S) = g(f −1 (S)), where g ∈ K PX and S ∈ PY . A coalgebra for this functor will be a map of the form α : X → K PX . 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).Big Toy Models Workshop on Informatic Penomena 2009 – 61
  • Comparison Proposition 25 Suppose we are given a Chu morphism f : C → C ′ , where C and C ′ are normal Chu spaces, such that f ∗ = f∗ . Then f∗ : GC → GC ′ is an −1 FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism f : GC → GC ′ , then (f, f −1 ) : C → C ′ is a Chu morphism.Big Toy Models Workshop on Informatic Penomena 2009 – 62
  • Comparison Proposition 25 Suppose we are given a Chu morphism f : C → C ′ , where C and C ′ are normal Chu spaces, such that f ∗ = f∗ . Then f∗ : GC → GC ′ is an −1 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 functor G : NChuK → FK −Coalg, with G(f, f −1 ) = f .Big Toy Models Workshop on Informatic Penomena 2009 – 62
  • Comparison Proposition 25 Suppose we are given a Chu morphism f : C → C ′ , where C and C ′ are normal Chu spaces, such that f ∗ = f∗ . Then f∗ : GC → GC ′ is an −1 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 functor G : NChuK → FK −Coalg, with G(f, f −1 ) = f . There is an evident inverse to this functor.Big Toy Models Workshop on Informatic Penomena 2009 – 62
  • Comparison Proposition 25 Suppose we are given a Chu morphism f : C → C ′ , where C and C ′ are normal Chu spaces, such that f ∗ = f∗ . Then f∗ : GC → GC ′ is an −1 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 functor G : 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.Big Toy Models Workshop on Informatic Penomena 2009 – 62
  • Discussion: Critique of CoalgebrasBig Toy Models Workshop on Informatic Penomena 2009 – 63
  • Discussion: Critique of Coalgebras • Assuming Chu spaces are normal is overly restrictive.Big Toy Models Workshop on Informatic Penomena 2009 – 63
  • Discussion: Critique of Coalgebras • 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.Big Toy Models Workshop on Informatic Penomena 2009 – 63
  • Discussion: Critique of Coalgebras • 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.Big Toy Models Workshop on Informatic Penomena 2009 – 63
  • Discussion: Critique of Coalgebras • 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.Big Toy Models Workshop on Informatic Penomena 2009 – 63
  • Discussion: In Praise of CoalgebrasIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 64
  • Discussion: In Praise of CoalgebrasIntroductionChu Spaces • The coalgebraic point of view can be described as state-based, butRepresenting Physical in a way that emphasizes that the meaning of states lies in theirSystems observable behaviour. Indeed, in the “universal model” we shallCharacterizing ChuMorphisms on construct, the states are determined exactly as the possibleQuantum Chu Spaces observable behaviours — we actually find a canonical solution forThe RepresentationTheorem what the state space should be in these terms. States areReducing The ValueSet identified exactly if they have the same observable behaviour.DiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 64
  • Discussion: In Praise of CoalgebrasIntroductionChu Spaces • The coalgebraic point of view can be described as state-based, butRepresenting Physical in a way that emphasizes that the meaning of states lies in theirSystems observable behaviour. Indeed, in the “universal model” we shallCharacterizing ChuMorphisms on construct, the states are determined exactly as the possibleQuantum Chu Spaces observable behaviours — we actually find a canonical solution forThe RepresentationTheorem what the state space should be in these terms. States areReducing The ValueSet identified exactly if they have the same observable behaviour.DiscussionChu Spaces and We can see this as a kind of reconciliation between the ontic andCoalgebras epistemic standpoints.Primer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 64
  • Discussion: In Praise of CoalgebrasIntroductionChu Spaces • The coalgebraic point of view can be described as state-based, butRepresenting Physical in a way that emphasizes that the meaning of states lies in theirSystems observable behaviour. Indeed, in the “universal model” we shallCharacterizing ChuMorphisms on construct, the states are determined exactly as the possibleQuantum Chu Spaces observable behaviours — we actually find a canonical solution forThe RepresentationTheorem what the state space should be in these terms. States areReducing The ValueSet identified exactly if they have the same observable behaviour.DiscussionChu Spaces and We can see this as a kind of reconciliation between the ontic andCoalgebras epistemic standpoints.Primer on coalgebraBasic Concepts • Coalgebras allow us to capture the ‘dynamics of measurement’ —Representing PhysicalSystems As what happens after a measurement — in a way that Chu spacesCoalgebrasComparison: A First don’t. They have ‘extension in time’.Try• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 64
  • Extension in TimeIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 65
  • Extension in TimeIntroduction Consider a coalgebraic representation of stochastic transducers:Chu Spaces F : X → Prob(O × X)IRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces where I is a fixed set of inputs, O a fixed set of outputs, and Prob(S) isThe Representation the set of probability distributions on S .TheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 65
  • Extension in TimeIntroduction Consider a coalgebraic representation of stochastic transducers:Chu Spaces F : X → Prob(O × X)IRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces where I is a fixed set of inputs, O a fixed set of outputs, and Prob(S) isThe Representation the set of probability distributions on S .TheoremReducing The ValueSet We can think of I as a set of questions, and O as a set of answersDiscussion (which we could standardize by only considering yes/no questions).Chu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTry• First Approximation• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 65
  • Extension in TimeIntroduction Consider a coalgebraic representation of stochastic transducers:Chu Spaces F : X → Prob(O × X)IRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu Spaces where I is a fixed set of inputs, O a fixed set of outputs, and Prob(S) isThe Representation the set of probability distributions on S .TheoremReducing The ValueSet We can think of I as a set of questions, and O as a set of answersDiscussion (which we could standardize by only considering yes/no questions).Chu Spaces andCoalgebras What we learn from this is thatPrimer on coalgebraBasic Concepts QM is less nondeterministic/probabilistic than stochastic transducersRepresenting PhysicalSystems AsCoalgebras since in QM if we know the preparation and the outcome of theComparison: A First measurement, we know (by the projection postulate) exactly what theTry• First Approximation resulting quantum state will be.• Comparison• Discussion: Critiqueof Coalgebras• Discussion: In Praise Big Toy Modelsof Coalgebras Workshop on Informatic Penomena 2009 – 65
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet Semantics in One CountryDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountry• CoalgebraicSemantics For OneSystem•Big Toy Models Well Behaved
  • Coalgebraic Semantics For One SystemBig Toy Models Workshop on Informatic Penomena 2009 – 67
  • Coalgebraic Semantics For One System We fix attention on a single Hilbert space H. This determines a set of question Q = L(H).Big Toy Models Workshop on Informatic Penomena 2009 – 67
  • Coalgebraic Semantics For One System 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: F Q : X → ({0} + (0, 1] × X)Q .Big Toy Models Workshop on Informatic Penomena 2009 – 67
  • Coalgebraic Semantics For One System 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: F Q : X → ({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.Big Toy Models Workshop on Informatic Penomena 2009 – 67
  • Well Behaved Functors Unlike the functors FK , the functors F Q 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 .Big Toy Models Workshop on Informatic Penomena 2009 – 68
  • Well Behaved Functors Unlike the functors FK , the functors F Q 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 labelled by probabilities.Big Toy Models Workshop on Informatic Penomena 2009 – 68
  • Representing One Quantum System As A CoalgebraBig Toy Models Workshop on Informatic Penomena 2009 – 69
  • Representing One Quantum System As A Coalgebra The F Q -coalgebra which is of primary interest to us is the map aH : H◦ → ({0} + (0, 1] × H◦ )Q defined by:   0, eH (ψ, S) = 0 aH (ψ)(S) =  (r, P ψ), e (ψ, S) = r > 0 S HBig Toy Models Workshop on Informatic Penomena 2009 – 69
  • Representing One Quantum System As A Coalgebra The F Q -coalgebra which is of primary interest to us is the map aH : H◦ → ({0} + (0, 1] × H◦ )Q defined by:   0, eH (ψ, S) = 0 aH (ψ)(S) =  (r, P ψ), e (ψ, S) = r > 0 S H The new ingredient compared with the Chu space representation of H is the state which results in the case of a ‘yes’ answer to the question, which is computed according to the Luders rule. ¨Big Toy Models Workshop on Informatic Penomena 2009 – 69
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The Value Externalising Contravariance AsSetDiscussion IndexingChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models• The Indexed
  • The Indexed CategoryIntroduction We define a functorChu Spaces F : Setop → CATRepresenting PhysicalSystems withCharacterizing ChuMorphisms onQuantum Chu Spaces Q → F Q −CoalgThe RepresentationTheorem and for f : Q′ → Q:Reducing The Value ′Set tf : F Q → F Q :: Θ → Θ ◦ fDiscussionChu Spaces andCoalgebras is a natural transformation, andPrimer on coalgebra ∗ Q Q′Basic Concepts F(f ) = f : Coalg−F → Coalg−FRepresenting Physical f ∗ : (A, α) → (A, tf ◦ α)Systems AsCoalgebras AComparison: A FirstTry is a functor.Semantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 71• The Indexed
  • The Indexed CategoryIntroduction We define a functorChu Spaces F : Setop → CATRepresenting PhysicalSystems withCharacterizing ChuMorphisms onQuantum Chu Spaces Q → F Q −CoalgThe RepresentationTheorem and for f : Q′ → Q:Reducing The Value ′Set tf : F Q → F Q :: Θ → Θ ◦ fDiscussionChu Spaces andCoalgebras is a natural transformation, andPrimer on coalgebra ∗ Q Q′Basic Concepts F(f ) = f : Coalg−F → Coalg−FRepresenting Physical f ∗ : (A, α) → (A, tf ◦ α)Systems AsCoalgebras AComparison: A FirstTry is a functor.Semantics in OneCountry Thus we get a strict indexed category of coalgebra categories, withExternalisingContravariance As contravariant indexing.Indexing Big Toy Models Workshop on Informatic Penomena 2009 – 71• The Indexed
  • The Grothendieck ConstructionIntroduction Where we have an indexed category, we can apply the GrothendieckChu Spaces construction to glue all the fibres together (and get a fibration).Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 72• The Indexed
  • The Grothendieck ConstructionIntroduction Where we have an indexed category, we can apply the GrothendieckChu Spaces construction to glue all the fibres together (and get a fibration).Representing PhysicalSystemsCharacterizing Chu Given a functorMorphisms onQuantum Chu Spaces I : Cop → CATThe RepresentationTheorem define I with objects (A, a), where A is an object of C and a is anReducing The Value object of I(A).SetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 72• The Indexed
  • The Grothendieck ConstructionIntroduction Where we have an indexed category, we can apply the GrothendieckChu Spaces construction to glue all the fibres together (and get a fibration).Representing PhysicalSystemsCharacterizing Chu Given a functorMorphisms onQuantum Chu Spaces I : Cop → CATThe RepresentationTheorem define I with objects (A, a), where A is an object of C and a is anReducing The Value object of I(A).SetDiscussion Arrows are (G, g) : (A, a) → (B, b), where G : B → A andChu Spaces andCoalgebras g : I(G)(a) → b.Primer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 72• The Indexed
  • The Grothendieck ConstructionIntroduction Where we have an indexed category, we can apply the GrothendieckChu Spaces construction to glue all the fibres together (and get a fibration).Representing PhysicalSystemsCharacterizing Chu Given a functorMorphisms onQuantum Chu Spaces I : Cop → CATThe RepresentationTheorem define I with objects (A, a), where A is an object of C and a is anReducing The Value object of I(A).SetDiscussion Arrows are (G, g) : (A, a) → (B, b), where G : B → A andChu Spaces andCoalgebras g : I(G)(a) → b.Primer on coalgebra Composition of (G, g) : (A, a) → (B, b) and (H, h) : (B, b) → (C, c)Basic ConceptsRepresenting Physical is given bySystems AsCoalgebras (G ◦ H, h ◦ I(G)(g)) : (A, a) → (C, c).Comparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 72• The Indexed
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The Value Indexed Comparison With ChuSetDiscussion SpacesChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models
  • Slicing and Dicing Chu Q For each Q, we define ChuK to be the subcategory of ChuK of Chu spaces (X, Q, e) and morphisms of the form (f∗ , idQ ).Big Toy Models Workshop on Informatic Penomena 2009 – 74
  • Slicing and Dicing Chu Q For each Q, we define ChuK 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 ˆ (− × Q, K) ˆ where K : 1 → Set picks out the object K .Big Toy Models Workshop on Informatic Penomena 2009 – 74
  • Slicing and Dicing Chu Q For each Q, we define ChuK 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 ˆ (− × Q, K) ˆ where K : 1 → Set picks out the object K . Given f : Q′ → Q, we define a functor Q′ ∗ f : ChuQ K → ChuK :: (X, Q, e) → (X, Q′ , e ◦ (1 × f )) and which is the identity on morphisms.Big Toy Models Workshop on Informatic Penomena 2009 – 74
  • Slicing and Dicing Chu Q For each Q, we define ChuK 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 ˆ (− × Q, K) ˆ where K : 1 → Set picks out the object K . Given f : Q′ → Q, we define a functor Q′ ∗ f : ChuQ K → ChuK :: (X, Q, e) → (X, Q′ , e ◦ (1 × f )) and which is the identity on morphisms. This gives an indexed category Chu : Setop → CATBig Toy Models Workshop on Informatic Penomena 2009 – 74
  • Grothendieck puts Chu back together againIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 75
  • Grothendieck puts Chu back together againIntroduction Proposition 27Chu SpacesRepresenting Physical Chu ∼ ChuK . =SystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 75
  • The Truncation FunctorIntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 76
  • The Truncation FunctorIntroduction The relationship between coalgebras and Chu spaces is further clarifiedChu Spaces by an indexed truncation functor T : F → Chu.Representing PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSetDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 76
  • The Truncation FunctorIntroduction The relationship between coalgebras and Chu spaces is further clarifiedChu Spaces by an indexed truncation functor T : F → Chu.Representing PhysicalSystemsCharacterizing Chu For each set Q there is a functorMorphisms on Q TQ : F Q −Coalg → ChuKQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet TQ (X, α) = (X, Q, e)Discussion where Chu Spaces andCoalgebras  0, α(x)(q) = 0Primer on coalgebra e(x, q) =  r, α(x)(q) = (r, x′ )Basic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 76
  • The Truncation FunctorIntroduction The relationship between coalgebras and Chu spaces is further clarifiedChu Spaces by an indexed truncation functor T : F → Chu.Representing PhysicalSystemsCharacterizing Chu For each set Q there is a functorMorphisms on Q TQ : F Q −Coalg → ChuKQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet TQ (X, α) = (X, Q, e)Discussion where Chu Spaces andCoalgebras  0, α(x)(q) = 0Primer on coalgebra e(x, q) =  r, α(x)(q) = (r, x′ )Basic ConceptsRepresenting PhysicalSystems As For f : Q′ → Q there is a natural transformationCoalgebrasComparison: A First ′Try τf : TQ → TQSemantics in OneCountry f Q Q′Externalising τ(X,α) = (idX , f ) : T (X, α) → T (X, α).Contravariance AsIndexing Big Toy Models Workshop on Informatic Penomena 2009 – 76
  • IntroductionChu SpacesRepresenting PhysicalSystemsCharacterizing ChuMorphisms onQuantum Chu SpacesThe RepresentationTheoremReducing The ValueSet A Universal ModelDiscussionChu Spaces andCoalgebrasPrimer on coalgebraBasic ConceptsRepresenting PhysicalSystems AsCoalgebrasComparison: A FirstTrySemantics in OneCountryExternalisingContravariance AsIndexing Big Toy Models
  • A Universal ModelBig Toy Models Workshop on Informatic Penomena 2009 – 78
  • A Universal Model We can now define a single coalgebra which is universal for quantum systems in the following sense:Big Toy Models Workshop on Informatic Penomena 2009 – 78
  • A Universal Model 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 F Q .Big Toy Models Workshop on Informatic Penomena 2009 – 78
  • A Universal Model 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 F Q . • 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 turn induces a functor f ∗ : F Q′ −Coalg → F Q −Coalg.Big Toy Models Workshop on Informatic Penomena 2009 – 78
  • A Universal Model 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 F Q . • 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 turn induces a functor f ∗ : F Q′ −Coalg → F Q −Coalg. • This functor can be applied to the coalgebra (K◦ , α) corresponding to the Hilbert space K to yield a coalgebra in F Q −Coalg.Big Toy Models Workshop on Informatic Penomena 2009 – 78
  • UniversalityBig Toy Models Workshop on Informatic Penomena 2009 – 79
  • Universality • Since (U, γ) is the final coalgebra in F Q −Coalg, there is a unique coalgebra homomorphism h : f ∗ (K◦ , α) → (U, γ).Big Toy Models Workshop on Informatic Penomena 2009 – 79
  • Universality • Since (U, γ) is the final coalgebra in F Q −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 fully abstract fashion, i.e. identifying states precisely according to observational equivalence.Big Toy Models Workshop on Informatic Penomena 2009 – 79
  • Universality • Since (U, γ) is the final coalgebra in F Q −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 fully abstract fashion, i.e. identifying states precisely according to observational equivalence. • This homomorphism is an arrow in the Grothendieck category.Big Toy Models Workshop on Informatic Penomena 2009 – 79
  • Universality • Since (U, γ) is the final coalgebra in F Q −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 fully abstract 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.Big Toy Models Workshop on Informatic Penomena 2009 – 79
  • Universality • Since (U, γ) is the final coalgebra in F Q −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 fully abstract 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!Big Toy Models Workshop on Informatic Penomena 2009 – 79