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Big Toy Models: Representing Physical Systems As Chu Spaces Samson Abramsky Oxford University Computing Laboratory Big Toy Models Workshop on Informatic Penomena 2009 – 1 Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Introduction Themes Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 3 Themes Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’. Exemplifies one of the main thrusts of our group in Oxford: methods and concepts which have been developed in Theoretical Computer Science are ripe for use in Physics. Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 3 Themes Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’. Exemplifies one of the main thrusts of our group in Oxford: methods and concepts which have been developed in Theoretical Computer Science are ripe for use in Physics. • Models vs. Axioms. Examples: sheaves and toposes, domain-theoretic models of the λ-calculus. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 3 Themes Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’. Exemplifies one of the main thrusts of our group in Oxford: methods and concepts which have been developed in Theoretical Computer Science are ripe for use in Physics. • Models vs. Axioms. Examples: sheaves and toposes, domain-theoretic models of the λ-calculus. • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of quantum states: A toy theory’. Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 3 Themes Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • (Foundations of) Mathematical Physics, ‘but not as we know it, Jim’. Exemplifies one of the main thrusts of our group in Oxford: methods and concepts which have been developed in Theoretical Computer Science are ripe for use in Physics. • Models vs. Axioms. Examples: sheaves and toposes, domain-theoretic models of the λ-calculus. • Toy Models. Rob Spekkens: ‘Evidence for the epistemic view of quantum states: A toy theory’. • Big toy models. Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 3 Chu Spaces Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 4 Chu Spaces Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces We should understand Chu spaces as providing a very general (and, we might reasonably say, rather simple) ‘logic of systems or structures’. Indeed, they have been proposed by Barwise and Seligman as the vehicle for a general logic of ‘distributed systems’ and information flow. Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 4 Chu Spaces Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces We should understand Chu spaces as providing a very general (and, we might reasonably say, rather simple) ‘logic of systems or structures’. Indeed, they have been proposed by Barwise and Seligman as the vehicle for a general logic of ‘distributed systems’ and information flow. This logic of Chu spaces was in no way biassed in its conception towards the description of quantum mechanics or any other kind of physical system. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 4 Chu Spaces Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion We should understand Chu spaces as providing a very general (and, we might reasonably say, rather simple) ‘logic of systems or structures’. Indeed, they have been proposed by Barwise and Seligman as the vehicle for a general logic of ‘distributed systems’ and information flow. This logic of Chu spaces was in no way biassed in its conception towards the description of quantum mechanics or any other kind of physical system. Just for this reason, it is interesting to see how much of quantum-mechanical structure and concepts can be absorbed and essentially determined by this more general systems logic. Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 4 Outline I Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 5 Outline I Introduction • Themes • Chu Spaces • Outline I • Outline II • Chu spaces as a setting. We can find natural representations of quantum (and other) systems as Chu spaces. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 5 Outline I Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems • Chu spaces as a setting. We can find natural representations of quantum (and other) systems as Chu spaces. • The general ‘logic’ of Chu spaces and morphisms allow us to ‘rationally reconstruct’ many key quantum notions: Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 5 Outline I Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Chu spaces as a setting. We can find natural representations of quantum (and other) systems as Chu spaces. • The general ‘logic’ of Chu spaces and morphisms allow us to ‘rationally reconstruct’ many key quantum notions: • States as rays of Hilbert spaces fall out as the biextensional collapse of the Chu spaces. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 5 Outline I Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Chu spaces as a setting. We can find natural representations of quantum (and other) systems as Chu spaces. • The general ‘logic’ of Chu spaces and morphisms allow us to ‘rationally reconstruct’ many key quantum notions: • States as rays of Hilbert spaces fall out as the biextensional collapse of the Chu spaces. • Chu morphisms are automatically the unitaries and antiunitaries — the physical symmetries of quantum systems. Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 5 Outline I Introduction • Themes • Chu Spaces • Outline I • Outline II Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra • Chu spaces as a setting. We can find natural representations of quantum (and other) systems as Chu spaces. • The general ‘logic’ of Chu spaces and morphisms allow us to ‘rationally reconstruct’ many key quantum notions: • States as rays of Hilbert spaces fall out as the biextensional collapse of the Chu spaces. • Chu morphisms are automatically the unitaries and antiunitaries — the physical symmetries of quantum systems. • This leads to a full and faithful representation of the groupoid of Hilbert spaces and their physical symmetries in Chu spaces over the unit interval. Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics One Big Toy inModels Workshop on Informatic Penomena 2009 – 5 Outline II 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? 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 rôle 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 rôle 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 rôle 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.html Big Toy Models Workshop on Informatic Penomena 2009 – 6 Introduction Chu Spaces • Chu Spaces • Definitions • Extensionality and Separability • Biextensional Collapse Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy Models Chu Spaces Chu Spaces 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. 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 Definitions 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. 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∗ , Big Toy Models (g ◦ f )∗ = f ∗ ◦ g ∗ . 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 Separability 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: 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 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 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). 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 Collapse Introduction Chu Spaces • Chu Spaces • Definitions • Extensionality and Separability • Biextensional Collapse Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then f∗ preserves ∼. That is, for all x1 , x2 ∈ X , x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ). Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy Models Workshop on Informatic Penomena 2009 – 11 Biextensional Collapse Introduction Chu Spaces • Chu Spaces • Definitions • Extensionality and Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then f∗ preserves ∼. That is, for all x1 , x2 ∈ X , x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ). Separability • Biextensional Collapse Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Proof For any a′ ∈ A′ : e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ). Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy Models Workshop on Informatic Penomena 2009 – 11 Biextensional Collapse Introduction Chu Spaces • Chu Spaces • Definitions • Extensionality and Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then f∗ preserves ∼. That is, for all x1 , x2 ∈ X , x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ). Separability • Biextensional Collapse Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Proof For any a′ ∈ A′ : e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ). We shall write eChuK , sChuK and bChuK for the full subcategories of ChuK determined by the extensional, separated and biextensional Chu spaces. Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy Models Workshop on Informatic Penomena 2009 – 11 Biextensional Collapse Introduction Chu Spaces • Chu Spaces • Definitions • Extensionality and Proposition 1 If f : (X, A, e) → (X ′ , A′ , e′ ) is a Chu morphism, then f∗ preserves ∼. That is, for all x1 , x2 ∈ X , x1 ∼ x2 ⇒ f∗ (x1 ) ∼ f∗ (x2 ). Separability • Biextensional Collapse Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces Proof For any a′ ∈ A′ : e′ (f∗ (x1 ), a′ ) = e(x1 , f ∗ (a′ )) = e(x2 , f ∗ (a′ )) = e′ (f∗ (x2 ), a′ ). The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy Models We shall write eChuK , sChuK and bChuK for the full subcategories of ChuK determined by the extensional, separated and biextensional Chu spaces. We shall mainly work with extensional and biextensional Chu spaces. Obviously bChuK is a full sub-category of eChuK . Proposition 2 The inclusion bChuK Q, the biextensional collapse.. ⊂ - eChuK has a left adjoint Workshop on Informatic Penomena 2009 – 11 Introduction Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One Representing Physical Systems The General Paradigm 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. 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 Spaces Introduction Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One Workshop on Informatic Penomena 2009 – 14 Representing Quantum Systems As Chu Spaces Introduction A quantum system with a Hilbert space H as its state space will be represented as (H◦ , L(H), eH ) Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces where Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One Workshop on Informatic Penomena 2009 – 14 Representing Quantum Systems As Chu Spaces Introduction A quantum system with a Hilbert space H as its state space will be represented as (H◦ , L(H), eH ) Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces where • H◦ is the set of non-zero vectors of H Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One Workshop on Informatic Penomena 2009 – 14 Representing Quantum Systems As Chu Spaces Introduction A quantum system with a Hilbert space H as its state space will be represented as (H◦ , L(H), eH ) Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces Characterizing Chu Morphisms on Quantum Chu Spaces where • H◦ is the set of non-zero vectors of H • L(H) is the set of closed subspaces of H — the ‘yes/no’ questions of QM The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One Workshop on Informatic Penomena 2009 – 14 Representing Quantum Systems As Chu Spaces Introduction A quantum system with a Hilbert space H as its state space will be represented as (H◦ , L(H), eH ) Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set where • H◦ is the set of non-zero vectors of H • L(H) is the set of closed subspaces of H — the ‘yes/no’ questions of QM • The evaluation function eH is the ‘statistical algorithm’ giving the basic predictive content of Quantum Mechanics: Discussion Chu Spaces and Coalgebras Primer on coalgebra hPS ψ | PS ψi kPS ψk2 hψ | PS ψi = = . eH (ψ, S) = 2 hψ | ψi hψ | ψi kψk Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One Workshop on Informatic Penomena 2009 – 14 Representing Quantum Systems As Chu Spaces Introduction A quantum system with a Hilbert space H as its state space will be represented as (H◦ , L(H), eH ) Chu Spaces Representing Physical Systems • The General Paradigm • Representing Quantum Systems As Chu Spaces Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set where • H◦ is the set of non-zero vectors of H • L(H) is the set of closed subspaces of H — the ‘yes/no’ questions of QM • The evaluation function eH is the ‘statistical algorithm’ giving the basic predictive content of Quantum Mechanics: Discussion Chu Spaces and Coalgebras Primer on coalgebra hPS ψ | PS ψi kPS ψk2 hψ | PS ψi = = . eH (ψ, S) = 2 hψ | ψi hψ | ψi kψk Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Big Toy inModels Semantics One We have thus directly transcribed the basic ingredients of the Dirac/von Neumann-style formulation of Quantum Mechanics into the definition of this Chu space. Workshop on Informatic Penomena 2009 – 14 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Characterizing Chu Morphisms on Quantum Chu Spaces Overview Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 16 Overview Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = We shall now see how the simple, discrete notions of Chu spaces suffice to determine the appropriate notions of state equivalence, and to pick out the physically significant symmetries on Hilbert space in a very striking fashion. This leads to a full and faithful representation of the category of quantum systems, with the groupoid structure of their physical symmetries, in the category of Chu spaces valued in the unit interval. Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 16 Overview Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality We shall now see how the simple, discrete notions of Chu spaces suffice to determine the appropriate notions of state equivalence, and to pick out the physically significant symmetries on Hilbert space in a very striking fashion. This leads to a full and faithful representation of the category of quantum systems, with the groupoid structure of their physical symmetries, in the category of Chu spaces valued in the unit interval. The arguments here make use of Wigner’s theorem and the dualities of projective geometry, in the modern form developed by Faure and Frölicher, Modern Projective Geometry (2000). • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 16 Overview Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: We shall now see how the simple, discrete notions of Chu spaces suffice to determine the appropriate notions of state equivalence, and to pick out the physically significant symmetries on Hilbert space in a very striking fashion. This leads to a full and faithful representation of the category of quantum systems, with the groupoid structure of their physical symmetries, in the category of Chu spaces valued in the unit interval. The arguments here make use of Wigner’s theorem and the dualities of projective geometry, in the modern form developed by Faure and Frölicher, Modern Projective Geometry (2000). The surprising point is that unitarity/anitunitarity is essentially forced by the mere requirement of being a Chu morphism. This even extends to surjectivity, which here is derived rather than assumed. Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 16 Biextensionaity Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 17 Biextensionaity Introduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ). Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 17 Biextensionaity Introduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ). Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = A basic property of the evaluation. Lemma 3 For ψ ∈ H◦ and S ∈ L(H): ψ ∈ S ⇐⇒ eH (ψ, S) = 1. Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 17 Biextensionaity Introduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ). Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: A basic property of the evaluation. Lemma 3 For ψ ∈ H◦ and S ∈ L(H): ψ ∈ S ⇐⇒ eH (ψ, S) = 1. From this, we can prove: Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but not separable. The equivalence classes of the relation ∼ on states are exactly the rays of H. That is: φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ. Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 17 Biextensionaity Introduction Given a Hilbert space H, consider the Chu space (H◦ , L(H), eH ). Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value A basic property of the evaluation. Lemma 3 For ψ ∈ H◦ and S ∈ L(H): ψ ∈ S ⇐⇒ eH (ψ, S) = 1. From this, we can prove: Proposition 4 The Chu space (H◦ , L(H), eH ) is extensional but not separable. The equivalence classes of the relation ∼ on states are exactly the rays of H. That is: φ ∼ ψ ⇐⇒ ∃λ ∈ C. φ = λψ. Thus we have recovered the standard notion of pure states as the rays of the Hilbert space from the general notion of state equivalence in Chu spaces. Workshop on Informatic Penomena 2009 – 17 Projectivity = Biextensionality Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 18 Projectivity = Biextensionality Introduction We shall now use some notions and results from projective geometry. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 18 Projectivity = Biextensionality Introduction We shall now use some notions and results from projective geometry. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces Given a vector ψ ∈ H◦ , we write ψ̄ = {λψ | λ ∈ C} for the ray which it generates. The rays are the atoms in the lattice L(H). • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 18 Projectivity = Biextensionality Introduction We shall now use some notions and results from projective geometry. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality Given a vector ψ ∈ H◦ , we write ψ̄ = {λψ | λ ∈ C} for the ray which it generates. The rays are the atoms in the lattice L(H). We write P(H) for the set of rays of H. By virtue of Proposition 4, we can write the biextensional collapse of (H◦ , L(H), eH ) given by Proposition 2 as (P(H), L(H), ēH) where ēH (ψ̄, S) = eH (ψ, S). • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 18 Projectivity = Biextensionality Introduction We shall now use some notions and results from projective geometry. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together Given a vector ψ ∈ H◦ , we write ψ̄ = {λψ | λ ∈ C} for the ray which it generates. The rays are the atoms in the lattice L(H). We write P(H) for the set of rays of H. By virtue of Proposition 4, we can write the biextensional collapse of (H◦ , L(H), eH ) given by Proposition 2 as (P(H), L(H), ēH) where ēH (ψ̄, S) = eH (ψ, S). We restate Lemma 3 for the biextensional case. Lemma 5 For ψ ∈ H◦ and S ∈ L(H): ēH (ψ̄, S) = 1 ⇐⇒ ψ̄ ⊆ S. The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 18 Characterizing Chu Morphisms 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), ēH) → (P(K), L(K), ēK ). 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), ēH) → (P(K), L(K), ēK ). Proposition 6 For ψ ∈ H◦ and S ∈ L(K): ψ̄ ⊆ f ∗ (S) ⇐⇒ f∗ (ψ̄) ⊆ S. Proof By Lemma 5: ψ̄ ⊆ f ∗ (S) ⇔ ēH (ψ̄, f ∗ (S)) = 1 ⇔ ēK (f∗ (ψ̄), S) = 1 ⇔ f∗ (ψ̄) ⊆ 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), ēH) → (P(K), L(K), ēK ). Proposition 6 For ψ ∈ H◦ and S ∈ L(K): ψ̄ ⊆ f ∗ (S) ⇐⇒ f∗ (ψ̄) ⊆ S. Proof By Lemma 5: ψ̄ ⊆ f ∗ (S) ⇔ ēH (ψ̄, f ∗ (S)) = 1 ⇔ ēK (f∗ (ψ̄), S) = 1 ⇔ f∗ (ψ̄) ⊆ S. Note that P(H) ⊆ L(H). Big Toy Models Workshop on Informatic Penomena 2009 – 19 Injectivity Assumption Big Toy Models Workshop on Informatic Penomena 2009 – 20 Injectivity Assumption Proposition 7 If f∗ is injective, then the following diagram commutes: P(H) f∗- ∩ P(K) ∩ (1) ? L(H) ∗ f ? L(K) That is, for all ψ ∈ H◦ : ψ̄ = f ∗ (f∗ (ψ̄)). Big Toy Models Workshop on Informatic Penomena 2009 – 20 Injectivity Assumption Proposition 7 If f∗ is injective, then the following diagram commutes: P(H) f∗- ∩ P(K) ∩ (1) ? L(H) ∗ f ? L(K) That is, for all ψ ∈ H◦ : ψ̄ = f ∗ (f∗ (ψ̄)). Proposition 6 implies that ψ̄ ⊆ f ∗ (f∗ (ψ̄)). For the converse, suppose that φ̄ ⊆ f ∗ (f∗ (ψ̄)). Applying Proposition 6 again, this implies that f∗ (φ̄) ⊆ f∗ (ψ̄). Since f∗ (φ̄) and f∗ (ψ̄) are atoms, this implies that f∗ (φ̄) = f∗ (ψ̄), which since f∗ is injective implies that φ̄ = ψ̄ . Thus the only atom below f ∗ (f∗ (ψ̄)) is ψ̄ . Since L(H) is atomistic, this implies that f ∗ (f∗ (ψ̄)) ⊆ ψ̄ . Proof Big Toy Models Workshop on Informatic Penomena 2009 – 20 Orthogonality is Preserved Another basic property of the evaluation. Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces Lemma 8 For any φ, ψ ∈ H◦ : ēH (φ̄, ψ̄) = 0 ⇐⇒ φ ⊥ ψ. • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 21 Orthogonality is Preserved Another basic property of the evaluation. Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality Lemma 8 For any φ, ψ ∈ H◦ : ēH (φ̄, ψ̄) = 0 ⇐⇒ φ ⊥ ψ. Proposition 9 If f∗ is injective, it preserves and reflects orthogonality. That is, for all φ, ψ ∈ H◦ : φ ⊥ ψ ⇐⇒ f∗ (φ̄) ⊥ f∗ (ψ̄). • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 21 Orthogonality is Preserved Another basic property of the evaluation. Introduction Chu Spaces Lemma 8 Representing Physical Systems ēH (φ̄, ψ̄) = 0 ⇐⇒ φ ⊥ ψ. Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: For any φ, ψ ∈ H◦ : Proposition 9 If f∗ is injective, it preserves and reflects orthogonality. That is, for all φ, ψ ∈ H◦ : φ ⊥ ψ ⇐⇒ f∗ (φ̄) ⊥ f∗ (ψ̄). Proof φ ⊥ ψ ⇐⇒ ēH (φ̄, ψ̄) = 0 Lemma 8 ⇐⇒ ēH (φ̄, f ∗ (f∗ (ψ̄))) = 0 Proposition 7 Surjectivity Comes for Free! • Putting The Pieces Together ⇐⇒ ēK (f∗ (φ̄), f∗ (ψ̄)) = 0 The Representation Theorem ⇐⇒ f∗ (φ̄) ⊥ f∗ (ψ̄) Big ToyThe Models Reducing Value Lemma 8. Workshop on Informatic Penomena 2009 – 21 Constructing the Left Adjoint Introduction We define a map f → : L(H) → L(K): Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces → f (S) = _ {f∗ (ψ̄) | ψ ∈ S◦ } where S◦ = S \ {0}. • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 22 Constructing the Left Adjoint Introduction We define a map f → : L(H) → L(K): Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality → f (S) = _ {f∗ (ψ̄) | ψ ∈ S◦ } where S◦ = S \ {0}. Lemma 10 The map f → is left adjoint to f ∗ : f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ). • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 22 Constructing the Left Adjoint Introduction We define a map f → : L(H) → L(K): Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value → f (S) = _ {f∗ (ψ̄) | ψ ∈ S◦ } where S◦ = S \ {0}. Lemma 10 The map f → is left adjoint to f ∗ : f → (S) ⊆ T ⇐⇒ S ⊆ f ∗ (T ). We can now extend the diagram (1): P(H) ∩ f∗- P(K) ∩ (2) f →- ? L(H) ⊥ L(K) f∗ Workshop on Informatic Penomena 2009 – 22 ? Using Projective Duality 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. 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 Big Toy Models f∗ is a total map of projective geometries. 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 Theorem Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 24 Wigner’s Theorem Introduction Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 by Chu Spaces Representing Physical Systems P(g)(ψ̄) = g(ψ). Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 24 Wigner’s Theorem Introduction Given a semilinear map g : V1 → V2 , we define Pg : PV1 → PV2 by Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value P(g)(ψ̄) = g(ψ). We can now state Wigner’s Theorem in the form we shall use it. Theorem 12 Let f : P(H) → P(K) be a total map of projective geometries, where dim H > 2. If f preserves orthogonality, meaning that φ̄ ⊥ ψ̄ ⇒ f (φ̄) ⊥ f (ψ̄) then there is a semilinear map g : H → K such that P(g) = f , and hg(φ) | g(ψ)i = σ(hφ | ψi), where σ is the homomorphism associated with g . Moreover, this homomorphism is either the identity or complex conjugation, so g is either linear or antilinear. The map g is unique up to a phase, i.e. a scalar of modulus 1. Workshop on Informatic Penomena 2009 – 24 Remarks Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 25 Remarks Introduction Chu Spaces Representing Physical Systems • Note that in our case, taking f∗ = f , Pg is just the action of the biextensional collapse functor on Chu morphisms. Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 25 Remarks Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = • Note that in our case, taking f∗ = f , Pg is just the action of the biextensional collapse functor on Chu morphisms. • Note that a total map of projective geometries must necessarily come from an injective map g on the underlying vector spaces, since P(g) maps rays to rays, and hence g must have trivial kernel. Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 25 Remarks Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Note that in our case, taking f∗ = f , Pg is just the action of the biextensional collapse functor on Chu morphisms. • Note that a total map of projective geometries must necessarily come from an injective map g on the underlying vector spaces, since P(g) maps rays to rays, and hence g must have trivial kernel. • For this reason, partial maps of projective geometries are considered in the Faure-Frölicher approach. However, we are simply following the ‘logic’ of Chu space morphisms here. • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing 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∗ (φ̄) ⊆ ψ̄ ; for otherwise, for some λ, g(φ) = λψ , and hence g(σ −1 (λ−1 )φ) = ψ . Then by Proposition 6, f ∗ (ψ̄) = {0}. It follows that for all φ ∈ H◦ , ēK (f∗ (φ̄), ψ̄) = ēH (φ̄, {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∗ (φ̄) ⊆ ψ̄ ; for otherwise, for some λ, g(φ) = λψ , and hence g(σ −1 (λ−1 )φ) = ψ . Then by Proposition 6, f ∗ (ψ̄) = {0}. It follows that for all φ ∈ H◦ , ēK (f∗ (φ̄), ψ̄) = ēH (φ̄, {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. hψ + χ | χi = hχ | χi = 6 0. This yields the required contradiction. Big Toy Models Workshop on Informatic Penomena 2009 – 26 Putting The Pieces Together Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 27 Putting The Pieces Together Introduction Chu Spaces We say that a map U : H → K is semiunitary if it is either unitary or antiunitary; that is, if it is a bijective map satisfying Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, hU φ | U ψi = σ(hφ | ψi) where σ is the identity if U is unitary, and complex conjugation if U is antiunitary. Note that semiunitaries preserve norm, so if U and V are semiunitaries and U = λV , then |λ| = 1. • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 27 Putting The Pieces Together Introduction Chu Spaces We say that a map U : H → K is semiunitary if it is either unitary or antiunitary; that is, if it is a bijective map satisfying Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces • Overview • Biextensionaity • Projectivity = Biextensionality • Characterizing Chu Morphisms • Injectivity Assumption • Orthogonality is Preserved • Constructing the Left Adjoint • Using Projective Duality • Wigner’s Theorem • Remarks • A Surprise: Surjectivity Comes for Free! • Putting The Pieces Together U (φ+ψ) = U φ+U ψ, U (λφ) = σ(λ)U φ, hU φ | U ψi = σ(hφ | ψi) where σ is the identity if U is unitary, and complex conjugation if U is antiunitary. Note that semiunitaries preserve norm, so if U and V are semiunitaries and U = λV , then |λ| = 1. Theorem 14 Let H, K be Hilbert spaces of dimension greater than 2. Consider a Chu morphism (f∗ , f ∗ ) : (P(H), L(H), ēH) → (P(K), L(K), ēK ). where f∗ is injective. Then there is a semiunitary U : H → K such that f∗ = P(U ). U is unique up to a phase. The Representation Theorem Big ToyThe Models Reducing Value Workshop on Informatic Penomena 2009 – 27 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts The Representation Theorem The Big Picture Big 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 Schrödinger 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 Remarks 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. 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 Functors Introduction We define a functor R : SymmH → eChu[0,1] : Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase R : U : H → K 7−→ (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK ) where U◦ is the restriction of U to H◦ . As noted in Proposition 2, the inclusion bChu[0,1] ⊂ - eChu[0,1] has a left adjoint, which we name Q. By Proposition 4, this can be defined on the image of R as follows: Q : (H◦ , L(H), eH ) 7→ (PH, L(H), ēH ) and for (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK ), Q : (U◦ , U −1 ) 7−→ (PU, U −1 ). Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 31 Not Quite Right Yet Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 32 Not Quite Right Yet Introduction Chu Spaces Representing Physical Systems We write emChu, bmChu for the subcategories of eChu[0,1] and bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is injective. The functors R and Q factor through these subcategories. Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 32 Not Quite Right Yet Representing Physical Systems We write emChu, bmChu for the subcategories of eChu[0,1] and bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is injective. The functors R and Q factor through these subcategories. Characterizing Chu Morphisms on Quantum Chu Spaces Proposition 15 Both Introduction Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an R : SymmH → emChu and Q : emChu → bmChu are well-defined functors. R is faithful but not full; Q is full but not faithful. embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 32 Not Quite Right Yet Representing Physical Systems We write emChu, bmChu for the subcategories of eChu[0,1] and bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is injective. The functors R and Q factor through these subcategories. Characterizing Chu Morphisms on Quantum Chu Spaces Proposition 15 Both Introduction Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set R : SymmH → emChu and Q : emChu → bmChu are well-defined functors. R is faithful but not full; Q is full but not faithful. This involves verifying that unitaries and antiunitaries U : H → K do indeed yield Chu morphisms! Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 32 Not Quite Right Yet Representing Physical Systems We write emChu, bmChu for the subcategories of eChu[0,1] and bChu[0,1] obtained by restricting to Chu morphisms f for which f∗ is injective. The functors R and Q factor through these subcategories. Characterizing Chu Morphisms on Quantum Chu Spaces Proposition 15 Both Introduction Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras R : SymmH → emChu and Q : emChu → bmChu are well-defined functors. R is faithful but not full; Q is full but not faithful. This involves verifying that unitaries and antiunitaries U : H → K do indeed yield Chu morphisms! The key property, for ψ ∈ H◦ and S ∈ L(H), is: PS (U ψ) = U (PU −1 (S) ψ). Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 32 Biextensionality and Scalar Factors Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 33 Biextensionality and Scalar Factors Introduction We can analyze the non-fullness of R more precisely as follows. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Proposition 16 Let (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK ) be a Chu morphism in the image of R. Given an arbitrary function f : H → C \ {0}, define f U : H◦ → K◦ by: f U (ψ) = f (ψ)U (ψ). Then (f U, U −1 ) ∼ (U◦ , U −1 ). Moreover, the ∼-equivalence class of U is exactly the set of functions of this form. Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 33 Biextensionality and Scalar Factors Introduction We can analyze the non-fullness of R more precisely as follows. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem • The Big Picture • Remarks • Functors • Not Quite Right Yet • Biextensionality and Scalar Factors • Projectivising The Symmetry Groupoid • Jes’ Right • PR is an embedding up to a phase Reducing The Value Set Discussion Proposition 16 Let (U◦ , U −1 ) : (H◦ , L(H), eH ) → (K◦ , L(K), eK ) be a Chu morphism in the image of R. Given an arbitrary function f : H → C \ {0}, define f U : H◦ → K◦ by: f U (ψ) = f (ψ)U (ψ). Then (f U, U −1 ) ∼ (U◦ , U −1 ). Moreover, the ∼-equivalence class of U is exactly the set of functions of this form. Thus before biextensional collapse, Chu morphisms can introduce arbitrary scalar factors pointwise. Once we move to the biextensional category, we know by Theorem 14 that our representation will be full, and essentially faithful — up to a global phase. This points to the need for a projective version of the symmetry groupoid. Chu Spaces and Coalgebras Primer on coalgebra Big Toy Models Basic Concepts Workshop on Informatic Penomena 2009 – 33 Projectivising The Symmetry Groupoid 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 λλ̄ = 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 λλ̄ = 1, U(1) is the unitary group on the one-dimensional Hilbert space. The circle group acts on the symmetry groupoid SymmH by scalar multiplication. For Hilbert spaces H, K we can define U(1) × SymmH(H, K) → SymmH(H, K) :: (λ, U ) 7→ λU. Moreover, this is a category action, meaning that (λU ) ◦ V = U ◦ (λV ) = λ(U ◦ V ). 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 λλ̄ = 1, U(1) is the unitary group on the one-dimensional Hilbert space. The circle group acts on the symmetry groupoid SymmH by scalar multiplication. For Hilbert spaces H, K we can define U(1) × SymmH(H, K) → SymmH(H, K) :: (λ, U ) 7→ λU. Moreover, this is a category action, meaning that (λU ) ◦ V = U ◦ (λV ) = λ(U ◦ V ). It follows that we can form a quotient category (in fact again a groupoid) with the same objects as SymmH, and in which the morphisms will be the orbits of this group action: U ∼ V ⇔ ∃λ ∈ U(1). U = λV. Big Toy Models Workshop on Informatic Penomena 2009 – 34 Jes’ Right 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. 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: SymmH > R P ∨ ∨ PSymmH > Big Toy Models PR > emChu Q ∨ ∨ >> bmChu 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: SymmH > R P ∨ ∨ PSymmH > Theorem 17 Big Toy Models PR > emChu Q ∨ ∨ >> bmChu The functor PR : PSymmH → bmChu is full and faithful. Workshop on Informatic Penomena 2009 – 35 PR is an embedding up to a phase Big Toy Models Workshop on Informatic Penomena 2009 – 36 PR is an embedding up to a phase • To see that PR is essentially injective on objects, we can use the representation theorems of Piron and Solèr, 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 Solèr, 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 Solèr, 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 Solèr, 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 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Reducing The Value Set Generalities Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 38 Generalities Introduction Chu Spaces We now return to the issue of whether it is necessary to use the full unit interval as the value set for our Chu spaces. Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 38 Generalities Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical We now return to the issue of whether it is necessary to use the full unit interval as the value set for our Chu spaces. We begin with some generalities. Given a function v : K → L, we define a functor Fv : ChuK → ChuL : Fv : (X, A, e) 7→ (X, A, v ◦ e) and Fv f = f for Chu morphisms f . Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 38 Generalities Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces We now return to the issue of whether it is necessary to use the full unit interval as the value set for our Chu spaces. We begin with some generalities. Given a function v : K → L, we define a functor Fv : ChuK → ChuL : The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions Fv : (X, A, e) 7→ (X, A, v ◦ e) and Fv f = f for Chu morphisms f . Proposition 18 Fv is a faithful functor. If v is injective, it is full. • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 38 The Question 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? 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? Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 40 Two Values? Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces We shall write n = {0, . . . , n − 1} for the finite ordinals. The most popular choice of value set for Chu spaces, by far, has been 2, and indeed many interesting categories can be strictly (and even concretely) represented in Chu2 . The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 40 Two Values? Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical We shall write n = {0, . . . , n − 1} for the finite ordinals. The most popular choice of value set for Chu spaces, by far, has been 2, and indeed many interesting categories can be strictly (and even concretely) represented in Chu2 . This makes the following question natural: Question 19 and faithful? Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 40 Two Values? Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values We shall write n = {0, . . . , n − 1} for the finite ordinals. The most popular choice of value set for Chu spaces, by far, has been 2, and indeed many interesting categories can be strictly (and even concretely) represented in Chu2 . This makes the following question natural: Question 19 and faithful? Is there a function v : [0, 1] → 2 such that Fv ◦ PR is full What we can show is that the most plausible candidates for such functions, yielding the two canonical forms of possibilistic semantics as a coarse-graining of probabilistic semantics, both in fact fail. Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 40 The Canonical Possibilistic Reductions Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 41 The Canonical Possibilistic Reductions Introduction Chu Spaces Representing Physical Systems Note that any function v : [0, 1] → {0, 1} must lose information either on 0 or on 1 — or both. Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 41 The Canonical Possibilistic Reductions Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Note that any function v : [0, 1] → {0, 1} must lose information either on 0 or on 1 — or both. In this sense, the two ‘sharpest’ mappings will be: v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1. Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 41 The Canonical Possibilistic Reductions Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions Note that any function v : [0, 1] → {0, 1} must lose information either on 0 or on 1 — or both. In this sense, the two ‘sharpest’ mappings will be: v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1. These are the two canonical reductions of probabilistic to possibilistic information: the first maps ‘definitely not’ to ‘no’, and anything else to ‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitely yes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’. • Two is Too Few • Other Case • Analysis • Three Values Suffice! Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Big Toy Models Representing Physical Workshop on Informatic Penomena 2009 – 41 The Canonical Possibilistic Reductions Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • Generalities • The Question • Two Values? • The Canonical Possibilistic Reductions • Two is Too Few • Other Case • Analysis • Three Values Suffice! Note that any function v : [0, 1] → {0, 1} must lose information either on 0 or on 1 — or both. In this sense, the two ‘sharpest’ mappings will be: v0 : 0 7→ 0, (0, 1] 7→ 1 v1 : [0, 1) 7→ 0, 1 7→ 1. These are the two canonical reductions of probabilistic to possibilistic information: the first maps ‘definitely not’ to ‘no’, and anything else to ‘yes’, which is to be read as ‘possibly yes’; the second maps ‘definitely yes’ to ‘yes’, and anything else to ‘no’, to be read as ‘possibly no’. Note that, under the first of these, we no longer have eH (ψ, S) = 1 ⇐⇒ ψ ∈ S Discussion Chu Spaces and Coalgebras while under the second, we no longer have Primer on coalgebra Basic Concepts Big Toy Models Representing Physical eH (ψ, S) = 0 ⇐⇒ ψ ⊥ S. Workshop on Informatic Penomena 2009 – 41 Two is Too Few Proposition 20 Big Toy Models For neither v = v0 nor v = v1 is Fv ◦ PR full. 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: ēH (ψ̄, 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: ēH (ψ̄, 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: ēH (ψ̄, 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: ēH (ψ̄, 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 Analysis 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 7→ 0, (0, 1) 7→ 2, 1 7→ 1 Thus we lose information only on the probabilities strictly between 0 and 1, which are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2. 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 7→ 0, (0, 1) 7→ 2, 1 7→ 1 Thus we lose information only on the probabilities strictly between 0 and 1, which are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2. Why is this adequate? Given a vector ψ and a subspace S , we can write ψ uniquely as θ + χ, where θ ∈ S and χ ∈ S ⊥ . For non-zero ψ , there are only three possibilities: 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 7→ 0, (0, 1) 7→ 2, 1 7→ 1 Thus we lose information only on the probabilities strictly between 0 and 1, which are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2. Why is this adequate? Given a vector ψ and a subspace S , we can write ψ uniquely as θ + χ, where θ ∈ S and χ ∈ S ⊥ . For non-zero ψ , there are only three possibilities: • θ = 0 and χ 6= 0, so eH (φ, S) = 0 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 7→ 0, (0, 1) 7→ 2, 1 7→ 1 Thus we lose information only on the probabilities strictly between 0 and 1, which are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2. Why is this adequate? Given a vector ψ and a subspace S , we can write ψ uniquely as θ + χ, where θ ∈ S and χ ∈ S ⊥ . For non-zero ψ , there are only three possibilities: • θ = 0 and χ 6= 0, so eH (φ, S) = 0 • θ 6= 0 and χ = 0, so eH (φ, S) = 1 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 7→ 0, (0, 1) 7→ 2, 1 7→ 1 Thus we lose information only on the probabilities strictly between 0 and 1, which are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2. Why is this adequate? Given a vector ψ and a subspace S , we can write ψ uniquely as θ + χ, where θ ∈ S and χ ∈ S ⊥ . For non-zero ψ , there are only three possibilities: • θ = 0 and χ 6= 0, so eH (φ, S) = 0 • θ 6= 0 and χ = 0, so eH (φ, S) = 1 • θ 6= 0 and χ 6= 0, so eH (ψ, S) ∈ (0, 1), and hence v ◦ eH (ψ, S) = 2. 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 7→ 0, (0, 1) 7→ 2, 1 7→ 1 Thus we lose information only on the probabilities strictly between 0 and 1, which are lumped together as ‘maybe’ — represented here, by arbitrary convention, by 2. Why is this adequate? Given a vector ψ and a subspace S , we can write ψ uniquely as θ + χ, where θ ∈ S and χ ∈ S ⊥ . For non-zero ψ , there are only three possibilities: • θ = 0 and χ 6= 0, so eH (φ, S) = 0 • θ 6= 0 and χ = 0, so eH (φ, S) = 1 • θ 6= 0 and χ 6= 0, so eH (ψ, S) ∈ (0, 1), and hence v ◦ eH (ψ, S) = 2. These are the only case discriminations which are used in 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 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Discussion Where Next? Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47 Where Next? Introduction Chu Spaces • Connections and contrasts between Chu spaces and coalgebras. Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47 Where Next? Introduction Chu Spaces Representing Physical Systems • Connections and contrasts between Chu spaces and coalgebras. • Mixed states — handled generally at the level of Chu spaces. Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47 Where Next? Introduction Chu Spaces • Connections and contrasts between Chu spaces and coalgebras. Representing Physical Systems • Mixed states — handled generally at the level of Chu spaces. Characterizing Chu Morphisms on Quantum Chu Spaces • Universal Chu spaces. The Representation Theorem Reducing The Value Set Discussion • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47 Where Next? Introduction Chu Spaces • Connections and contrasts between Chu spaces and coalgebras. Representing Physical Systems • Mixed states — handled generally at the level of Chu spaces. Characterizing Chu Morphisms on Quantum Chu Spaces • Universal Chu spaces. The Representation Theorem • Linear and other type theories. Reducing The Value Set Discussion • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47 Where Next? Introduction Chu Spaces • Connections and contrasts between Chu spaces and coalgebras. Representing Physical Systems • Mixed states — handled generally at the level of Chu spaces. Characterizing Chu Morphisms on Quantum Chu Spaces • Universal Chu spaces. The Representation Theorem Reducing The Value Set Discussion • Linear and other type theories. • Local logics. • Where Next? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 47 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As Chu Spaces and Coalgebras Chu Spaces and Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As Workshop on Informatic Penomena 2009 – 49 Chu Spaces and Coalgebras Introduction Chu Spaces • Coalgebras over Set; ‘universal coalgebra’. Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As Workshop on Informatic Penomena 2009 – 49 Chu Spaces and Coalgebras Introduction Chu Spaces • Coalgebras over Set; ‘universal coalgebra’. Representing Physical Systems • Each of these general systems models has been studied Characterizing Chu Morphisms on Quantum Chu Spaces extensively. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As Workshop on Informatic Penomena 2009 – 49 Chu Spaces and Coalgebras Introduction Chu Spaces • Coalgebras over Set; ‘universal coalgebra’. Representing Physical Systems • Each of these general systems models has been studied Characterizing Chu Morphisms on Quantum Chu Spaces extensively. Their connections have not been studied at all. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As Workshop on Informatic Penomena 2009 – 49 Chu Spaces and Coalgebras Introduction Chu Spaces • Coalgebras over Set; ‘universal coalgebra’. Representing Physical Systems • Each of these general systems models has been studied Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras extensively. Their connections have not been studied at all. • They have complementary merits and deficiencies. • Chu spaces have, coalgebras lack: contravariance. • Coalgebras have, Chu spaces lack: extension in time. • Symmetry vs. rigidity. Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As Workshop on Informatic Penomena 2009 – 49 Chu Spaces and Coalgebras Introduction Chu Spaces • Coalgebras over Set; ‘universal coalgebra’. Representing Physical Systems • Each of these general systems models has been studied Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras • Chu Spaces and Coalgebras extensively. Their connections have not been studied at all. • They have complementary merits and deficiencies. • Chu spaces have, coalgebras lack: contravariance. • Coalgebras have, Chu spaces lack: extension in time. • Symmetry vs. rigidity. Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Big Toy Models Contravariance As • Interesting formal consequences: • Indexed structure (‘externalising contravariance’) • Grothendieck construction: new description of Chu spaces. • Truncation functors. Workshop on Informatic Penomena 2009 – 49 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Primer on coalgebra Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces Category theory allows us to dualize algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions, coalgebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51 Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Category theory allows us to dualize algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions, coalgebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: • Programming over infinite data structures: streams, lazy lists, infinite trees . . . Discussion Chu Spaces and Coalgebras Primer on coalgebra • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51 Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Category theory allows us to dualize algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions, coalgebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: • Programming over infinite data structures: streams, lazy lists, infinite trees . . . Discussion Chu Spaces and Coalgebras • A novel notion of coinduction Primer on coalgebra • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51 Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Category theory allows us to dualize algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions, coalgebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: • Programming over infinite data structures: streams, lazy lists, infinite trees . . . Discussion Chu Spaces and Coalgebras • A novel notion of coinduction Primer on coalgebra • Modelling state-based computations of all kinds • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51 Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Category theory allows us to dualize algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions, coalgebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: • Programming over infinite data structures: streams, lazy lists, infinite trees . . . Discussion Chu Spaces and Coalgebras • A novel notion of coinduction Primer on coalgebra • Modelling state-based computations of all kinds • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras • The key notion of bisimulation equivalence between processes. Comparison: A First Try Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51 Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Category theory allows us to dualize algebras to obtain a notion of coalgebras of an endofunctor. However, while algebras abstract a familiar set of notions, coalgebras open up a new and rather unexpected territory, and provides an effective abstraction and mathematical theory for a central class of computational phenomena: • Programming over infinite data structures: streams, lazy lists, infinite trees . . . Discussion Chu Spaces and Coalgebras • A novel notion of coinduction Primer on coalgebra • Modelling state-based computations of all kinds • Coalgebras Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • The key notion of bisimulation equivalence between processes. • A general coalgebraic logic can be read off from the functor, and used to specify and reason about properties of systems. Semantics in One Country Externalising Contravariance As Big Toy Models Workshop on Informatic Penomena 2009 – 51 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras • The Final Coalgebra Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Basic Concepts F -Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Let F : C −→ C be a functor. An F -coalgebra is a pair (A, γ : A −→ F A) for some object A of C. We say that A is the carrier of the coalgebra, while γ is the behaviour map. An F -coalgebra homomorphism from (A, γ : A −→ F A) to (B, δ : B −→ F B) is an arrow h : A −→ B such that Reducing The Value Set Discussion A γ- FA Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras h Fh ? B ? δ - FB F -coalgebras and their homomorphisms form a category F −Coalg. • The Final Coalgebra Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 53 Final F -coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces An F -coalgebra (C, γ) is final if for every F -coalgebra (A, α) there is a unique homomorphism from (A, α) to (C, γ). Proposition 22 If a final F -coalgebra exists, it is unique up to isomorphism. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Proposition 23 (Lambek Lemma) If γ : A −→ F A is final, it is an isomorphism Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras • The Final Coalgebra Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 54 Labelled Transition Systems Introduction Chu Spaces Let A be a set of actions. A labelled transition system over A is a coalgebra for the functor Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem LTA : Set −→ Set :: X 7→ Pf (A × X). Such a coalgebra γ : S −→ Pf (A × S) Reducing The Value Set Discussion Chu Spaces and Coalgebras can be understood operationally as follows: • S is a set of states Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras • The Final Coalgebra • For each state s ∈ S , γ(s) specifies the possible transitions from a that state. We write s −→ s′ if (a, s′ ) ∈ γ(s). We think of such a transition as consisting of the system performing the action a, and changing state from s to s′ . Note that we regard actions as directly observable, while states are not. Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 55 Transition Graphs as Coalgebras Introduction Chu Spaces Representing Physical Systems Note that any labelled transition graph (or “state machine”) with labels in A is a coalgebra for LTA . Examples 1. a Characterizing Chu Morphisms on Quantum Chu Spaces b 1 c 2 The Representation Theorem Reducing The Value Set This corresponds to the coalgebra ({1, 2}, γ) Discussion γ : 1 7→ {(a, 1), (b, 2)}, Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition 2. c b 1 Systems • Transition Graphs as Coalgebras • The Final Coalgebra 2 7→ {(c, 2)} 1 7→ {(b, 2), (c, 1)}, a 2 a 2 7→ {(a, 1), (a, 3)}, 3 3 7→ ∅ Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 56 The Final Coalgebra Introduction The key point is this. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces Proposition 24 For any set A of actions, there is a final LTA -coalgebra (ProcA , out). The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras • The Final Coalgebra Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 57 The Final Coalgebra Introduction The key point is this. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Proposition 24 For any set A of actions, there is a final LTA -coalgebra (ProcA , out). We think of elements of the final coalgebra as processes. The final coalgebra provides a “universal semantics” for transition systems, which is “fully abstract” with respect to observable behaviour. Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras • The Final Coalgebra Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 57 The Final Coalgebra Introduction The key point is this. Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Proposition 24 For any set A of actions, there is a final LTA -coalgebra (ProcA , out). We think of elements of the final coalgebra as processes. The final coalgebra provides a “universal semantics” for transition systems, which is “fully abstract” with respect to observable behaviour. All of this generalizes to a large class of endofunctors. Primer on coalgebra Basic Concepts • F -Coalgebras • Final F -coalgebras • Labelled Transition Systems • Transition Graphs as Coalgebras • The Final Coalgebra Representing Physical Systems As Coalgebras Big Toy Models Comparison: A First Workshop on Informatic Penomena 2009 – 57 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras • Coalgebras as Models of Physical Systems Comparison: A First Try Semantics in One Country Big Toy Models Externalising Representing Physical Systems As Coalgebras Coalgebras as Models of Physical Systems Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras • Coalgebras as Models of Physical Systems Comparison: A First Try Semantics in One Country Big Toy Models Externalising Workshop on Informatic Penomena 2009 – 59 Coalgebras as Models of Physical Systems Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces Recall our basic setup: systems are modelled by a set of states S , of questions Q, and an evaluation e : S × Q → [0, 1]. The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras • Coalgebras as Models of Physical Systems Comparison: A First Try Semantics in One Country Big Toy Models Externalising Workshop on Informatic Penomena 2009 – 59 Coalgebras as Models of Physical Systems Introduction Chu Spaces Representing Physical Systems Recall our basic setup: systems are modelled by a set of states S , of questions Q, and an evaluation e : S × Q → [0, 1]. Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Problems: Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras • Coalgebras as Models of Physical Systems Comparison: A First Try Semantics in One Country Big Toy Models Externalising Workshop on Informatic Penomena 2009 – 59 Coalgebras as Models of Physical Systems Introduction Chu Spaces Representing Physical Systems Recall our basic setup: systems are modelled by a set of states S , of questions Q, and an evaluation e : S × Q → [0, 1]. Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Problems: • In Chu spaces, we get to specify Q as well as S . How do we do this with coalgebras? Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras • Coalgebras as Models of Physical Systems Comparison: A First Try Semantics in One Country Big Toy Models Externalising Workshop on Informatic Penomena 2009 – 59 Coalgebras as Models of Physical Systems Introduction Chu Spaces Representing Physical Systems Recall our basic setup: systems are modelled by a set of states S , of questions Q, and an evaluation e : S × Q → [0, 1]. Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras • Coalgebras as Models of Physical Systems Problems: • In Chu spaces, we get to specify Q as well as S . How do we do this with coalgebras? • Q is contravariant (the maps f ∗ go backwards.). Coalgebras are based on covariant functors. (We could work with domains, but there are drawbacks). Comparison: A First Try Semantics in One Country Big Toy Models Externalising Workshop on Informatic Penomena 2009 – 59 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Comparison: A First Try First Approximation Fix a set K . We can define a functor on Set: FK : X 7→ 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 7→ 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 7→ 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 7→ 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∗−1 . Then f∗ : GC → GC ′ is an FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism f : GC → GC ′ , then (f, f −1 ) : C → C ′ is a Chu morphism. 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∗−1 . Then f∗ : GC → GC ′ is an FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism f : GC → GC ′ , then (f, f −1 ) : C → C ′ is a Chu morphism. Let NChuK be the category of normal Chu spaces and Chu morphisms of the form (f, f −1 ). Then by the Proposition, G extends to a 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∗−1 . Then f∗ : GC → GC ′ is an FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism f : GC → GC ′ , then (f, f −1 ) : C → C ′ is a Chu morphism. Let NChuK be the category of normal Chu spaces and Chu morphisms of the form (f, f −1 ). Then by the Proposition, G extends to a 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∗−1 . Then f∗ : GC → GC ′ is an FK -algebra homomorphism. Conversely, given any FK -algebra homomorphism f : GC → GC ′ , then (f, f −1 ) : C → C ′ is a Chu morphism. Let NChuK be the category of normal Chu spaces and Chu morphisms of the form (f, f −1 ). Then by the Proposition, G extends to a functor G : NChuK → FK −Coalg, with G(f, f −1 ) = f . There is an evident inverse to this functor. Proposition 26 Big Toy Models NChuK and FK −Coalg are isomorphic categories. Workshop on Informatic Penomena 2009 – 62 Discussion: Critique of Coalgebras Big 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 Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 64 Discussion: In Praise of Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • The coalgebraic point of view can be described as state-based, but in a way that emphasizes that the meaning of states lies in their observable behaviour. Indeed, in the “universal model” we shall construct, the states are determined exactly as the possible observable behaviours — we actually find a canonical solution for what the state space should be in these terms. States are identified exactly if they have the same observable behaviour. Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 64 Discussion: In Praise of Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • The coalgebraic point of view can be described as state-based, but in a way that emphasizes that the meaning of states lies in their observable behaviour. Indeed, in the “universal model” we shall construct, the states are determined exactly as the possible observable behaviours — we actually find a canonical solution for what the state space should be in these terms. States are identified exactly if they have the same observable behaviour. Discussion Chu Spaces and Coalgebras Primer on coalgebra We can see this as a kind of reconciliation between the ontic and epistemic standpoints. Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 64 Discussion: In Praise of Coalgebras Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set • The coalgebraic point of view can be described as state-based, but in a way that emphasizes that the meaning of states lies in their observable behaviour. Indeed, in the “universal model” we shall construct, the states are determined exactly as the possible observable behaviours — we actually find a canonical solution for what the state space should be in these terms. States are identified exactly if they have the same observable behaviour. Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try We can see this as a kind of reconciliation between the ontic and epistemic standpoints. • Coalgebras allow us to capture the ‘dynamics of measurement’ — what happens after a measurement — in a way that Chu spaces don’t. They have ‘extension in time’. • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 64 Extension in Time Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 65 Extension in Time Introduction Consider a coalgebraic representation of stochastic transducers: Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem F : X 7→ Prob(O × X)I where I is a fixed set of inputs, O a fixed set of outputs, and Prob(S) is the set of probability distributions on S . Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 65 Extension in Time Introduction Consider a coalgebraic representation of stochastic transducers: Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion F : X 7→ Prob(O × X)I where I is a fixed set of inputs, O a fixed set of outputs, and Prob(S) is the set of probability distributions on S . We can think of I as a set of questions, and O as a set of answers (which we could standardize by only considering yes/no questions). Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models Workshop on Informatic Penomena 2009 – 65 Extension in Time Introduction Consider a coalgebraic representation of stochastic transducers: Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras F : X 7→ Prob(O × X)I where I is a fixed set of inputs, O a fixed set of outputs, and Prob(S) is the set of probability distributions on S . We can think of I as a set of questions, and O as a set of answers (which we could standardize by only considering yes/no questions). What we learn from this is that Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try • First Approximation • Comparison • Discussion: Critique of Coalgebras • Discussion: In Praise ofBig Coalgebras Toy Models QM is less nondeterministic/probabilistic than stochastic transducers since in QM if we know the preparation and the outcome of the measurement, we know (by the projection postulate) exactly what the resulting quantum state will be. Workshop on Informatic Penomena 2009 – 65 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country • Coalgebraic Semantics For One System Models •Big WellToy Behaved Semantics in One Country Coalgebraic Semantics For One System 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). 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 7→ ({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 7→ ({0} + (0, 1] × X)Q . A coalgebra for this functor is then a map α : X → ({0} + (0, 1] × X)Q The interpretation is that X is a set of states; the coalgebra map sends its state to its behaviour, which is a function from questions in Q to the probability that the answer is ‘yes’; and, if the probability is not 0, to the successor state following a ‘yes’ answer. 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 Coalgebra Big 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 Big 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 Lüders rule. Big Toy Models Workshop on Informatic Penomena 2009 – 69 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Externalising Contravariance As Indexing The Indexed Category Introduction We define a functor F : Setop → CAT Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem with Q 7→ F Q −Coalg and for f : Q′ → Q: Reducing The Value Set ′ tf : F Q → F Q :: Θ 7→ Θ ◦ f Discussion Chu Spaces and Coalgebras is a natural transformation, and Primer on coalgebra ∗ F(f ) = f : Coalg−F Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Q → Coalg−F Q′ f ∗ : (A, α) 7→ (A, tfA ◦ α) is a functor. Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Workshop on Informatic Penomena 2009 – 71 The Indexed Category Introduction We define a functor F : Setop → CAT Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem with Q 7→ F Q −Coalg and for f : Q′ → Q: Reducing The Value Set ′ tf : F Q → F Q :: Θ 7→ Θ ◦ f Discussion Chu Spaces and Coalgebras is a natural transformation, and Primer on coalgebra ∗ F(f ) = f : Coalg−F Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Q → Coalg−F Q′ f ∗ : (A, α) 7→ (A, tfA ◦ α) is a functor. Thus we get a strict indexed category of coalgebra categories, with contravariant indexing. Workshop on Informatic Penomena 2009 – 71 The Grothendieck Construction Introduction Chu Spaces Where we have an indexed category, we can apply the Grothendieck construction to glue all the fibres together (and get a fibration). Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Workshop on Informatic Penomena 2009 – 72 The Grothendieck Construction Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Where we have an indexed category, we can apply the Grothendieck construction to glue all the fibres together (and get a fibration). Given a functor I : Cop → CAT R define I with objects (A, a), where A is an object of C and a is an object of I(A). Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Workshop on Informatic Penomena 2009 – 72 The Grothendieck Construction Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Where we have an indexed category, we can apply the Grothendieck construction to glue all the fibres together (and get a fibration). Given a functor I : Cop → CAT R define I with objects (A, a), where A is an object of C and a is an object of I(A). Arrows are (G, g) : (A, a) → (B, b), where G : B → A and g : I(G)(a) → b. Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Workshop on Informatic Penomena 2009 – 72 The Grothendieck Construction Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Where we have an indexed category, we can apply the Grothendieck construction to glue all the fibres together (and get a fibration). Given a functor I : Cop → CAT R define I with objects (A, a), where A is an object of C and a is an object of I(A). Arrows are (G, g) : (A, a) → (B, b), where G : B → A and g : I(G)(a) → b. Composition of (G, g) : (A, a) → (B, b) and (H, h) : (B, b) → (C, c) is given by (G ◦ H, h ◦ I(G)(g)) : (A, a) → (C, c). Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models • The Indexed Workshop on Informatic Penomena 2009 – 72 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models Indexed Comparison With Chu Spaces 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 (− × Q, K̂) where K̂ : 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 (− × Q, K̂) where K̂ : 1 → Set picks out the object K . Given f : Q′ → Q, we define a functor ∗ f : ChuQ K → Q′ ChuK :: (X, Q, e) 7→ (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 (− × Q, K̂) where K̂ : 1 → Set picks out the object K . Given f : Q′ → Q, we define a functor ∗ f : ChuQ K → Q′ ChuK :: (X, Q, e) 7→ (X, Q′ , e ◦ (1 × f )) and which is the identity on morphisms. This gives an indexed category Chu : Setop → CAT Big Toy Models Workshop on Informatic Penomena 2009 – 74 Grothendieck puts Chu back together again Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models Workshop on Informatic Penomena 2009 – 75 Grothendieck puts Chu back together again Introduction Chu Spaces Representing Physical Systems Proposition 27 Z Chu ∼ = ChuK . Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models Workshop on Informatic Penomena 2009 – 75 The Truncation Functor Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models Workshop on Informatic Penomena 2009 – 76 The Truncation Functor Introduction Chu Spaces The relationship between coalgebras and Chu spaces is further clarified by an indexed truncation functor T : F → Chu. Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models Workshop on Informatic Penomena 2009 – 76 The Truncation Functor Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The relationship between coalgebras and Chu spaces is further clarified by an indexed truncation functor T : F → Chu. For each set Q there is a functor TQ : F Q −Coalg → ChuQ K The Representation Theorem TQ (X, α) = (X, Q, e) Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras where   0, α(x)(q) = 0 e(x, q) =  r, α(x)(q) = (r, x′ ) Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models Workshop on Informatic Penomena 2009 – 76 The Truncation Functor Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The relationship between coalgebras and Chu spaces is further clarified by an indexed truncation functor T : F → Chu. For each set Q there is a functor TQ : F Q −Coalg → ChuQ K The Representation Theorem TQ (X, α) = (X, Q, e) Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras where   0, α(x)(q) = 0 e(x, q) =  r, α(x)(q) = (r, x′ ) For f : Q′ → Q there is a natural transformation ′ Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models τf : TQ → TQ f τ(X,α) Q = (idX , f ) : T (X, α) → T Q′ (X, α). Workshop on Informatic Penomena 2009 – 76 Introduction Chu Spaces Representing Physical Systems Characterizing Chu Morphisms on Quantum Chu Spaces The Representation Theorem Reducing The Value Set Discussion Chu Spaces and Coalgebras Primer on coalgebra Basic Concepts Representing Physical Systems As Coalgebras Comparison: A First Try Semantics in One Country Externalising Contravariance As Indexing Big Toy Models A Universal Model A Universal Model 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: 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 ′ ∗ Q induces a functor f : F −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 ′ ∗ Q induces a functor f : F −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 Universality 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). 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

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