Braided notions of dialogue categories Paul-Andr Mellis CNRS & - PowerPoint PPT Presentation

Braided notions of dialogue categories Paul-André Melliès CNRS & Université Paris Denis Diderot Groupe de travail Sémantique Laboratoire PPS 24 janvier 2012

Where is the flow of logic? Looking for a connection between proofs and knots

Revisiting proof-nets in linear logic Claim: the traditional distinction between proof nets ↔ proof structures deserves to be understood from this topological point of view. 3

Sequent calculus In linear logic, the two equivalent proofs π 1 π 2 π 2 π 3 · · · · π 3 π 1 · · · · · · · · · · · · ⊢ A ⊢ B , C · ⊢ B , C ⊢ D · ⊢ A ⊗ B , C ⊢ D ⊢ A ⊢ B , C ⊗ D ⊢ A ⊗ B , C ⊗ D ⊢ A ⊗ B , C ⊗ D 4

& Proof nets are interpreted as the same proof net: π 1 π 2 π 3 A B C D 5

& Sequentialization by deformation π 2 π 1 π 3 π 2 π 3 · · π 1 · · · · · · · ⊢ B , C ⊢ D ⊢ A ⊢ B , C ⊗ D ⊢ A ⊗ B , C ⊗ D A C B D 6

& Sequentialization by deformation π 2 π 1 π 3 π 1 π 2 · · π 3 · · · · · · ⊢ A ⊢ B , C · ⊢ A ⊗ B , C ⊢ D ⊢ A ⊗ B , C ⊗ D A C B D 7

& & & & Multiplicative proof nets axiom axiom axiom B* A* B A A* A B B* & A* A & A A* A B B A Unfortunately, proof nets are not exactly string diagrams... 8

Tensorial logic tensorial logic = a logic of tensor and negation = linear logic without A � ¬¬ A = the syntax of linear continuations = the syntax of dialogue games Provides a synthesis of linear logic and game semantics Research program: recast there the various aspects of linear logic 9

Tensorial logic Γ ⊢ A A , ∆ ⊢ B axiom cut A ⊢ A Γ , ∆ ⊢ B Γ , A ⊢ ⊥ Γ ⊢ A left ¬ right ¬ Γ , ¬ A ⊢ ⊥ Γ ⊢ ¬ A Γ , A , B ⊢ C Γ ⊢ A ∆ ⊢ B right ⊗ left ⊗ Γ , ∆ ⊢ A ⊗ B Γ , A ⊗ B ⊢ C Γ ⊢ A right true left true ⊢ true Γ , true ⊢ A 10

Dialogue categories A monoidal category with a left duality A natural bijection between the set of maps A ⊗ B −→ ⊥ and the set of maps B −→ A ⊸ ⊥ A familiar situation in tensorial algebra 11

Dialogue categories A monoidal category with a right duality A natural bijection between the set of maps A ⊗ B −→ ⊥ and the set of maps A −→ ⊥ � B A familiar situation in tensorial algebra 12

Dialogue categories Definition. A dialogue category is a monoidal category C equipped with ⊲ an object ⊥ ⊲ two natural bijections : C ( A ⊗ B , ⊥ ) C ( B , A ⊸ ⊥ ) ϕ A , B −→ : C ( A ⊗ B , ⊥ ) C ( A , ⊥ � B ) ψ A , B −→ 13

� � � � Cyclic dialogue categories A dialogue category equipped with a family of bijections wheel A , B : C ( A ⊗ B , ⊥ ) −→ C ( B ⊗ A , ⊥ ) natural in A and B making the diagram associativity C (( B ⊗ C ) ⊗ A , ⊥ ) � C ( A ⊗ ( C ⊗ B ) , ⊥ ) wheel A , B ⊗ C wheel B , C ⊗ A C ( A ⊗ ( B ⊗ C )) C (( C ⊗ A ) ⊗ B , ⊥ ) associativity associativity wheel A ⊗ B , C C (( A ⊗ B ) ⊗ C , ⊥ ) � C ( C ⊗ ( A ⊗ B ) , ⊥ ) commutes. 14

Cyclic dialogue categories The wheel should be understood diagrammatically as: wheel x , y : f �→ f x y y x 15

The coherence diagram f wheel x wheel , y z y , z x y z x wheel x y ,z f f x y z z x y 16

� � � � � � An equivalent formulation A dialogue category equipped with a natural isomorphism turn A : A ⊸ ⊥ −→ ⊥ � B making the diagram below commute: ⊥ eval eval ( ⊥ � A ) ⊗ A B ⊗ ( B ⊸ ⊥ ) turn − 1 turn A B ( A ⊸ ⊥ ) ⊗ A B ⊗ ( ⊥ � B ) eval eval turn A ⊗ B � B ⊗ ( ⊥ � ( A ⊗ B )) ⊗ A B ⊗ (( A ⊗ B ) ⊸ ⊥ ) ⊗ A 17

Balanced dialogue categories A braiding γ A , B : A ⊗ B −→ B ⊗ A A twist θ A : A −→ A B A A B A A 18

� � Main theorem Every category C of atomic formulas induces a functor [ − ] such that [ − ] � free-ribbon ( C ∗ ) free-dialogue ( C ) C where C ∗ is the category C extended with an object ∗ . Theorem. The functor [ − ] is faithful. Equality of proofs reduces to equality of knots modulo deformation 19

String Diagrams A notation by Roger Penrose 20

Monoidal Categories A monoidal category is a category C equipped with a functor: ⊗ : C × C −→ C an object: I and three natural transformations: α ( A ⊗ B ) ⊗ C A ⊗ ( B ⊗ C ) −→ ρ λ I ⊗ A A A ⊗ I A −→ −→ satisfying a series of coherence properties. 21

String Diagrams A morphism f : A ⊗ B ⊗ C −→ D ⊗ E is depicted as: D E f C A B 22

Composition f g The morphism A −→ B −→ C is depicted as C C g g ◦ f = B f A A A Vertical composition 23

Tensor product f g The morphism ( A −→ D ) is depicted as −→ B ) ⊗ ( C D B ⊗ D B f ⊗ g g = f C A ⊗ C A Horizontal tensor product 24

Example D B f D A f ⊗ id D 25

Example D B f g C A ( f ⊗ id D ) ◦ ( id A ⊗ g ) 26

Example D B g f C A ( id B ⊗ g ) ◦ ( f ⊗ id C ) 27

Meaning preserved by deformation D D B B g f = g f C C A A ( f ⊗ id D ) ◦ ( id A ⊗ g ) = ( id B ⊗ g ) ◦ ( f ⊗ id C ) 28

The functorial approach to knot invariants Ribbon categories 29

Braided categories A monoidal category C equipped with a family of isomorphisms : A ⊗ B −→ B ⊗ A γ A , B natural in A and B , represented pictorially as the positive braiding B A B A 30

Braided categories As expected, the inverse map γ − 1 : B ⊗ A −→ A ⊗ B A , B is represented pictorially as the negative braiding A B A B 31

� � � � Coherence diagram for braids [1] γ � ( B ⊗ C ) ⊗ A A ⊗ ( B ⊗ C ) α α ( A ⊗ B ) ⊗ C B ⊗ ( C ⊗ A ) B ⊗ γ γ ⊗ C α � B ⊗ ( A ⊗ C ) ( B ⊗ A ) ⊗ C 32

Same coherence diagram in string diagrams y z x y z x = x y z x y z 33

� � � � Coherence diagram for braids [2] γ � C ⊗ ( A ⊗ B ) ( A ⊗ B ) ⊗ C α − 1 α − 1 A ⊗ ( B ⊗ C ) ( C ⊗ A ) ⊗ B γ ⊗ B A ⊗ γ α − 1 � ( A ⊗ C ) ⊗ B A ⊗ ( C ⊗ B ) 34

Same coherence diagram in string diagrams z x y z x y = x y z x y z 35

Balanced categories A braided monoidal category C equipped with a twist : A −→ A θ A defined as a natural family of isomorphisms, and depicted as A A 36

� � � � Coherence for twists The twist θ is required to satisfy the equality id I θ I = and to make the diagram γ A , B A ⊗ B B ⊗ A θ A ⊗ B θ B ⊗ θ A A ⊗ B B ⊗ A γ B , A commute for all objects A and B . 37

Coherence for twists x y = θ x ⊗ y y x 38

Duality A dual pair A ⊣ B is defined as a pair of morphisms η : I −→ A ⊗ B ε : B ⊗ A −→ I which are depicted as B A B A 39

Coherence for duality The two morphisms η and ε should satisfy the “zig-zag” equalities: A B B A = = A B A B In that case, A is called a right dual of B . 40

� � � Ribbon categories Definition. A ribbon category is a balanced category C where ⊲ every object A has a right dual A ∗ ⊲ the diagram A ∗ ⊗ θ A A ∗ ⊗ A A ∗ ⊗ A θ A ∗ ⊗ A ε A ∗ ⊗ A ε � I commutes for all objects A . 41

Ribbon categories Remark. – In a ribbon category, the object A ∗ is also a left dual of A . * x * ε ’ x x x ε = = x x * x x * η η ’ 42

Ribbon categories Hence – the equations below are satisfied in every ribbon category x x x ε ε ’ * x = = * x η ’ η x x x 43

The free ribbon category Theorem [Shum 1994] The free ribbon category free-ribbon ( C ) generated by a category C has ⊲ as objects: signed sequences ( A ε 1 1 , . . . , A ε k k ) of objects of C , ⊲ as morphisms: framed tangles with links labelled by maps in C . 44

The free ribbon category So, a typical morphism in the category free-ribbon ( C ) ( A + ) ( B + , C − , D + ) −→ looks like this: B + C − D + g f A + where f : A → B and g : C → D are morphisms in the category C . 45

� � Knot invariants Every ribbon category D induces a knot invariant [ − ] free-ribbon ( C ) � D C The free ribbon category is a category of framed tangles 46

Jones polynomial invariant � x 4 + y 2 2 x 2 − x 4 + x 2 y 2 x 2 + 1 2 x 2 47

A construction of ribbon categories Categories of modules over Hopf algebras 48

Bialgebras A bialgebra in a braided category is an object H equipped with four maps µ : H ⊗ H → H η : I → H δ : H → H ⊗ H ε : H → I depicted as H H H H η µ ε δ H H H H defining a monoid and a comonoid, and satisfying the four equalities... 49

Bialgebras H H H H H H H H = = H H H H = = id H H H H 50

Antipode A Hopf algebra is a bialgebra equipped with a morphism S : H −→ H satisfying the equality: H H H = = S S H H H 51