✏ ✏ O An algebraic presentation of dialogue categories Paul-André Melliès ú August 22, 2012 Abstract In this paper, we describe an algebraic presentation of the notion of helical dialogue chirality. In particular, the helix structure enables us to decompose the dual of the left negation as the right negation of the dual. 1 Motivations The study of dialogue categories and chiralities leads to the following co- herence diagrams for the axiom combinator: axiom [ n ] / ( L (( a 7 m ) 7 n ) 6 n ú ) 6 m ú L ( a 7 m ) 6 m ú associativity L ( a 7 ( m 7 n )) 6 ( n ú 6 m ú ) axiom [ m ] monoidality axiom [ m 7 n ] / L ( a 7 ( m 7 n )) 6 ( m 7 n ) ú La commutes for all objects a, m, n and morphisms f : m æ n of the category A . In string diagrams: ∗ CNRS, Laboratoire PPS, UMR 7126, Université Paris Diderot, Sorbonne Paris Cité, F-75205 Paris, France. This work has been partly supported by the ANR Project RECRE. 1
/ / / / * * * * n m L m n n m L m n ( a ) = L L L L false true ( b ) = L L From the logical point of view, the two coherence diagrams ( ?? ) and ( ?? ) should be understood as η -expansion laws for the axiom link. Typically, the purpose of the η -expansion law ( ?? ) is to decompose the link axiom [ m 7 n ] into the more elementary links axiom [ m ] and axiom [ n ] . A natural question is whether there exists a similar η -expansion law which decomposes the axiom link right . axiom [ Rm ] L ( a 7 Rm ) 6 ( Rm ) ú La associated to the negation of m into the axiom link associated to the ob- ject m of the category B . To that purpose, it appears necessary to start from the left axiom link left . axiom [ ú m ] / equivalence / ( ú m ) ú 6 L ( ú m ) m 6 L ( ú m ) L ( true ) associated to the object ú m living this time in the category A . Then, the helical structure isomorphism L ( ú m ) ( Rm ) ú This defines a morphism η map L ( a 7 RL ( true )) L ( a 7 R ( m 6 ( Rm ) ú )) La 2
O ✏ ✏ / O At this point, we want a map distributivity L ( a 7 R ( m 6 ( Rm ) ú )) L ( a 7 Rm ) 6 ( Rm ) ú This requires to develop a general theory of these distributivity laws, de- scribing the coherence diagrams. 2 Helical dialogue categories and chiralities We construct a 2-category of helical dialogue categories, helical functors and helical natural transformations. 2.1 A 2-category of helical dialogue categories We define a 2-category HelCat with • helical dialogue categories as 0-cells, • helical functors as 1-cells, • dialogue natural transformations as 2-cells. The 0-dimensional cells. Recall from [6] that a helical dialogue cate- gory C is defined as a dialogue category equipped with a family of bijections : C ( x ¢ y, ‹ ) ≠æ C ( y ¢ x, ‹ ) wheel x,y natural in x and y and required to make the diagram associativity / C ( y ¢ ( z ¢ x ) , ‹ ) C (( y ¢ z ) ¢ x, ‹ ) wheel x,y ¢ z wheel y,z ¢ x (1) C ( x ¢ ( y ¢ z ) , ‹ ) C (( z ¢ x ) ¢ y, ‹ ) associativity associativity wheel x ¢ y,z / C ( z ¢ ( x ¢ y ) , ‹ ) C (( x ¢ y ) ¢ z, ‹ ) commute for all objects x, y, z of the category C . 3
7 ✏ ✏ ✏ / The 1-dimensional cells. A helical functor between two helical dialogue categories is a lax monoidal functor : ≠æ F C D equipped with a morphism ‹ F : F ( ‹ ) ≠æ ‹ such that the diagram F coercion / D ( F ( x ) ¢ F ( y ) , ‹ ) C ( x ¢ y, ‹ ) D ( F ( x ¢ y ) , F ( ‹ )) wheel F ( x ) ,F ( y ) wheel x,y F / D ( F ( y ¢ x ) , F ( ‹ )) coercion / D ( F ( y ) ¢ F ( x ) , ‹ ) C ( y ¢ x, ‹ ) commutes for all objects x , y of the category C . In this diagram, the two co- ercion maps are deduced by precomposing with the lax monoidal structure of the functor F : F ( x ) ¢ F ( y ) ≠æ F ( x ¢ y ) m x,y and by postcomposing with the map ‹ F . The 2-dimensional cells. A dialogue natural transformation : ( F, ‹ F ) ∆ ( G, ‹ G ) θ is defined as a natural transformation : ∆ θ F G making the diagram F ( ‹ ) ‹ F ' ‹ θ ‹ ‹ G G ( ‹ ) commute. The composition law and identities of the 2-category HelCat are defined as expected. 4
o ✏ / o ✏ ✏ � ✏ 2.2 A 2-category of helical dialogue chiralities We construct a 2-category HelChir with helical dialogue chiralities as 0- cells. The 0-dimensional cells. Definition 1 (Helical chirality) A helical chirality is a dialogue chirality equipped with a family of bijections È ú b | a ú Í σ a,b : È a | b Í ≠æ natural in a, b , and making the diagram below commute: χ R a 2 ,a 1 ,b / σ / È ú ( b 6 a ú È a 1 | b 6 a ú 2 ) | a ú È a 2 7 ú b | a ú È a 1 7 a 2 | b Í 2 Í 1 Í 1 Í χ R σ ú b,a 2 ,a 1 È ú b | ( a 1 7 a 2 ) ú Í 1 6 ( ú b ) ú Í È a 2 | a ú χ R a 1 , ú b,a ú σ È ú b | a ú 2 6 a ú 2 È ú b 7 a 1 | a ú È ú ( a ú 1 6 b ) | a ú È a 2 | a ú 1 Í 2 Í 2 Í 1 6 b Í (2) The 1-dimensional cells. A 1-dimensional cell in HelChir F : ( A 1 , B 1 ) ≠æ ( A 2 , B 2 ) is defined as a quadruple F = ( F • , F ¶ ,  F, F ) consisting of a lax monoidal functor F • : A 1 ≠æ A 2 , an oplax monoidal functor F ¶ : B 1 ≠æ B 2 , a mo- noidal natural isomorphism F • A 1 A 2 ( ≠ ) ú  ( ≠ ) ú F (3) B op (0 , 1) / B op (0 , 1) 1 2 F op (0 , 1) ¶ 5
✏ O ✏ / / ✏ ✏ ✏ ✏ / / ✏ O / + 3 together with a natural transformation: F • / A 2 A 1 (4) R R F B 1 B 2 F ¶ making the two diagrams È a | b 6 m ú Í χ m È a 7 m | b Í F a 7 m,b F a,b 6 m ú È F • ( a ) | F ¶ ( b 6 m ú ) Í È F • ( a 7 m ) | F ¶ ( b ) Í (5) monoidality of F ¶ È F • ( a ) | F ¶ ( b ) 6 F ¶ ( m ú ) Í monoidality of F •  F χ F • ( m ) / È F • ( a ) | F ¶ ( b ) 6 F • ( m ) ú Í È F • ( a ) 7 F • ( m ) | F ¶ ( b ) Í F a,b È a | b Í È F • ( a ) | F ¶ ( b ) Í (6) σ a,b σ F • ( a ) ,F ¶ ( b ) F ú b,a ú Â È ú b | a ú Í / È ú ( F ¶ ( b )) | ( F • ( a )) ú Í / È F • ( ú b ) | F ¶ ( a ú ) Í F commute for all objects a, m in A 1 and b in B 1 . Here, the map : È a | b Í ≠æ È F • ( a ) | F ¶ ( b ) Í F a,b is defined as the composite F a,b / È F • ( a ) | F ¶ ( b ) Í È a | b Í F • F A 1 ( a, Rb ) A 2 ( F • ( a ) , F • ( Rb )) A 2 ( F • ( a ) , RF ¶ ( b )) 6
7 / ✏ ✏ ✏ � ✏ ↵ ◆ 7 ) ( - 5 ↵ ◆ � O O ↵ ◆ - 5 / > O ↵ ◆ O ' The 2-dimensional cells. A 2-dimensional cell in HelChir θ : F ∆ G : ( A 1 , B 1 ) ≠æ ( A 2 , B 2 ) is defined as a pair ( θ • , θ ¶ ) of monoidal natural transformations θ • : F • ∆ G • and θ ¶ : G ¶ ∆ F ¶ satisfying the two equations below: F • F • A 1 A 2 A 1 A 2 θ •  F G • ( ≠ ) ú ( ≠ ) ú ( ≠ ) ú ( ≠ ) ú = (7) F op (0 , 1)  ¶ G B op (0 , 1) B op (0 , 1) 5 B op (0 , 1) B op θ op (0 , 1) 1 2 1 2 ¶ G op (0 , 1) G op (0 , 1) ¶ ¶ F • θ • F • / A 2 / A 2 A 1 A 1 G • = (8) R R R R F G G ¶ B 1 B 2 B 1 B 2 F ¶ θ ¶ F ¶ 2.3 Equivalence The 2-functor induces a biequivalence between the 2-categories HelCat and HelChir . 3 Helical chiralities revisited In this section, we study another formulation of helical chiralities. 3.1 Helical dialogue chiralities (bis) Definition 2 (Helical chirality bis) A helical chirality is a pair of mo- noidal categories ( A , 7 , true ) ( B , 6 , false ) 7
✏ / / " O ✏ O c / equipped with a monoidal equivalence ( ≠ ) ú B op (0 , 1) monoidal A equivalence ú ( ≠ ) with two families of bijections È a | b 6 m ú Í χ R : È a 7 m | b Í ≠æ m,a,b È a | m ú 6 b Í χ L : È m 7 a | b Í ≠æ m,a,b natural in a , b and m , where A op ◊ B È ≠ | ≠ Í = A ( ≠ , R ( ≠ ) ) : ≠æ Set The families χ L and χ R are moreover required to make the diagrams below commute: χ R È a | b 6 ( m 7 n ) ú Í m 7 n È a 7 ( m 7 n ) | b Í (9) associativity associativity monoidality of negation χ R χ R / È a 7 m | b 6 n ú Í / È a | ( b 6 n ú ) 6 m ú Í n m È ( a 7 m ) 7 n | b Í χ L m 7 n È a | ( m 7 n ) ú 6 b Í È ( m 7 n ) 7 a | b Í (10) associativity associativity monoidality of negation χ L χ L / È n 7 a | m ú 6 b Í / È a | n ú 6 ( m ú 6 b ) Í m n È m 7 ( n 7 a ) | b Í together with the additional coherence diagram between χ L and χ R : χ R χ L È m 7 a | b 6 n ú Í / È a | m ú 6 ( b 6 n ú ) Í n m È ( m 7 a ) 7 n | b Í (11) associativity associativity χ L χ R / È a | ( m ú 6 b ) 6 n ú Í / È a 7 n | m ú 6 b Í m n È m 7 ( a 7 n ) | b Í Proposition 1 The two notions of helical chirality formulated in Defini- tions 1 and 2 are equivalent. 8
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