Zooms for Effects Dominique Duval Udine, September 11., 2009 IFIP W.G.1.3. meeting
Outline Introduction Diagrammatic logics Parameterization Sequential product Conclusion
Motivations Wanted. A framework for the semantics of effects. Monads. For two kinds of morphisms: ◮ in general f : X → Y “stands for” some f ′ : X → T ( Y ) ◮ sometimes v : X → Y is pure, then v ′ = η ◦ v Wanted. Several kinds of objects, of arrows, of equations,... each kind “stands for” something...
In this talk A category of logics ◮ objects: “logics” with models and proofs ◮ morphisms: “stands for” should be a morphism
“stands for”? E.g., a monad. ◮ in general f : X → Y “stands for” some f ′ : X → T ( Y ) f f T ( Y ) X Y X Far Near
“stands for” is part of a “zoom” E.g., a monads ◮ in general f : X → Y “stands for” some f ′ : X → T ( Y ) ◮ sometimes v : X → Y is pure, then v ′ = η ◦ v f v X Y “Decorated” f f ′ v v ′ T ( Y ) X Y X = v Y η Far Near
“zooms” are spans Dec Far Near f v X Y Decorated f f ′ v ′ v T ( Y ) X Y X = v Y η Far Near Slogan. First be wrong, then add corrections, in order to finally get right.
This talk ◮ Diagrammatic logics (categories...) with Christian Lair . ◮ Zooms for parameterization with C´ esar Dom´ ınguez . ◮ A zoom for sequential product with Jean-Guillaume Dumas and Jean-Claude Reynaud .
Outline Introduction Diagrammatic logics Parameterization Sequential product Conclusion
A diagrammatic logic Definition. A logic L is a functor with a full and faithful right adjoint R : L S T ⊥ ff R In addition, this is induced by a morphism of limit sketches. Properties. ◮ R makes T a full subcategory of S ◮ L ( R (Θ)) ∼ = Θ for each theory Θ ◮ S and T have colimits, and L preserves colimits
Models L S T ⊥ ff R Definitions. ◮ S is the category of specifications ◮ T is the category of theories ◮ Σ presents Θ when Θ ∼ = L (Σ) . ◮ Σ and Σ ′ are equivalent when L (Σ) ∼ = L (Σ ′ ) . Models. Mod (Σ , Θ) = T [ L (Σ) , Θ] ∼ = S [Σ , R (Θ)] The models form a category iff T is a 2-category.
Proofs Theorem. [Gabriel-Zisman 1967] (for homotopy theory) Up to equivalence, L is a localization: it adds inverses to some morphisms in S . Definition. An entailment is τ : Σ → Σ ′ in S such that L ( τ ) is invertible in T . Then Σ and Σ ′ are equivalent. Hence: the bicategory of fractions S2 . Definition. A proof is a fraction. in S2 : σ Σ ′ Σ Σ 1 1 τ in S : in T : ( L τ ) − 1 L σ σ Σ ′ L Σ ′ Σ Σ 1 L Σ 1 L Σ 1 1 τ L τ
Morphisms of logics Definition. A morphism of logics F : L 1 → L 2 is a pair of functors ( F S , F T ) such that: L 1 T 1 S 1 ∼ F S F T = L 2 S 2 T 2 In addition, they are induced by morphisms of limit sketches. Definition. A 2-morphism of logics ℓ : F ⇒ F ′ : L 1 → L 2 is a pair of natural transformations ( ℓ S , ℓ T ) such that: L 1 S 1 T 1 l S l T F ′ = F ′ F S ⇒ F T ⇒ S T L 2 S 2 T 2
Altogether... ◮ A 2-category of logics DiaLog with a 2-functor that focuses on the theories: DiaLog → Cat ( L : S → T ) �→ T ◮ “Everything” happens in the bicategory of fractions: a specification Σ should be seen up to equivalence.
Outline Introduction Diagrammatic logics Parameterization Sequential product Conclusion
Parameterization Starting point: Sergeraert’s software for effective homology. Goal: formalize the process of: ◮ adding a parameter to some operations ◮ then passing a value (an argument) to the parameter A kind of benchmark, that may be treated with monads ( T ( X ) = X A ), hidden algebras, coalgebras, institutions...
Parameterization and diagrammatic logics ◮ Parameterization: a zoom ◮ Parameter passing: a zoom and a 2-morphism
Example: Differential monoids A specification of monoids Mon : ◮ type G ◮ operations prd : G 2 → G , e : → G ◮ equations prd ( x , prd ( y , z )) = prd ( prd ( x , y ) , z ) , prd ( x , e ) = x , prd ( e , x ) = x A specification of differential monoids DMon : ◮ Mon with ◮ operation dif : G → G ◮ equations dif ( prd ( x , y )) = prd ( dif ( x ) , dif ( y )) , dif ( e ) = e , dif ( dif ( x )) = e
A specification of decorated differential monoids DecDMon : ◮ Mon with ◮ operation dif : G → G ◮ equations dif ( prd ( x , y )) = prd ( dif ( x ) , dif ( y )) , dif ( e ) = e , dif ( dif ( x )) = e A specification for monoids with a parameterized differential ParDMon : ◮ Mon with ◮ type A ◮ operation dif ′ : A × G → G ◮ equations dif ′ ( p , ( prd ( x , y ))) = prd ( dif ′ ( p , x ) , dif ′ ( p , y )) , dif ′ ( p , e ) = e , dif ′ ( p , dif ′ ( p , x )) = e
A zoom for parametererizing Dec Far Near prd G 2 G dif G G DecDMon G 2 prd G prd G 2 A × G 2 G dif dif ′ G G A × G G DMon ParDMon
Parameter passing Each parameterized differential monoid PM together with an argument α ∈ PM ( A ) ⇒ a differential monoid M α with: – the same underlying monoid as PM – the differential x �→ M α ( dif )( x ) = PM ( dif ′ )( α, x ) In the specifications: Add a constant a : 1 → A in the “near” logic.
A zoom for parameter passing... Dec Far Near dif G G DecDMon dif ′ dif a × G A × G G G G G DMon ParDMon
...with a 2-morphism of logics Dec ⇒ Far Near dif G G DecDMon dif a × G A × G dif ′ G G G G DMon ParDMon ↑ dif G G
Outline Introduction Diagrammatic logics Parameterization Sequential product Conclusion
Sequential product Goal: formalize the fact that the order of evaluation of the arguments does matter when there are effects. Monads: the strength. In the framework of diagrammatic logics: A zoom, from an ordinay product to a sequential product. There are two kinds of morphisms And two kinds of equations!
About products X = X 1 × X 2 , Y = Y 1 × Y 2 , Z = Y 1 × X 2 . Without effects: g × f = ( id × g ) ◦ ( f × id ) id f f X 1 Y 1 X 1 Y 1 Y 1 = = = id × g = f × id f × g X Y X Z Y = = = id g g X 2 Y 2 X 2 X 2 Y 2
A zoom for the sequential product Dec Far Near f X 1 Y 1 = f × v X Y ≈ v X 2 Y 2 f f X 1 Y 1 S × X 1 S × X 2 = = f × v f × v X Y S × X S × Y = = v v X 2 Y 2 X 2 Y 2
Outline Introduction Diagrammatic logics Parameterization Sequential product Conclusion
Zooms Dec expansion undecoration Far Near PROOFS MODELS
THANK YOU!
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