Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction Enriched Lawvere Theories theories for Operational Semantics Lawvere theories enriched theories enrichment enriched categories enriched products John C. Baez enriched theories Christian Williams V-theories examples change of University of California, Riverside semantics change of base preserving theories SYCO 4, May 22 2019 applications combinators change of base Conclusion
Introduction Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction theories Lawvere theories enriched theories enrichment How do we integrate syntax and semantics? enriched categories enriched products object type enriched theories V-theories morphism term examples ∗ 2-morphism rewrite ∗ change of semantics change of base preserving theories applications combinators change of base Conclusion
Operational semantics Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction theories algebraic theories : denotational semantics Lawvere theories enriched theories enrichment ( ab ) c = a ( bc ) enriched categories enriched products enriched theories enriched theories : operational semantics V-theories examples change of semantics change of base preserving theories applications combinators change of base Conclusion
Lawvere theories Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Th(Mon) Introduction type M monoid theories Lawvere theories enriched theories M 2 operations m : → M multiplication enrichment e : 1 → M identity enriched categories enriched products enriched theories equations V-theories examples change of M 3 M 2 semantics change of base preserving theories applications M 2 M 2 M 2 M combinators change of base Conclusion 1 × M M M M × 1
Enriched theories Enriched Lawvere Theories for Operational Semantics John C. Baez Th(PsMon) Christian Williams type M pseudomonoid Introduction theories Lawvere theories M 2 operations ⊗ : → M multiplication enriched theories enrichment I: 1 → M identity enriched categories enriched products enriched theories rewrites V-theories examples M 3 M 2 change of semantics ⇓ α change of base preserving theories M 2 M 2 M 2 M applications combinators ⇓ λ ρ ⇓ change of base 1 × M M M M × 1 Conclusion equations pentagon, triangle identities
Enriched categories Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction Let V be monoidal. A V -enriched category has hom-objects theories in V; composition and identity are morphisms in V, as are the Lawvere theories enriched theories components of a V-functor and a V-natural transformation: enrichment enriched categories V -category C( a , b ) ∈ V enriched products enriched theories V-theories V -functor F ab : C( a , b ) → D( F ( a ) , F ( b )) ∈ V examples change of semantics change of base V -transformation ϕ a : 1 V → D( F ( a ) , G ( a )) ∈ V . preserving theories applications These form the 2-category VCat. combinators change of base Conclusion
Our enriching category Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Let V be a cartesian closed category: Introduction V( a × b , c ) ∼ = V( a , [ b , c ]) . theories Lawvere theories enriched theories Then V ∈ VCat. enrichment enriched categories enriched products Let V ∈ CCC fc (1) , meaning assume and choose: enriched theories V-theories examples � n V := 1 V . change of semantics n change of base preserving theories Let N V := { n V | n ∈ N } ⊂ full V applications combinators change of base A V := N op and – our “arities”. Conclusion V
Enriched products Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams The V -product of ( a i ) ∈ C is an object � i a i ∈ C equipped Introduction with a V-natural isomorphism theories Lawvere theories i a i ) ∼ enriched theories C( − , � = � i C( − , a i ) . enrichment enriched categories enriched products A V-functor F : C → D preserves V-products if the enriched theories V-theories “projections” induce a V-natural isomorphism: examples change of i a i )) ∼ semantics D( − , F ( � = � i D( − , F ( a i )) . change of base preserving theories applications combinators Let VCat fp be the 2-category of V-categories with finite change of base V-products and V-functors preserving them. Conclusion
Enriched Lawvere theories Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction theories Lawvere theories Definition enriched theories enrichment A V -theory is a V-category T ∈ VCat fp whose objects are enriched categories finite V-products of a distinguished object. enriched products enriched theories V-theories examples A morphism of V-theories is a V-functor F : T → T ′ ∈ VCat fp . These and V-natural transformations change of semantics change of base form the 2-category of V-theories, VLaw. preserving theories applications combinators change of base Conclusion
Enriched models Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction theories Lawvere theories Definition enriched theories enrichment A context is a V-category C ∈ VCat fp . enriched categories A model of T is a V-functor enriched products enriched theories V-theories µ : T → C ∈ VCat fp . examples change of semantics The category of models is Mod(T , C) := VCat fp (T , C). change of base preserving theories applications combinators change of base Conclusion
Example: monoidal categories Enriched Lawvere Theories for Operational Semantics Let V = Cat. John C. Baez Th(PsMon) Christian Williams type M pseudomonoid Introduction theories Lawvere theories M 2 operations ⊗ : → M multiplication enriched theories enrichment I: 1 → M identity enriched categories enriched products enriched theories rewrites V-theories examples M 3 M 2 change of semantics ⇓ α change of base preserving theories M 2 M 2 M 2 M applications combinators ⇓ λ ρ ⇓ change of base 1 × M M M M × 1 Conclusion equations pentagon, triangle identities
Example: cartesian object Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Let V = Cat. Introduction Th(Cart) theories type X cartesian object Lawvere theories enriched theories enrichment X 2 → X operations m : product enriched categories enriched products e : 1 → X terminal element enriched theories V-theories examples rewrites △ : id X = ⇒ m ◦ ∆ X unit of m ⊢ ∆ X change of π : ∆ X ◦ m = ⇒ id X 2 counit of m ⊢ ∆ X semantics change of base ⊤ : id X = ⇒ e ◦ ! X unit of e ⊢ ! X preserving theories ǫ : ! X ◦ e = ⇒ id 1 counit of e ⊢ ! X applications combinators change of base equations triangle identities Conclusion
Change of base Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction Let F : V → W preserve finite products, and C ∈ VCat. theories Lawvere theories Then F induces a change of base : enriched theories enrichment enriched categories F ∗ (C)( a , b ) := F (C( a , b )) . enriched products enriched theories V-theories This gives a 2-functor examples change of semantics F ∗ : VCat → WCat . change of base preserving theories Enrichment provides semantics, so change of base should applications combinators preserve theories to be a change of semantics . change of base Conclusion
Preservation of theories Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams Introduction Theorem theories Let F : V → W ∈ CCC fc (1) . Lawvere theories Then F is a change of semantics: enriched theories enrichment enriched categories F ∗ preserves theories. For every V -theory τ V : A V → T , enriched products enriched theories V-theories F ∗ ( τ V ) ∼ examples τ W := A W − → F ∗ (A V ) − − − − → F ∗ (T) is a W -theory. change of semantics F ∗ preserves models. For every model µ : T → C , change of base preserving theories applications F ∗ ( µ ): F ∗ (T) → F ∗ (C) is a model of ( F ∗ (T) , τ W ) . combinators change of base Conclusion
Change of semantics Enriched Lawvere Theories for Operational Semantics John C. Baez Christian Williams There is a “spectrum” of semantics: Introduction theories Lawvere theories enriched theories enrichment FC FP FS enriched categories enriched products Gph Cat Pos Set ⊣ ⊣ ⊣ enriched theories V-theories UC UP UG examples change of semantics change of base maps small-step to big-step operational semantics. FC ∗ preserving theories FP ∗ maps big-step to full-step operational semantics. applications combinators maps full-step to denotational semantics. FS ∗ change of base Conclusion
Recommend
More recommend
