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Models of Classical Linear Logic via Bifibrations of Polycategories N. Blanco and N. Zeilberger School of Computer Science University of Birmingham, UK SYCO5, September 2019 N. Blanco and N. Zeilberger ( School of Computer Science University

  1. Models of Classical Linear Logic via Bifibrations of Polycategories N. Blanco and N. Zeilberger School of Computer Science University of Birmingham, UK SYCO5, September 2019 N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 1 / 27

  2. Outline Multicategories and Monoidal categories 1 Opfibration of Multicategories 2 Polycategories and Linearly Distributive Categories 3 Bifibration of polycategories 4 N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 2 / 27

  3. Multicategories and Monoidal categories Outline Multicategories and Monoidal categories 1 N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 3 / 27

  4. Multicategories and Monoidal categories Tensor product of vector spaces In linear algebra: universal property C A , B A ⊗ B In category theory as a structure: a monoidal product ⊗ Universal property of tensor product needs many-to-one maps Category with many-to-one maps ⇒ Multicategory N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 3 / 27

  5. Multicategories and Monoidal categories Multicategory 1 Definition A multicategory M has: A collection of objects Γ finite list of objects and A objects Set of multimorphisms M (Γ; A ) Identities id A : A → A f : Γ → A g : Γ 1 , A , Γ 2 → B Composition: g ◦ i f : Γ 1 , Γ , Γ 2 → B With usual unitality and associativity and: interchange law: ( g ◦ f 1 ) ◦ f 2 = ( g ◦ f 2 ) ◦ f 1 where f 1 and f 2 are composed in two different inputs of g 1 Tom Leinster. Higher Operads, Higher Categories . 2004. N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 4 / 27

  6. Multicategories and Monoidal categories Representable multicategory Definition A multimorphism u : Γ → A is universal if any multimap f : Γ 1 , Γ , Γ 2 → B factors uniquely through u . Definition A multicategory is representable if for any finite list Γ = ( A i ) there is a universal map Γ → � A i . B Γ 1 , A 1 , ..., A n , Γ 2 → B Γ 1 , A 1 ⊗ ... ⊗ A n , Γ 2 → B Γ 1 , A 1 , ..., A n , Γ 2 Γ 1 , A 1 ⊗ ... ⊗ A n , Γ 2 N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 5 / 27

  7. Multicategories and Monoidal categories Representable multicategories and Monoidal categories Let C be a monoidal category. There is an underlying representable multicategory − → C whose: objects are the objects of C multimorphisms f : A 1 , ..., A n → B are morphisms f : A 1 ⊗ ... ⊗ A n → B in C Conversely any representable multicategory is the underlying multicategory of some monoidal category. N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 6 / 27

  8. Multicategories and Monoidal categories Finite dimensional vector spaces and multilinear maps Theorem The multicategory − − − − → FVect of finite dimensional vector spaces and multilinear maps is representable. Definition For normed vect. sp. ( A i , � − � A i ) , ( B , � − � B ), f : A 1 , ..., A n → B is short/contractive if for any � x = x 1 , ..., x n , � f ( � x ) � B ≤ � � x i � A i i Theorem The multicategory − − − − → FBan 1 of finite dimensional Banach spaces and short multilinear maps is representable. Its tensor product is equipped with the projective crossnorm. N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 7 / 27

  9. Multicategories and Monoidal categories Projective crossnorm 2 Definition Given two normed vector spaces ( A , � − � A ) and ( B , � − � B ) we defined a normed on A ⊗ B called the projective crossnorm as follows: � � u � A ⊗ B = inf � a i � A � b i � B u = � a i ⊗ b i i i Proposition Any (well-behaved) norm � − � on A ⊗ B is smaller than the projective one: � u � ≤ � u � A ⊗ B , ∀ u ∈ A ⊗ B How does this related to the fact that it is the norm of the tensor product? 2 Raymond A. Ryan. Introduction to Tensor Products of Banach Spaces . 2002. N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 8 / 27

  10. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B U C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  11. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B U C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  12. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B U C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  13. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B U C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  14. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B U C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  15. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B U C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  16. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � C − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B Remark �−� , �−� → �−� A ⊗ B U opcartesian lifting C − − − − → FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  17. Multicategories and Monoidal categories Lifting the tensor product of − FVect to − − − − → − − − → FBan 1 � − � − − − − → FBan 1 � − � A , � − � B � − � A ⊗ B Remark id A ⊗ B is contractive, U i.e. � u � ≤ � u � A ⊗ B A ⊗ B − − − − → id A ⊗ B FVect A , B A ⊗ B N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 9 / 27

  18. Opfibration of Multicategories Outline Opfibration of Multicategories 2 N. Blanco and N. Zeilberger ( School of Computer Science University of Birmingham, UK ) Models of Classical Linear Logic via Bifibrations of Polycategories SYCO5, September 2019 10 / 27

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