A type theory for cartesian closed bicategories Marcelo Fiore and - PowerPoint PPT Presentation

A type theory for cartesian closed bicategories Marcelo Fiore and Philip Saville* University of Cambridge Department of Computer Science and Technology * now at University of Edinburgh School of Informatics 9th July 2019 1 / 25

Cartesian closed bicategories Cartesian closed categories ‘up to isomorphism’. Examples: - Generalised species and cartesian distributors particularly for applications in higher category theory (Fiore, Gambino, Hyland, Winskel), (Fiore & Joyal) - Categorical algebra (operads) (Gambino & Joyal) - Game semantics (concurrent games) (Yamada & Abramsky, Winskel et al. , Paquet) 2 / 25

Internal monoids In a category with finite products: 1 e m Ý Ñ M Ð Ý M ˆ M e ˆ M M ˆ e 1 ˆ M M ˆ M M ˆ 1 Unit law m M – – – M ˆ m p M ˆ M q ˆ M M ˆ p M ˆ M q Assoc. law M ˆ M m ˆ M m M ˆ M M m 3 / 25

Internal monoids In a category with finite products: In Set : monoids 1 e m Ý Ñ M Ð Ý M ˆ M In Cat : strict monoidal categories e ˆ M M ˆ e 1 ˆ M M ˆ M M ˆ 1 Unit law m M – – – M ˆ m p M ˆ M q ˆ M M ˆ p M ˆ M q Assoc. law M ˆ M m ˆ M m M ˆ M M m 3 / 25

Internal pseudomonoids In Cat : 1 e m Ý Ñ M Ð Ý M ˆ M e ˆ M M ˆ e 1 ˆ M M ˆ M M ˆ 1 Unit 2-cells ρ λ m – – M » » data M ˆ m » p M ˆ M q ˆ M M ˆ p M ˆ M q M ˆ M Assoc. 2-cell α m m ˆ M – M ˆ M M m 4 / 25

Internal pseudomonoids In Cat : 1 e m Ý Ñ M Ð Ý M ˆ M e ˆ M M ˆ e 1 ˆ M M ˆ M M ˆ 1 Unit 2-cells ρ λ m – – M » » data M ˆ m » p M ˆ M q ˆ M M ˆ p M ˆ M q M ˆ M Assoc. 2-cell α m m ˆ M – M ˆ M M m + triangle and pentagon laws ù monoidal category 4 / 25

Internal pseudomonoids In Cat : 1 e m Ý Ñ M Ð Ý M ˆ M ...likewise in any fp-bicategory e ˆ M M ˆ e Unit 2-cells 1 ˆ M M ˆ M M ˆ 1 ρ λ m – – M » » data M ˆ m » p M ˆ M q ˆ M M ˆ p M ˆ M q M ˆ M Assoc. 2-cell α m m ˆ M – M ˆ M M m + triangle and pentagon laws ù monoidal category 4 / 25

In a CCC every r X ñ X s becomes a monoid: ´ ¯ Id X ˝ 1 Ý Ý Ñ r X ñ X s Ð Ý r X ñ X s ˆ r X ñ X s In a cc-bicategory every r X ñ X s becomes a pseudomonoid: ? ´ Id X ˝ ¯ 1 Ý Ý Ñ r X ñ X s Ð Ý r X ñ X s ˆ r X ñ X s need to check coherence laws ( i.e. triangle + pentagon) 5 / 25

Coherence Programme: 1. Construct a type theory Λ ˆ , Ñ for cartesian closed bicategories ps (this work) , 2. Use NBE to prove the type theory is coherent bicategorical version of [Fiore2002] (my thesis) , 6 / 25

Coherence Programme: 1. Construct a type theory Λ ˆ , Ñ for cartesian closed bicategories ps (this work) , 2. Use NBE to prove the type theory is coherent bicategorical version of [Fiore2002] (my thesis) , Application: Algebraic structure definable in every CCC ñ algebraic pseudo-structure definable in every cc-bicategory 6 / 25

Desiderata A type theory Λ ˆ , Ñ that: ps 7 / 25

Desiderata A type theory Λ ˆ , Ñ that: ps 1. Generalises the simply-typed lambda calculus, 2. Is reasonable for calculations, 3. Is sound and complete 7 / 25

Desiderata A type theory Λ ˆ , Ñ that: ps 1. Generalises the simply-typed lambda calculus, 2. Is reasonable for calculations, 3. Is sound and complete i.e. freeness property for the syntactic model. 7 / 25

Bicategories 8 / 25

Bicategories - Objects X P ob p B q , 8 / 25

Bicategories - Objects X P ob p B q , ` ˘ - Hom-categories B p X , Y q , ‚ , id : 8 / 25

Bicategories - Objects X P ob p B q , ` ˘ - Hom-categories B p X , Y q , ‚ , id : f 1-cells X Ý Ñ Y f 2-cells X ó α Y f 1 8 / 25

Bicategories - Objects X P ob p B q , ` ˘ - Hom-categories B p X , Y q , ‚ , id : f 1-cells X Ý Ñ Y f 2-cells X ó α Y f f 1 ó α X Y f 1 ó α 1 f 2 8 / 25

Bicategories - Objects X P ob p B q , ` ˘ - Hom-categories B p X , Y q , ‚ , id : f 1-cells X Ý Ñ Y f 2-cells X ó α Y f 1 - Functors Id X 1 Ý Ý Ñ B p X , X q ˝ X , Y , Z B p Y , Z q ˆ B p X , Y q Ý Ý Ý Ý Ñ B p X , Z q 8 / 25

Bicategories - Objects X P ob p B q , ` ˘ - Hom-categories B p X , Y q , ‚ , id : f 1-cells X Ñ Y Ý f 2-cells X ó α Y f 1 - Functors Id X 1 Ý Ý Ñ B p X , X q ˝ X , Y , Z B p Y , Z q ˆ B p X , Y q Ý Ý Ý Ý Ñ B p X , Z q g f X Y Z ó α ó β f 1 g 1 8 / 25

Bicategories - Objects X P ob p B q , ` ˘ - Hom-categories B p X , Y q , ‚ , id : f 1-cells X Ý Ñ Y f 2-cells X ó α Y f 1 - Functors Id X 1 Ý Ý Ñ B p X , X q ˝ X , Y , Z B p Y , Z q ˆ B p X , Y q Ý Ý Ý Ý Ñ B p X , Z q - Invertible 2-cells a h , g , f p h ˝ g q ˝ f ù ù ù ñ h ˝ p g ˝ f q l f Id X ˝ f ù ñ f r g g ˝ Id X ù ñ g subject to a triangle law and pentagon law. 8 / 25

Cartesian closed bicategories Bicategories B equipped with biuniversal 1-cells 9 / 25

Cartesian closed bicategories Bicategories B equipped with biuniversal 1-cells (fp) π i : Π n p A 1 , . . . , A n q Ñ A i p 1 ď i ď n q (cc) eval : p A ñ B q ˆ A Ñ B NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters! 9 / 25

Cartesian closed bicategories Bicategories B equipped with biuniversal 1-cells (fp) π i : Π n p A 1 , . . . , A n q Ñ A i p 1 ď i ď n q (cc) eval : p A ñ B q ˆ A Ñ B inducing families of equivalences ś n B p X , Π n p A 1 , . . . , A n qq » i “ 1 B p X , A i q B p X , A “ ⊲ B q » B p X ˆ A , B q NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters! 9 / 25

Cartesian closed bicategories Bicategories B equipped with biuniversal 1-cells (fp) π i : Π n p A 1 , . . . , A n q Ñ A i p 1 ď i ď n q (cc) eval : p A ñ B q ˆ A Ñ B inducing families of equivalences p π 1 ˝´ ,...,π n ˝´q ś n B p X , Π n p A 1 , . . . , A n qq » i “ 1 B p X , A i q % x´ ,..., “y (tupling) eval A , B ˝p´ˆ A q ⊲ B q B p X , A “ » B p X ˆ A , B q % λ (currying) NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters! 9 / 25

Substitution and composition In any CCC: � x k r u 1 { x 1 , . . . , u n { x n s � “ � u k � “ π k ˝ x � u 1 � , . . . , � u n � y 10 / 25

Substitution and composition In any CCC: � x k r u 1 { x 1 , . . . , u n { x n s � “ � u k � “ π k ˝ x � u 1 � , . . . , � u n � y In any cc-bicategory: � x k r u 1 { x 1 , . . . , u n { x n s � “ � u k � – π k ˝ x � u 1 � , . . . , � u n � y 10 / 25

Substitution and composition In any CCC: � x k r u 1 { x 1 , . . . , u n { x n s � “ � u k � “ π k ˝ x � u 1 � , . . . , � u n � y In any cc-bicategory: � x k r u 1 { x 1 , . . . , u n { x n s � “ � u k � – π k ˝ x � u 1 � , . . . , � u n � y Question: what is bicategorical substitution? 10 / 25

An algebraic theory with substitution: 11 / 25

An algebraic theory with substitution: - Sorts S , 11 / 25

An algebraic theory with substitution: - Sorts S , - Constants x 1 : X 1 , . . . , x n : X n $ t p x 1 , . . . , x n q : Y , 11 / 25

An algebraic theory with substitution: - Sorts S , - Constants x 1 : X 1 , . . . , x n : X n $ t p x 1 , . . . , x n q : Y , - Variables x 1 : X 1 , . . . , x n : X n $ x i : X i p 1 ď i ď n q , 11 / 25

An algebraic theory with substitution: - Sorts S , - Constants x 1 : X 1 , . . . , x n : X n $ t p x 1 , . . . , x n q : Y , - Variables x 1 : X 1 , . . . , x n : X n $ x i : X i p 1 ď i ď n q , - A substitution rule t , p u 1 , . . . , u n q ÞÑ t r u i { x i s 11 / 25

An algebraic theory with substitution: - Sorts S , - Constants x 1 : X 1 , . . . , x n : X n $ t p x 1 , . . . , x n q : Y , - Variables x 1 : X 1 , . . . , x n : X n $ x i : X i p 1 ď i ď n q , - A substitution rule t , p u 1 , . . . , u n q ÞÑ t r u i { x i s such that x k r u i { x i s “ u k p 1 ď k ď n q t r x i { x i s “ t t r u i { x i sr v j { y j s “ t r u i r v j { y j s{ x i s 11 / 25

Abstract clone p S , C q = abstract theory of substitution: 12 / 25

Abstract clone p S , C q = abstract theory of substitution: - Sorts S , 12 / 25

Abstract clone p S , C q = abstract theory of substitution: - Sorts S , t - Hom-sets C p X 1 , . . . , X n ; Y q of operations X 1 , . . . , X n Ý Ñ Y , 12 / 25

Abstract clone p S , C q = abstract theory of substitution: - Sorts S , t - Hom-sets C p X 1 , . . . , X n ; Y q of operations X 1 , . . . , X n Ý Ñ Y , p p i q X 1 ,..., Xn - Projections X 1 , . . . , X n Ý Ý Ý Ý Ý Ñ X i p 1 ď i ď n q , 12 / 25

Abstract clone p S , C q = abstract theory of substitution: - Sorts S , t - Hom-sets C p X 1 , . . . , X n ; Y q of operations X 1 , . . . , X n Ý Ñ Y , p p i q X 1 ,..., Xn - Projections X 1 , . . . , X n Ý Ý Ý Ý Ý Ñ X i p 1 ď i ď n q , - Substitution mappings C p X 1 , . . . , X n ; Y q ˆ ś n i “ 1 C p Γ; X i q Ñ C p Γ; Y q t , p u 1 , . . . , u n q ÞÑ t r u 1 , . . . , u n s 12 / 25