Coherence and normalisation-by-evaluation for bicategorical cartesian closed structure Marcelo Fiore : and Philip Saville* : University of Cambridge Department of Computer Science and Technology *University of Edinburgh School of Informatics 1 / 27
Cartesian closed bicategories: - Generalised species and cartesian distributors (linear logic, higher category theory) (Fiore, Gambino, Hyland, Winskel), (Fiore & Joyal) - Categorical algebra (operads) (Gambino & Joyal) - Game semantics (concurrent games) (Yamada & Abramsky, Winskel et al. , Paquet) - ‘STLC with explicit substitution and invertible βη -rewrites’ (free cartesian closed bicategory) (Fiore & S., LICS 2019) 2 / 27
Cartesian closed bicategories: - Generalised species and cartesian distributors (linear logic, higher category theory) (Fiore, Gambino, Hyland, Winskel), (Fiore & Joyal) - Categorical algebra (operads) (Gambino & Joyal) - Game semantics (concurrent games) (Yamada & Abramsky, Winskel et al. , Paquet) - ‘STLC with explicit substitution and invertible βη -rewrites’ (free cartesian closed bicategory) (Fiore & S., LICS 2019) In the paper: two proofs of coherence ù ‘all (pasting) diagrams commute’ 2 / 27
Cartesian closed bicategories: - Generalised species and cartesian distributors (linear logic, higher category theory) (Fiore, Gambino, Hyland, Winskel), (Fiore & Joyal) - Categorical algebra (operads) (Gambino & Joyal) - Game semantics (concurrent games) (Yamada & Abramsky, Winskel et al. , Paquet) - ‘STLC with explicit substitution and invertible βη -rewrites’ (free cartesian closed bicategory) (Fiore & S., LICS 2019) In the paper: two proofs of coherence ù ‘all (pasting) diagrams commute’ Consequences: 2 / 27
Cartesian closed bicategories: - Generalised species and cartesian distributors (linear logic, higher category theory) (Fiore, Gambino, Hyland, Winskel), (Fiore & Joyal) - Categorical algebra (operads) (Gambino & Joyal) - Game semantics (concurrent games) (Yamada & Abramsky, Winskel et al. , Paquet) - ‘STLC with explicit substitution and invertible βη -rewrites’ (free cartesian closed bicategory) (Fiore & S., LICS 2019) In the paper: two proofs of coherence ù ‘all (pasting) diagrams commute’ Consequences: 1. Modulo ” there is at most one rewrite t ù βη t 1 2 / 27
Cartesian closed bicategories: - Generalised species and cartesian distributors (linear logic, higher category theory) (Fiore, Gambino, Hyland, Winskel), (Fiore & Joyal) - Categorical algebra (operads) (Gambino & Joyal) - Game semantics (concurrent games) (Yamada & Abramsky, Winskel et al. , Paquet) - ‘STLC with explicit substitution and invertible βη -rewrites’ (free cartesian closed bicategory) (Fiore & S., LICS 2019) In the paper: two proofs of coherence ù ‘all (pasting) diagrams commute’ Consequences: 1. Modulo ” there is at most one rewrite t ù βη t 1 2. Constructions in cc-bicategories simplify to STLC 2 / 27
Cartesian closed bicategories 3 / 27
Composition by universal property ñ bicategory In a category C with pullbacks: 1. objects: objects of C , 2. 1-cells A ù B : spans p A Ð X Ñ B q , X 3. 2-cells: commutative squares A h B Y Composition defined by pullback: X ˆ B Y x X Y A B C 4 / 27
Composition by universal property ñ bicategory In a category C with pullbacks: 1. objects: objects of C , 2. 1-cells A ù B : spans p A Ð X Ñ B q , X 3. 2-cells: commutative squares A h B Y Composition defined by pullback: ù associative up to iso X ˆ B Y x X Y A B C 4 / 27
Bicategories 5 / 27
Bicategories - Objects X , Y , Z , . . . 5 / 27
Bicategories - Objects X , Y , Z , . . . - Morphisms ( 1-cells ) f : X Ñ Y 5 / 27
Bicategories - Objects X , Y , Z , . . . - Morphisms ( 1-cells ) f : X Ñ Y f - 2-cells X Y ó α f 1 5 / 27
Bicategories - Objects X , Y , Z , . . . - Morphisms ( 1-cells ) f : X Ñ Y f - 2-cells X Y ó α f 1 - Invertible 2-cells witnessing the axioms 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 . 5 / 27
Bicategories - Objects X , Y , Z , . . . - Morphisms ( 1-cells ) f : X Ñ Y f - 2-cells X Y ó α f 1 - Invertible 2-cells witnessing the axioms 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. 5 / 27
Bicategories - Objects X , Y , Z , . . . - Morphisms ( 1-cells ) f : X Ñ Y f - 2-cells X Y ó α f 1 - Invertible 2-cells witnessing the axioms 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. ù ‘Monoidal category with many objects’ 5 / 27
Bicategories everywhere (not all cartesian closed!) 1. Monoidal category = one-object bicategory 2. 2-category = bicategory with a , l , r all id 3. Span p C q 4. Proof-relevant relations 5. Profunctors (distributors) 6. Polynomial functors (W-types, ornaments, containers, . . . ) 7. Concurrent games 6 / 27
Cartesian closed bicategories Bicategories B equipped with 7 / 27
Cartesian closed bicategories Bicategories B equipped with families of equivalences B p X , A 1 ˆ A 2 q » B p X , A 1 q ˆ B p X , A 2 q B p X , A “ ⊲ B q » B p X ˆ A , B q NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters! 7 / 27
Cartesian closed bicategories Bicategories B equipped with families of equivalences p π 1 ˝´ ,π 2 ˝´q B p X , A 1 ˆ A 2 q » B p X , A 1 q ˆ B p X , A 2 q % x´ , “y (pairing) eval A , B ˝p´ˆ A q B p X , A “ ⊲ B q » B p X ˆ A , B q % λ (currying) NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters! 7 / 27
Cartesian closed bicategories Bicategories B equipped with families of equivalences p π 1 ˝´ ,π 2 ˝´q B p X , A 1 ˆ A 2 q » B p X , A 1 q ˆ B p X , A 2 q % x´ , “y (pairing) eval A , B ˝p´ˆ A q B p X , A “ ⊲ B q » B p X ˆ A , B q % λ (currying) – – π i ˝ x f 1 , f 2 y ù ñ f i ù ñ x π 1 ˝ g , π 2 ˝ g y g 7 / 27
Cartesian closed bicategories Bicategories B equipped with families of equivalences p π 1 ˝´ ,π 2 ˝´q B p X , A 1 ˆ A 2 q » B p X , A 1 q ˆ B p X , A 2 q % x´ , “y (pairing) eval A , B ˝p´ˆ A q B p X , A “ ⊲ B q » B p X ˆ A , B q % λ (currying) – – π i ˝ x f 1 , f 2 y ù ñ f i g ù ñ x π 1 ˝ g , π 2 ˝ g y – – eval A , B ˝ p λ f ˆ A q ù ñ f ù ñ λ p eval A , B ˝ p g ˆ A qq g 7 / 27
Cartesian closed bicategories Bicategories B equipped with families of equivalences p π 1 ˝´ ,π 2 ˝´q B p X , A 1 ˆ A 2 q » B p X , A 1 q ˆ B p X , A 2 q % x´ , “y (pairing) eval A , B ˝p´ˆ A q B p X , A “ ⊲ B q » B p X ˆ A , B q % λ (currying) – – π i ˝ x f 1 , f 2 y ù ñ f i ù ñ x π 1 ˝ g , π 2 ˝ g y g π i ˝ x π 1 ˝ g , π 2 ˝ g y β ˆ π i ˝ η ˆ i π i ˝ g π i ˝ g id 7 / 27
Coherence 8 / 27
Mac Lane-style proof theory, rewriting theory "all diagrams commute" coherence coherence-by-strictification 2-cat. universal algebra, Yoneda embedding B » B 9 / 27
Mac Lane-style proof theory, rewriting theory "all diagrams commute" “denotational semantics” (1) use an internal language (2) prove normalisation by semantic argument coherence coherence-by-strictification 2-cat. universal algebra, Yoneda embedding B » B 9 / 27
Mac Lane-style proof theory, rewriting theory "all diagrams commute" “denotational semantics” (1) use an internal language (2) prove normalisation by semantic argument coherence in the paper coherence-by-strictification 2-cat. universal algebra, Yoneda embedding B » B 9 / 27
“denotational semantics” (1) use an internal language (2) prove normalisation by semantic argument coherence ‚ builds on categorical & type-theoretic intuition ‚ once set up about as hard as categorical proof 9 / 27
Theorem (Coherence of cc-bicategories) For any f , f 1 : X Ñ Y in the free cc-bicategory on a graph, there exists at most one 2-cell τ : f ñ f 1 . 10 / 27
Theorem (Coherence of cc-bicategories) For any f , f 1 : X Ñ Y in the free cc-bicategory on a graph, there exists at most one 2-cell τ : f ñ f 1 . Consequence 1: modulo ” there is at most one rewrite t ù βη t 1 10 / 27
Theorem (Coherence of cc-bicategories) For any f , f 1 : X Ñ Y in the free cc-bicategory on a graph, there exists at most one 2-cell τ : f ñ f 1 . Consequence 1: modulo ” there is at most one rewrite t ù βη t 1 Consequence 2: can use STLC for constructions in cc-bicategories 10 / 27
Theorem (Coherence of cc-bicategories) For any f , f 1 : X Ñ Y in the free cc-bicategory on a graph, there exists at most one 2-cell τ : f ñ f 1 . Consequence 1: modulo ” there is at most one rewrite t ù βη t 1 Consequence 2: can use STLC for constructions in cc-bicategories 11 / 27
The internal language for cc-bicategories, Λ ˆ , Ñ (Fiore & S., 2019) ps Judgements c.f. Seely, Hilken, Hirschowitz Terms Γ $ t : A (1-cells) Γ $ τ : t ñ t 1 : A Rewrites (2-cells) Γ $ τ ” τ 1 : t ñ t 1 : A Equations 12 / 27
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