Coherence for cartesian closed bicategories, via - PowerPoint PPT Presentation

The internal language for cc-bicategories, Λ ˆ , Ñ ps Judgements c.f. 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 Features - Weak composition enforced by explicit substitution - Usual STLC operations on terms and rewrites - STLC embeds as explicit-substitution free fragment - Free property for syntactic model ù A logic of program transformations (modulo equations) there exists a rewrite t ñ t 1 t “ βη t 1 iff 12 / 41

The internal language for cc-bicategories, Λ ˆ , Ñ ps Judgements c.f. 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 Features - Weak composition enforced by explicit substitution - Usual STLC operations on terms and rewrites - STLC embeds as explicit-substitution free fragment - Free property for syntactic model ù A logic of program transformations (modulo equations) there exists a rewrite t ñ t 1 t “ βη t 1 iff 12 / 41

The internal language for cc-bicategories, Λ ˆ , Ñ ps Judgements c.f. 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 Features - Weak composition enforced by explicit substitution - Usual STLC operations on terms and rewrites - STLC embeds as explicit-substitution free fragment - Free property for syntactic model ù A logic of program transformations (modulo equations) there exists a rewrite t ñ t 1 t “ βη t 1 iff ù coherence of cc-bicategories = coherence property of 2-cells 12 / 41

Theorem (Coherence) For any t and t 1 in Λ ˆ , Ñ , there exists at most one rewrite τ such ps that Γ $ τ : t ñ t 1 : A . 13 / 41

Theorem (Coherence) For any t and t 1 in Λ ˆ , Ñ , there exists at most one rewrite τ such ps that Γ $ τ : t ñ t 1 : A . Corollary 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 . 13 / 41

Theorem (Coherence) For any t and t 1 in Λ ˆ , Ñ , there exists at most one rewrite τ such ps that Γ $ τ : t ñ t 1 : A . Corollary 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 . ù can use STLC for constructions in cc-bicategories! 13 / 41

Theorem (Coherence) For any t and t 1 in Λ ˆ , Ñ , there exists at most one rewrite τ such ps that Γ $ τ : t ñ t 1 : A . Corollary 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 . ù can use STLC for constructions in cc-bicategories! 1. prove result in STLC 13 / 41

Theorem (Coherence) For any t and t 1 in Λ ˆ , Ñ , there exists at most one rewrite τ such ps that Γ $ τ : t ñ t 1 : A . Corollary 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 . ù can use STLC for constructions in cc-bicategories! 1. prove result in STLC 2. βη -equalities ù 2-cells 13 / 41

Theorem (Coherence) For any t and t 1 in Λ ˆ , Ñ , there exists at most one rewrite τ such ps that Γ $ τ : t ñ t 1 : A . Corollary 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 . ù can use STLC for constructions in cc-bicategories! 1. prove result in STLC 2. βη -equalities ù 2-cells 3. coherence guaranteed 13 / 41

Normalisation-by-evaluation 14 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote Normal terms nf p t q “ quote p � t � q “ . . . (long- βη ) 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q Normal terms nf p t q “ quote p � t � q “ . . . (long- βη ) 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q unquote Normal terms Neutral terms nf p t q “ quote p � t � q “ . . . includes variables (long- βη ) defined by mutual induction 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q unquote Normal terms Neutral terms nf p t q “ quote p � t � q “ . . . includes variables (long- βη ) For $ λ x . t : A “ ⊲ B , get � t � : � A � Ñ � B � . 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q unquote Normal terms Neutral terms nf p t q “ quote p � t � q “ . . . includes variables (long- βη ) For $ λ x . t : A “ ⊲ B , get � t � : � A � Ñ � B � . For fresh x : unquote p x q : � A � 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q unquote Normal terms Neutral terms nf p t q “ quote p � t � q “ . . . includes variables (long- βη ) For $ λ x . t : A “ ⊲ B , get � t � : � A � Ñ � B � . For fresh x : � t � p unquote p x qq : � B � 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q unquote Normal terms Neutral terms nf p t q “ quote p � t � q “ . . . includes variables (long- βη ) For $ λ x . t : A “ ⊲ B , get � t � : � A � Ñ � B � . For fresh x : quote p � t � p unquote p x qqq : B 15 / 41

Normalisation-by-evaluation Aim: find canonical representatives within each βη -equivalence class Strategy: � ´ � Semantics Syntax βη -equivalent terms t , t 1 , t 2 , . . . � t � “ � t 1 � “ � t 2 � “ . . . quote nf p´q unquote Normal terms Neutral terms nf p t q “ quote p � t � q “ . . . includes variables (long- βη ) For $ λ x . t : A “ ⊲ B , get � t � : � A � Ñ � B � . For fresh x : nf p λ x . t q : “ λ x . quote p � t � p unquote p x qqq : A “ ⊲ B 15 / 41

Normalisation-by-evaluation, categorically 16 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations p Γ $ t : A q ÞÑ s � Γ $ t : A � s any interpretation of base types; 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations p Γ $ t : A q ÞÑ s � Γ $ t : A � s any interpretation of base types; s � ´ � : neuts A ñ C p s � ´ � , s � A � q C any CCC s � ´ � : norms A ñ C p s � ´ � , s � A � q 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations p Γ $ t : A q ÞÑ s � Γ $ t : A � s any interpretation of base types; s � ´ � : neuts A ñ C p s � ´ � , s � A � q C any CCC s � ´ � : norms A ñ C p s � ´ � , s � A � q Strategy: 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations p Γ $ t : A q ÞÑ s � Γ $ t : A � s any interpretation of base types; s � ´ � : neuts A ñ C p s � ´ � , s � A � q C any CCC s � ´ � : norms A ñ C p s � ´ � , s � A � q Strategy: 1. define a glueing category G 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations p Γ $ t : A q ÞÑ s � Γ $ t : A � s any interpretation of base types; s � ´ � : neuts A ñ C p s � ´ � , s � A � q C any CCC s � ´ � : norms A ñ C p s � ´ � , s � A � q Strategy: 1. define a glueing category G 2. pick an interpretation e � ´ � in G 17 / 41

Normalisation-by-evaluation, categorically (Fiore) Syntax as indexed presheaves over a category of contexts Con : + neuts A : Γ ÞÑ t neutral terms t such that Γ $ t : A u : Con Ñ Set norms A : Γ ÞÑ t normal terms t such that Γ $ t : A u syntax glued to semantics by natural transformations p Γ $ t : A q ÞÑ s � Γ $ t : A � s any interpretation of base types; s � ´ � : neuts A ñ C p s � ´ � , s � A � q C any CCC s � ´ � : norms A ñ C p s � ´ � , s � A � q Strategy: 1. define a glueing category G 2. pick an interpretation e � ´ � in G 3. define quote and unquote as maps in this category 17 / 41

The glued category G p C , s q nat. trans Objects functor p Con Ñ Set P ù ñ C p s � ´ � , X q C ccc, P C s : BaseTypes Ñ Ob C 18 / 41

The glued category G p C , s q nat. trans Objects functor p Con Ñ Set P ù ñ C p s � ´ � , X q C ccc, P C s : BaseTypes Ñ Ob C Morphisms nat. trans P ñ P 1 κ P 1 P p p 1 C p s � ´ � , X 1 q C p s � ´ � , X q k ˝ p´q arrow X Ñ X 1 18 / 41

The glued category G p C , s q nat. trans Objects functor p Con Ñ Set P ù ñ C p s � ´ � , X q C ccc, P C s : BaseTypes Ñ Ob C Morphisms nat. trans P ñ P 1 κ P 1 "syntax" P p p 1 "interpretation" C p s � ´ � , X 1 q C p s � ´ � , X q "semantics" k ˝ p´q arrow X Ñ X 1 18 / 41

The category G p C , s q nat. trans functor Objects p Con Ñ Set P ù ñ C p s � ´ � , X q C ccc, s : BaseTypes Ñ Ob C P C Morphisms "syntax" κ P 1 P "interpretation" p p 1 C p s � ´ � , X 1 q C p s � ´ � , X q "semantics" k ˝ p´q 19 / 41

The category G p C , s q nat. trans functor Objects p Con Ñ Set P ù ñ C p s � ´ � , X q C ccc, s : BaseTypes Ñ Ob C P C Morphisms "syntax" κ P 1 P "interpretation" p p 1 C p s � ´ � , X 1 q C p s � ´ � , X q "semantics" k ˝ p´q ù Cartesian closed, with a strict CCC-functor G p C , s q Ñ C 19 / 41

The category G p C , s q nat. trans functor Objects p Con Ñ Set P ù ñ C p s � ´ � , X q C ccc, s : BaseTypes Ñ Ob C P C Morphisms "syntax" κ P 1 P "interpretation" p p 1 C p s � ´ � , X 1 q C p s � ´ � , X q "semantics" k ˝ p´q ù Cartesian closed, with a strict CCC-functor G p C , s q Ñ C ù for every type A , , s � ´ � neuts A ù ù ù ñ C p s � ´ � , s � A � q . - P G p C , s q s � ´ � norms A ù ù ù ñ C p s � ´ � , s � A � q 19 / 41

Interpretation in the glued category G p C , s q C any ccc s : BaseTypes Ñ Ob C e : BaseTypes Ñ Ob p G p C , s qq s � ´ � ` ˘ β ÞÑ ù ù ù ñ C p s � ´ � , s � β � q neuts β 20 / 41

Interpretation in the glued category G p C , s q C any ccc s : BaseTypes Ñ Ob C e : BaseTypes Ñ Ob p G p C , s qq s � ´ � ` ˘ β ÞÑ ù ù ù ñ C p s � ´ � , s � β � q neuts β ν A ` ˘ e � A � : “ e � A � ù ñ C p s � ´ � , s � A � q by strict preservation 20 / 41

Interpretation in the glued category G p C , s q C any ccc s : BaseTypes Ñ Ob C e : BaseTypes Ñ Ob p G p C , s qq s � ´ � ` ˘ β ÞÑ ù ù ù ñ C p s � ´ � , s � β � q neuts β e � A � ν A ` ˘ s � ´ � , s � A � q C 20 / 41

Interpretation in the glued category G p C , s q C any ccc s : BaseTypes Ñ Ob C e : BaseTypes Ñ Ob p G p C , s qq s � ´ � ` ˘ β ÞÑ ù ù ù ñ C p s � ´ � , s � β � q neuts β e � Γ $ t : A � e � Γ � e � A � ν Γ ν A ` ˘ ` ˘ C s � ´ � , s � Γ � C s � ´ � , s � A � s � Γ $ t : A � ˝p´q 20 / 41

Interpretation in the glued category G p C , s q C any ccc s : BaseTypes Ñ Ob C e : BaseTypes Ñ Ob p G p C , s qq s � ´ � ` ˘ β ÞÑ ù ù ù ñ C p s � ´ � , s � β � q neuts β e � Γ $ t : A � e � Γ � e � A � ν Γ ν A ` ˘ ` ˘ C s � ´ � , s � Γ � C s � ´ � , s � A � s � Γ $ t : A � ˝p´q quote A neuts A e � A � s � ´ � ν A ` ˘ ` ˘ C s � ´ � , s � A � C s � ´ � , s � A � preserves βη 20 / 41

Interpretation in the glued category G p C , s q C any ccc s : BaseTypes Ñ Ob C e : BaseTypes Ñ Ob p G p C , s qq s � ´ � ` ˘ β ÞÑ ù ù ù ñ C p s � ´ � , s � β � q neuts β e � Γ $ t : A � e � Γ � e � A � ν Γ ν A ` ˘ ` ˘ C s � ´ � , s � Γ � C s � ´ � , s � A � s � Γ $ t : A � ˝p´q quote A unquote A neuts A e � A � e � A � norms A s � ´ � ν A ν A s � ´ � ` ˘ ` ˘ ` ˘ ` ˘ C s � ´ � , s � A � C s � ´ � , s � A � C s � ´ � , s � A � C s � ´ � , s � A � preserves βη preserves βη 20 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i unquote Γ e � Γ $ t : A � quote A neuts Γ norms A e � Γ � e � A � s � ´ � ν Γ ν A s � ´ � C p s � ´ � , s � Γ � q C p s � ´ � , s � Γ � q C p s � ´ � , s � A � q C p s � ´ � , s � A � q s � Γ $ t : A � ˝ p´q 21 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i unquote Γ e � Γ $ t : A � quote A neuts Γ norms A e � Γ � e � A � s � ´ � ν Γ ν A s � ´ � C p s � ´ � , s � Γ � q C p s � ´ � , s � Γ � q C p s � ´ � , s � A � q C p s � ´ � , s � A � q s � Γ $ t : A � ˝ p´q ù recall nf p λ x . t q : “ λ x . quote p � t � p unquote p x qqq 21 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm t unquote Γ e � Γ $ t : A � quote A neuts Γ norms A e � Γ � e � A � s � ´ � ν Γ ν A s � ´ � C p s � ´ � , s � Γ � q C p s � ´ � , s � Γ � q C p s � ´ � , s � A � q C p s � ´ � , s � A � q s � Γ $ t : A � ˝ p´q ù recall nf p λ x . t q : “ λ x . quote p � t � p unquote p x qqq 21 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . 22 / 41

Normalising p Γ $ t : A q neuts Γ : “ Π p x i : A i qP Γ neuts A i unquote Γ : “ Π p x i : A i qP Γ unquote A i norm Γ t p x i : A i q i p x i : A i q i nf p t q unquote Γ e � Γ $ t : A � quote A neuts Γ p Γ q e � Γ � p Γ q e � A � p Γ q norms A p Γ q ν Γ ν A s � ´ � C p s � Γ � , s � Γ � q C p s � Γ � , s � A � q s � ´ � s � Γ $ t : A � ˝ p´q id s � Γ � s � Γ $ t : A � s � Γ $ t : A � Define nf p t q : “ norm Γ t pp Γ $ x i : A i q i “ 1 ,..., n q . Since s � Γ $ nf p t q : A � “ s � Γ $ t : A � in every model, nf p t q “ βη t . 22 / 41

Normalisation-by-evaluation for Λ ˆ , Ñ ps 23 / 41

Strategy 1. define a glueing bicategory G 2. pick an interpretation e � ´ � in G 3. define quote and unquote as maps in this bicategory 24 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps inside Λ ˆ , Ñ A : “ neuts STLC + A ps : Con Ñ Set norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps inside Λ ˆ , Ñ A : “ neuts STLC + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps inside Λ ˆ , Ñ A : “ neuts STLC + A ps : Con Ñ Cat norms ps inside Λ ˆ , Ñ A : “ norms STLC A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù s � t � s � nf p t q � 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps A : “ neuts STLC inside Λ ˆ , Ñ + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù depends on t – s � t � s � nf p t q � 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps A : “ neuts STLC inside Λ ˆ , Ñ + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù depends on t – s � t � s � nf p t q � s � t 1 � s � nf p t 1 q � – depends on t 1 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps A : “ neuts STLC inside Λ ˆ , Ñ + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù depends on t – s � t � s � nf p t q � s � τ � s � t 1 � s � nf p t 1 q � – depends on t 1 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps A : “ neuts STLC inside Λ ˆ , Ñ + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù depends on t – s � t � s � nf p t q � s � τ � s � t 1 � s � nf p t 1 q � – depends on t 1 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps A : “ neuts STLC inside Λ ˆ , Ñ + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù depends on t – nf p t q t τ t 1 nf p t 1 q – depends on t 1 25 / 41

Strategy ù re-use universal properties (quote, unquote, G p C , s q ,. . . bicategorically) ù use embedding of STLC into Λ ˆ , Ñ ps neuts ps A : “ neuts STLC inside Λ ˆ , Ñ + A ps : Con Ñ Cat norms ps A : “ norms STLC inside Λ ˆ , Ñ A ps neuts ps A p Γ q , norms ps A p Γ q sets of terms as discrete categories ù If τ exists, it’s unique depends on t – nf p t q t τ t 1 nf p t 1 q – depends on t 1 25 / 41

The glued cc-bicategory G p B , s q B any cc-bicat s any interpretation of base types Want: for every type A , s � ´ � , neuts ps ù ù ù ñ B p s � ´ � , s � A � q . A - P G p B , s q s � ´ � norms ps ù ù ù ñ B p s � ´ � , s � A � q A 26 / 41

The glued cc-bicategory G p B , s q pseudo nat. trans. Objects pseudofunctor p p , p q Con Ñ Cat P ù ù ù ñ B p s � ´ � , X q B cc-bicat, P B s : BaseTypes Ñ Ob B 27 / 41

The glued cc-bicategory G p B , s q pseudo nat. trans. Objects pseudofunctor p p , p q Con Ñ Cat P ù ù ù ñ B p s � ´ � , X q B cc-bicat, P B s : BaseTypes Ñ Ob B pseudo nat. trans. P ñ P 1 1-cells κ P 1 P w k p – p 1 B p s � ´ � , X 1 q B p s � ´ � , X q k ˝ p´q arrow X Ñ X 1 27 / 41