Coherence for cartesian closed bicategories, via - - PowerPoint PPT Presentation

Coherence for cartesian closed bicategories, via normalisation-by-evaluation

Marcelo Fiore: and Philip Saville*

:University of Cambridge

Department of Computer Science and Technology *University of Edinburgh School of Informatics

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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)

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Coherence

3 / 41

Internal monoids

In a category with finite products: 1 e Ý Ñ M

m

Ð Ý M ˆ M Unit law 1 ˆ M M ˆ 1 M ˆ M M

e ˆ M M ˆ e – – m

• Assoc. law

pM ˆ Mq ˆ M M ˆ pM ˆ Mq M ˆ M M ˆ M M

– M ˆ m m m ˆ M m

4 / 41

Internal monoids

In a category with finite products: 1 e Ý Ñ M

m

Ð Ý M ˆ M In Set: monoids In Cat: strict monoidal categories Unit law 1 ˆ M M ˆ 1 M ˆ M M

e ˆ M M ˆ e – – m

• Assoc. law

pM ˆ Mq ˆ M M ˆ pM ˆ Mq M ˆ M M ˆ M M

– M ˆ m m m ˆ M m

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Internal pseudomonoids

In Cat: 1 e Ý Ñ M

m

Ð Ý M ˆ M Unit 2-cells 1 ˆ M M ˆ 1 M ˆ M M

e ˆ M M ˆ e » » m

• Assoc. 2-cell

pM ˆ Mq ˆ M M ˆ pM ˆ Mq M ˆ M M ˆ M M

» M ˆ m m m ˆ M m λ

–

α

–

ρ

– data

5 / 41

Internal pseudomonoids

In Cat: 1 e Ý Ñ M

m

Ð Ý M ˆ M Unit 2-cells 1 ˆ M M ˆ 1 M ˆ M M

e ˆ M M ˆ e » » m

• Assoc. 2-cell

pM ˆ Mq ˆ M M ˆ pM ˆ Mq M ˆ M M ˆ M M

» M ˆ m m m ˆ M m λ

–

α

–

ρ

– + triangle and pentagon laws ù monoidal category data

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Internal pseudomonoids

In Cat: 1 e Ý Ñ M

m

Ð Ý M ˆ M ...likewise in any fp-bicategory Unit 2-cells 1 ˆ M M ˆ 1 M ˆ M M

e ˆ M M ˆ e » » m

• Assoc. 2-cell

pM ˆ Mq ˆ M M ˆ pM ˆ Mq M ˆ M M ˆ M M

» M ˆ m m m ˆ M m λ

–

α

–

ρ

– + triangle and pentagon laws ù monoidal category data

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In a CCC every rX ñ Xs becomes a monoid: ´ 1

IdX

Ý Ý Ñ rX ñ Xs

˝

Ð Ý rX ñ Xs ˆ rX ñ Xs ¯ ? In a cc-bicategory every rX ñ Xs becomes a pseudomonoid: ´ 1

IdX

Ý Ý Ñ rX ñ Xs

˝

Ð Ý rX ñ Xs ˆ rX ñ Xs ¯

need to check coherence laws (i.e. triangle + pentagon)

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coherence Mac Lane-style

"all diagrams commute" proof theory, rewriting theory

coherence-by-strictification

B » B 2-cat. universal algebra, Yoneda embedding

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coherence Mac Lane-style

"all diagrams commute" proof theory, rewriting theory

coherence-by-strictification

B » B 2-cat. universal algebra, Yoneda embedding

“denotational semantics”

(1) construct an internal language (2) prove normalisation by semantic argument

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coherence “denotational semantics”

(1) construct an internal language (2) prove normalisation by semantic argument

‚ builds on categorical & type-theoretic intuition ‚ once set up about as hard as categorical proof

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Cartesian closed bicategories

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Composition by UMP ñ bicategory

In a category C with pullbacks:

• 1. objects: objects of C,
• 2. 1-cells A ù B: spans pA Ð X Ñ Bq,
• 3. 2-cells: commutative squares

X A B Y

h

Composition defined by pullback: X ˆB Y X Y A B C

x

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Composition by UMP ñ bicategory

In a category C with pullbacks:

• 1. objects: objects of C,
• 2. 1-cells A ù B: spans pA Ð X Ñ Bq,
• 3. 2-cells: commutative squares

X A B Y

h

Composition defined by pullback: ù associative up to iso X ˆB Y X Y A B C

x

9 / 41

Bicategories

10 / 41

Bicategories

• Objects X P obpBq,

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Bicategories

• Objects X P obpBq,
• Hom-categories

` BpX, Y q, ‚, id ˘ :

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Bicategories

• Objects X P obpBq,
• Hom-categories

` BpX, Y q, ‚, id ˘ :

1-cells X

f

Ý Ñ Y 2-cells X Y

f

óα

f 1

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Bicategories

• Objects X P obpBq,
• Hom-categories

` BpX, Y q, ‚, id ˘ :

1-cells X

f

Ý Ñ Y 2-cells X Y

f

óα

f 1

X Y

f f 1 óα óα1 f 2

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Bicategories

• Objects X P obpBq,
• Hom-categories

` BpX, Y q, ‚, id ˘ :

1-cells X

f

Ý Ñ Y 2-cells X Y

f

óα

f 1

• Identities IdX : X Ñ X and composition

BpY , Zq ˆ BpX, Y q

˝X,Y ,Z

Ý Ý Ý Ý Ñ BpX, Zq

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Bicategories

• Objects X P obpBq,
• Hom-categories

` BpX, Y q, ‚, id ˘ :

1-cells X

f

Ý Ñ Y 2-cells X Y

f

óα

f 1

• Identities IdX : X Ñ X and composition

BpY , Zq ˆ BpX, Y q

˝X,Y ,Z

Ý Ý Ý Ý Ñ BpX, Zq X Y Z

f óα f 1 g óβ g1

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Bicategories

• Objects X P obpBq,
• Hom-categories

` BpX, Y q, ‚, id ˘ :

1-cells X

f

Ý Ñ Y 2-cells X Y

f

óα

f 1

• Identities IdX : X Ñ X and composition

BpY , Zq ˆ BpX, Y q

˝X,Y ,Z

Ý Ý Ý Ý Ñ BpX, Zq

• Invertible 2-cells

ph ˝ gq ˝ f

ah,g,f

ù ù ù ñ h ˝ pg ˝ f q IdX ˝ f

lf

ù ñ f g ˝ IdX

rg

ù ñ g subject to a triangle law and pentagon law.

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Cartesian closed bicategories

Bicategories B equipped with

11 / 41

Cartesian closed bicategories

Bicategories B equipped with families of equivalences

B ` X, śn

i“1Ai

˘ śn

i“1 BpX, Aiq

» BpX, A “

⊲ Bq

BpX ˆ A, Bq » NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters!

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Cartesian closed bicategories

Bicategories B equipped with families of equivalences

B ` X, śn

i“1Ai

˘ śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

(tupling)

BpX, A “

⊲ Bq

BpX ˆ A, Bq

evalA,B˝p´ˆAq

% »

λ

(currying)

NB: Differ from the ‘cartesian bicategories’ of Carboni and Walters!

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Cartesian closed bicategories

Bicategories B equipped with families of equivalences

B ` X, śn

i“1Ai

˘ śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

(tupling)

BpX, A “

⊲ Bq

BpX ˆ A, Bq

evalA,B˝p´ˆAq

% »

λ

(currying)

πi ˝ xf1, . . . , fny

–

ù ñ fi g

–

ù ñ xg ˝ π1, . . . , g ˝ πny

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Cartesian closed bicategories

Bicategories B equipped with families of equivalences

B ` X, śn

i“1Ai

˘ śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

(tupling)

BpX, A “

⊲ Bq

BpX ˆ A, Bq

evalA,B˝p´ˆAq

% »

λ

(currying)

πi ˝ xf1, . . . , fny

–

ù ñ fi g

–

ù ñ xg ˝ π1, . . . , g ˝ πny evalA,B ˝ pλf ˆ Aq

–

ù ñ f g

–

ù ñ λpevalA,B ˝ pg ˆ Aqq

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The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

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The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

Features

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The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

Features

• Weak composition enforced by explicit substitution

x1 : A1, . . . , xn : An $ t : B p∆ $ ui : Aiqi“1.,n ∆ $ t txi ÞÑ uiu : B

x1 : A1, . . . , xn : An $ τ : t ñ t1 : B p∆ $ σi : ui ñ u1

i : Aiqi“1,...,n

∆ $ τ txi ÞÑ σiu : t txi ÞÑ uiu ñ t1 xi ÞÑ u1

i

( : B

ù binds the variables x1, . . . , xn

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The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

Features

• Weak composition enforced by explicit substitution
• Usual STLC operations on terms and rewrites

12 / 41

The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

Features

• Weak composition enforced by explicit substitution
• Usual STLC operations on terms and rewrites
• STLC embeds as explicit-substitution free fragment

12 / 41

The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

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

Γ A

Γ$t:A óΓ $ τ : t ñ t1 : A Γ$t1:A

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The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

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 t1 iff t “βη t1

12 / 41

The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

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 t1 iff t “βη t1

12 / 41

The internal language for cc-bicategories, Λˆ,Ñ

ps

Judgements c.f. Hilken, Hirschowitz

Terms Γ $ t : A (1-cells) Rewrites Γ $ τ : t ñ t1 : A (2-cells) Equations Γ $ τ ” τ 1 : t ñ t1 : A

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 t1 iff t “βη t1 ù coherence of cc-bicategories = coherence property of 2-cells

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Theorem (Coherence)

For any t and t1 in Λˆ,Ñ

ps

, there exists at most one rewrite τ such that Γ $ τ : t ñ t1 : A.

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Theorem (Coherence)

For any t and t1 in Λˆ,Ñ

ps

, there exists at most one rewrite τ such that Γ $ τ : t ñ t1 : 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.

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Theorem (Coherence)

For any t and t1 in Λˆ,Ñ

ps

, there exists at most one rewrite τ such that Γ $ τ : t ñ t1 : 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 t1 in Λˆ,Ñ

ps

, there exists at most one rewrite τ such that Γ $ τ : t ñ t1 : 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

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Theorem (Coherence)

For any t and t1 in Λˆ,Ñ

ps

, there exists at most one rewrite τ such that Γ $ τ : t ñ t1 : 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

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Theorem (Coherence)

For any t and t1 in Λˆ,Ñ

ps

, there exists at most one rewrite τ such that Γ $ τ : t ñ t1 : 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

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Normalisation-by-evaluation

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Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

´ quote

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

´ nfp´q quote

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

Neutral terms

includes variables defined by mutual induction

´ nfp´q quote unquote

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

Neutral terms

includes variables

´ nfp´q quote unquote

For $ λx.t : A “

⊲ B, get t : A Ñ B.

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

Neutral terms

includes variables

´ nfp´q quote unquote

For $ λx.t : A “

⊲ B, get t : A Ñ B. For fresh x:

unquotepxq : A

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

Neutral terms

includes variables

´ nfp´q quote unquote

For $ λx.t : A “

⊲ B, get t : A Ñ B. For fresh x:

tpunquotepxqq : B

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

Neutral terms

includes variables

´ nfp´q quote unquote

For $ λx.t : A “

⊲ B, get t : A Ñ B. For fresh x:

quoteptpunquotepxqqq : B

15 / 41

Normalisation-by-evaluation

Aim: find canonical representatives within each βη-equivalence class Strategy:

Syntax

βη-equivalent terms t, t1, t2, . . .

Semantics

t “ t1 “ t2 “ . . .

Normal terms

nfptq “ quoteptq “ . . . (long-βη)

Neutral terms

includes variables

´ nfp´q quote unquote

For $ λx.t : A “

⊲ B, get t : A Ñ B. For fresh x:

nfpλx.tq :“ λx.quoteptpunquotepxqqq : A “

⊲ B

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Normalisation-by-evaluation, categorically

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Normalisation-by-evaluation, categorically (Fiore)

Syntax as indexed presheaves over a category of contexts Con: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set

17 / 41

Normalisation-by-evaluation, categorically (Fiore)

Syntax as indexed presheaves over a category of contexts Con: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations

17 / 41

Normalisation-by-evaluation, categorically (Fiore)

Syntax as indexed presheaves over a category of contexts Con: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations pΓ $ t : Aq ÞÑ sΓ $ 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: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations pΓ $ t : Aq ÞÑ sΓ $ t : A s´ : neutsA ñ Cps´, sAq s´ : normsA ñ Cps´, sAq

s any interpretation of base types; C any CCC

17 / 41

Normalisation-by-evaluation, categorically (Fiore)

Syntax as indexed presheaves over a category of contexts Con: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations pΓ $ t : Aq ÞÑ sΓ $ t : A s´ : neutsA ñ Cps´, sAq s´ : normsA ñ Cps´, sAq

s any interpretation of base types; C any CCC

Strategy:

17 / 41

Normalisation-by-evaluation, categorically (Fiore)

Syntax as indexed presheaves over a category of contexts Con: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations pΓ $ t : Aq ÞÑ sΓ $ t : A s´ : neutsA ñ Cps´, sAq s´ : normsA ñ Cps´, sAq

s any interpretation of base types; C any CCC

Strategy:

• 1. define a glueing category G

17 / 41

Normalisation-by-evaluation, categorically (Fiore)

Syntax as indexed presheaves over a category of contexts Con: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations pΓ $ t : Aq ÞÑ sΓ $ t : A s´ : neutsA ñ Cps´, sAq s´ : normsA ñ Cps´, sAq

s any interpretation of base types; C any CCC

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: neutsA : Γ ÞÑ tneutral terms t such that Γ $ t : Au normsA : Γ ÞÑ tnormal terms t such that Γ $ t : Au + : Con Ñ Set syntax glued to semantics by natural transformations pΓ $ t : Aq ÞÑ sΓ $ t : A s´ : neutsA ñ Cps´, sAq s´ : normsA ñ Cps´, sAq

s any interpretation of base types; C any CCC

Strategy:

• 1. define a glueing category G
• 2. pick an interpretation e´ in G
• 3. define quote and unquote as maps in this category

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The glued category GpC, sq

Objects

C ccc, s : BaseTypes Ñ ObC

P

p

ù ñ Cps´, Xq

functor Con Ñ Set

• nat. trans

P C

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The glued category GpC, sq

Objects

C ccc, s : BaseTypes Ñ ObC

P

p

ù ñ Cps´, Xq

functor Con Ñ Set

• nat. trans

P C

Morphisms P P1 Cps´, Xq Cps´, X 1q

arrow X Ñ X 1

• nat. trans P ñ P1

κ p p1 k ˝ p´q

18 / 41

The glued category GpC, sq

Objects

C ccc, s : BaseTypes Ñ ObC

P

p

ù ñ Cps´, Xq

functor Con Ñ Set

• nat. trans

P C

Morphisms P P1 Cps´, Xq Cps´, X 1q

arrow X Ñ X 1

• nat. trans P ñ P1

κ p p1 k ˝ p´q

"syntax" "semantics" "interpretation"

18 / 41

The category GpC, sq

Objects

C ccc, s : BaseTypes Ñ ObC

P

p

ù ñ Cps´, Xq

functor Con Ñ Set

• nat. trans

P C

Morphisms P P1 Cps´, Xq Cps´, X 1q

κ p p1 k ˝ p´q

"syntax" "semantics" "interpretation"

19 / 41

The category GpC, sq

Objects

C ccc, s : BaseTypes Ñ ObC

P

p

ù ñ Cps´, Xq

functor Con Ñ Set

• nat. trans

P C

Morphisms P P1 Cps´, Xq Cps´, X 1q

κ p p1 k ˝ p´q

"syntax" "semantics" "interpretation"

ù Cartesian closed, with a strict CCC-functor GpC, sq Ñ C

19 / 41

The category GpC, sq

Objects

C ccc, s : BaseTypes Ñ ObC

P

p

ù ñ Cps´, Xq

functor Con Ñ Set

• nat. trans

P C

Morphisms P P1 Cps´, Xq Cps´, X 1q

κ p p1 k ˝ p´q

"syntax" "semantics" "interpretation"

ù Cartesian closed, with a strict CCC-functor GpC, sq Ñ C ù for every type A, neutsA

s´

ù ù ù ñ Cps´, sAq normsA

s´

ù ù ù ñ Cps´, sAq , .

• P GpC, sq

19 / 41

Interpretation in the glued category GpC, sq

C any ccc s : BaseTypes Ñ ObC

e : BaseTypes Ñ ObpGpC, sqq β ÞÑ ` neutsβ

s´

ù ù ù ñ Cps´, sβq ˘

20 / 41

Interpretation in the glued category GpC, sq

C any ccc s : BaseTypes Ñ ObC

e : BaseTypes Ñ ObpGpC, sqq β ÞÑ ` neutsβ

s´

ù ù ù ñ Cps´, sβq ˘ eA :“ ` eA

νA

ù ñ Cps´, sAq ˘

by strict preservation

20 / 41

Interpretation in the glued category GpC, sq

C any ccc s : BaseTypes Ñ ObC

e : BaseTypes Ñ ObpGpC, sqq β ÞÑ ` neutsβ

s´

ù ù ù ñ Cps´, sβq ˘ eA C ` s´, sAq ˘

νA

20 / 41

Interpretation in the glued category GpC, sq

C any ccc s : BaseTypes Ñ ObC

e : BaseTypes Ñ ObpGpC, sqq β ÞÑ ` neutsβ

s´

ù ù ù ñ Cps´, sβq ˘ eΓ eA C ` s´, sΓ ˘ C ` s´, sA ˘

νΓ eΓ$t:A νA sΓ$t:A˝p´q

20 / 41

Interpretation in the glued category GpC, sq

C any ccc s : BaseTypes Ñ ObC

e : BaseTypes Ñ ObpGpC, sqq β ÞÑ ` neutsβ

s´

ù ù ù ñ Cps´, sβq ˘ eΓ eA C ` s´, sΓ ˘ C ` s´, sA ˘

νΓ eΓ$t:A νA sΓ$t:A˝p´q

neutsA eA C ` s´, sA ˘ C ` s´, sA ˘

quoteA s´ νA preserves βη

20 / 41

Interpretation in the glued category GpC, sq

C any ccc s : BaseTypes Ñ ObC

e : BaseTypes Ñ ObpGpC, sqq β ÞÑ ` neutsβ

s´

ù ù ù ñ Cps´, sβq ˘ eΓ eA C ` s´, sΓ ˘ C ` s´, sA ˘

νΓ eΓ$t:A νA sΓ$t:A˝p´q

neutsA eA C ` s´, sA ˘ C ` s´, sA ˘

quoteA s´ νA preserves βη

eA normsA C ` s´, sA ˘ C ` s´, sA ˘

unquoteA νA s´ preserves βη

20 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

neutsΓ Cps´, sΓq eΓ Cps´, sΓq eA Cps´, sAq normsA Cps´, sAq

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

21 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

neutsΓ Cps´, sΓq eΓ Cps´, sΓq eA Cps´, sAq normsA Cps´, sAq

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

ù recall nfpλx.tq :“ λx.quoteptpunquotepxqqq

21 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

neutsΓ Cps´, sΓq eΓ Cps´, sΓq eA Cps´, sAq normsA Cps´, sAq

normt unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

ù recall nfpλx.tq :“ λx.quoteptpunquotepxqqq

21 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

22 / 41

Normalising pΓ $ t : Aq

neutsΓ :“ Πpxi:AiqPΓ neutsAi unquoteΓ :“ Πpxi:AiqPΓ unquoteAi

pxi : Aiqi pxi : Aiqi neutsΓpΓq eΓpΓq CpsΓ, sΓq idsΓ eApΓq CpsΓ, sAq sΓ $ t : A sΓ $ t : A normsApΓq nfptq

normΓ

t

unquoteΓ quoteA s´ νA νΓ eΓ $ t : A sΓ $ t : A ˝ p´q s´

Define nfptq :“ normΓ

t ppΓ $ xi : Aiqi“1,...,nq.

Since sΓ $ nfptq : A “ sΓ $ t : A in every model, nfptq “βη t.

22 / 41

Normalisation-by-evaluation for Λˆ,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, GpC, sq,. . . bicategorically)

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Set

neutsps

A pΓq, normsps A pΓq sets of terms 25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories 25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù st snfptq

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù st snfptq

depends on t –

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù st snfptq st1 snfpt1q

depends on t – – depends on t1

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù st snfptq st1 snfpt1q

sτ depends on t – – depends on t1

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù st snfptq st1 snfpt1q

sτ depends on t – – depends on t1

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù t nfptq t1 nfpt1q

τ – depends on t – depends on t1

25 / 41

Strategy

ù re-use universal properties

(quote, unquote, GpC, sq,. . . bicategorically)

ù use embedding of STLC into Λˆ,Ñ

ps

neutsps

A :“ neutsSTLC A

inside Λˆ,Ñ

ps

normsps

A :“ normsSTLC A

inside Λˆ,Ñ

ps

+ : Con Ñ Cat

neutsps

A pΓq, normsps A pΓq sets of terms as discrete categories

ù If τ exists, it’s unique t nfptq t1 nfpt1q

τ – depends on t – depends on t1

25 / 41

The glued cc-bicategory GpB, sq

B any cc-bicat s any interpretation of base types

Want: for every type A, neutsps

A s´

ù ù ù ñ Bps´, sAq normsps

A s´

ù ù ù ñ Bps´, sAq , .

• P GpB, sq

26 / 41

The glued cc-bicategory GpB, sq

Objects

B cc-bicat, s : BaseTypes Ñ ObB

P

pp,pq

ù ù ù ñ Bps´, Xq

pseudofunctor Con Ñ Cat pseudo

• nat. trans.

P B

27 / 41

The glued cc-bicategory GpB, sq

Objects

B cc-bicat, s : BaseTypes Ñ ObB

P

pp,pq

ù ù ù ñ Bps´, Xq

pseudofunctor Con Ñ Cat pseudo

• nat. trans.

P B

1-cells P P1 Bps´, Xq Bps´, X 1q

arrow X Ñ X 1 pseudo nat. trans. P ñ P1 κ p p1 k ˝ p´q wk

–

27 / 41

The glued cc-bicategory GpB, sq

Objects

B cc-bicat, s : BaseTypes Ñ ObB

P

pp,pq

ù ù ù ñ Bps´, Xq

pseudofunctor Con Ñ Cat pseudo

• nat. trans.

P B

1-cells P P1 Bps´, Xq Bps´, X 1q

arrow X Ñ X 1 pseudo nat. trans. P ñ P1 κ p p1 k ˝ p´q wk

–

"syntax" "semantics" "interpretation"

27 / 41

The glued cc-bicategory GpB, sq

Objects

B cc-bicat, s : BaseTypes Ñ ObB

P

pp,pq

ù ù ù ñ Bps´, Xq

pseudofunctor Con Ñ Cat pseudo

• nat. trans.

P B

1-cells P P1 Bps´, Xq Bps´, X 1q

κ p p1 k ˝ p´q wk

–

"syntax" "semantics" "interpretation"

2-cells P P1 Bps´, Xq Bps´, X 1q

κ κ1 k ˝ p´q k1 ˝ p´q

ó Θ ó ξ ˝ p´q

27 / 41

The glued cc-bicategory GpB, sq

P P1 Bps´, Xq Bps´, X 1q

κ ó Θ κ1 wk1

–

k1˝p´q

= P P1 Bps´, Xq Bps´, X 1q

κ wk

–

k˝p´q ó ξ ˝ p´q k1˝p´q

28 / 41

Interpretation in GpB, sq

B any cc-bicat s : BaseTypes Ñ ObB

e : BaseTypes Ñ ObpGpB, sqq β ÞÑ ` neutsps

β s´

ù ù ù ñ Bps´, sβq ˘

29 / 41

Interpretation in GpB, sq

B any cc-bicat s : BaseTypes Ñ ObB

e : BaseTypes Ñ ObpGpB, sqq β ÞÑ ` neutsps

β s´

ù ù ù ñ Bps´, sβq ˘ eA :“ ` eA

νA

ù ñ Bps´, sAq ˘

by strict preservation

29 / 41

Interpretation in GpB, sq

B any cc-bicat s : BaseTypes Ñ ObB

e : BaseTypes Ñ ObpGpB, sqq β ÞÑ ` neutsps

β s´

ù ù ù ñ Bps´, sβq ˘ eA B ` s´, sA ˘

νA

29 / 41

Interpretation in GpB, sq

B any cc-bicat s : BaseTypes Ñ ObB

e : BaseTypes Ñ ObpGpB, sqq β ÞÑ ` neutsps

β s´

ù ù ù ñ Bps´, sβq ˘ eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt

–

νΓ eΓ$t:A νA sΓ$t:A˝p´q

29 / 41

Interpretation in GpB, sq

B any cc-bicat s : BaseTypes Ñ ObB

e : BaseTypes Ñ ObpGpB, sqq β ÞÑ ` neutsps

β s´

ù ù ù ñ Bps´, sβq ˘ eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt

–

νΓ eΓ$t:A νA sΓ$t:A˝p´q

neutsps

A

eA B ` s´, sΓ ˘ B ` s´, sA ˘

quoteA s´ qA

–

νA preserves βη mod –

29 / 41

Interpretation in GpB, sq

B any cc-bicat s : BaseTypes Ñ ObB

e : BaseTypes Ñ ObpGpB, sqq β ÞÑ ` neutsps

β s´

ù ù ù ñ Bps´, sβq ˘ eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt

–

νΓ eΓ$t:A νA sΓ$t:A˝p´q

neutsps

A

eA B ` s´, sΓ ˘ B ` s´, sA ˘

quoteA s´ qA

–

νA preserves βη mod –

eA normsps

A

B ` s´, sΓ ˘ B ` s´, sA ˘

unquoteA νA uA

–

s´ preserves βη mod –

29 / 41

neutsps

Γ

Bps´, sΓq eΓ Bps´, sΓq eA Bps´, sAq normsps

A

Bps´, sAq

unquoteΓ quoteA s´ eΓ $ t : A sΓ $ t : A ˝ p´q s´ uΓ

–

wt

–

qA

–

30 / 41

neutsps

Γ

Bps´, sΓq eΓ Bps´, sΓq eA Bps´, sAq normsps

A

Bps´, sAq

normt unquoteΓ quoteA s´ eΓ $ t : A sΓ $ t : A ˝ p´q s´ uΓ

–

wt

–

qA

–

30 / 41

neutsps

Γ

Bps´, sΓq eΓ Bps´, sΓq eA Bps´, sAq normsps

A

Bps´, sAq

normt unquoteΓ quoteA s´ eΓ $ t : A sΓ $ t : A ˝ p´q s´ uΓ

–

wt

–

qA

– Instantiate at Γ, pass in variables ù st – snfptq

30 / 41

neutsps

Γ

Bps´, sΓq eΓ Bps´, sΓq eA Bps´, sAq normsps

A

Bps´, sAq

normt unquoteΓ quoteA s´ eΓ $ t : A sΓ $ t : A ˝ p´q s´ uΓ

–

wt

–

qA

– Instantiate at Γ, pass in variables ù st – snfptq st snfptq st1 snfpt1q

– –

30 / 41

neutsps

Γ

Bps´, sΓq eΓ Bps´, sΓq eA Bps´, sAq normsps

A

Bps´, sAq

normt unquoteΓ quoteA s´ eΓ $ t : A sΓ $ t : A ˝ p´q s´ uΓ

–

wt

–

qA

– Instantiate at Γ, pass in variables ù st – snfptq st snfptq st1 snfpt1q

sτ – –

30 / 41

Proving coherence

31 / 41

Interpreting pΓ $ τ : t ñ t1 : Aq in GpB, sq

eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt

–

νΓ eΓ$t:A νA sΓ$t:A˝p´q

32 / 41

Interpreting pΓ $ τ : t ñ t1 : Aq in GpB, sq

eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt

–

νΓ eΓ$t:A νA sΓ$t:A˝p´q

eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt1

–

νΓ eΓ$t1:A νA sΓ$t1:A˝p´q

32 / 41

Interpreting pΓ $ τ : t ñ t1 : Aq in GpB, sq

eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt

–

νΓ eΓ$t:A νA sΓ$t:A˝p´q

eΓ eA B ` s´, sΓ ˘ B ` s´, sA ˘

wt1

–

νΓ eΓ$t1:A νA sΓ$t1:A˝p´q

eΓ eA

eΓ$t:A ó eτ eΓ$t1:A

Bps´, sΓq Bps´, sAq

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

32 / 41

Cylinder condition for pΓ $ τ : t ñ t1 : Aq

eΓ eA Bps´, sΓq Bps´, sAq

eΓ$t:A ó eτ eΓ$t1:A wt1

–

sΓ$t1:A˝p´q

= eΓ eA Bps´, sΓq Bps´, sAq

eΓ$t:A wt

–

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

33 / 41

neutsps

Γ

eΓ eA normsps

A

Bps´, sΓq Bps´, sAq

unquoteΓ s´ uΓ

–

wt

–

eΓ$t:A quoteA s´ qA

–

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

= neutsps

Γ

eΓ eA normsps

A

Bps´, sΓq Bps´, sAq

unquoteΓ s´ uΓ

–

eΓ$t:A ó eτ eΓ$t1:A quoteA s´ qA

–

wt1

–

sΓ$t1:A˝p´q

34 / 41

Bps´, sΓq Bps´, sAq

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

= Bps´, sΓq Bps´, sAq neutsps

Γ

eΓ eA normsps

A

Bps´, sΓq Bps´, sAq

s´ w´1

t

–

sΓ$t:A˝p´q uΓ

–

s´ qA

–

unquoteΓ

s´ uΓ

–

eΓ$t:A ó eτ eΓ$t1:A

quoteA

s´ qA

–

wt1

–

sΓ$t1:A˝p´q

35 / 41

BpsΓ, sΓq BpsΓ, sAq

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

= BpsΓ, sΓq BpsΓ, sAq neutsps

Γ pΓq

eΓpΓq eApΓq normsps

A pΓq

BpsΓ, sΓq BpsΓ, sAq

s´ w´1

t

–

sΓ$t:A˝p´q uΓ

–

s´ qA

–

s´ uΓ

–

eΓ$t:A ó eτ eΓ$t1:A s´ qA

–

wt1

–

sΓ$t1:A˝p´q

36 / 41

In HompCon, Catq neutsps

Γ

eΓ eA normsps

A unquoteΓ eΓ$t:A ó eτ eΓ$t1:A quoteA

37 / 41

In Cat neutsps

Γ p∆q

eΓp∆q eAp∆q normsps

A p∆q unquoteΓ eΓ$t:A ó eτ eΓ$t1:A quoteA

a set!

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In Cat neutsps

Γ p∆q

eΓp∆q eAp∆q normsps

A p∆q unquoteΓ eΓ$t:A ó eτ eΓ$t1:A quoteA

a set!

Composite can only be the identity

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BpsΓ, sΓq BpsΓ, sAq

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

= BpsΓ, sΓq BpsΓ, sAq neutsps

Γ pΓq

eΓpΓq eApΓq normsps

A pΓq

BpsΓ, sΓq BpsΓ, sAq

s´ w´1

t

–

sΓ$t:A˝p´q uΓ

–

s´ qA

–

s´ uΓ

–

eΓ$t:A ó eτ eΓ$t1:A s´ qA

–

wt1

–

sΓ$t1:A˝p´q

independent of τ!

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idsΓ P BpsΓ, sΓq BpsΓ, sAq ù sτ

sΓ$t:A˝p´q ó sτ ˝ p´q sΓ$t1:A˝p´q

= BpsΓ, sΓq BpsΓ, sAq neutsps

Γ pΓq

eΓpΓq eApΓq normsps

A pΓq

BpsΓ, sΓq BpsΓ, sAq

s´ w´1

t

–

sΓ$t:A˝p´q uΓ

–

s´ qA

–

s´ uΓ

–

eΓ$t:A ó eτ eΓ$t1:A s´ qA

–

wt1

–

sΓ$t1:A˝p´q

independent of τ!

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Proposition

For any cc-bicategory B and interpretation s : BaseTypes Ñ ObB of base types, the interpretation sΓ $ τ : t ñ t1 : A depends only on t and t1.

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Proposition

For any cc-bicategory B and interpretation s : BaseTypes Ñ ObB of base types, the interpretation sΓ $ τ : t ñ t1 : A depends only on t and t1. ù For any derivable τ, τ 1 : t ñ1: A, sΓ $ τ : t ñ t1 : A “ sΓ $ τ 1 : t ñ t1 : A

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Proposition

For any cc-bicategory B and interpretation s : BaseTypes Ñ ObB of base types, the interpretation sΓ $ τ : t ñ t1 : A depends only on t and t1. ù For any derivable τ, τ 1 : t ñ1: A, sΓ $ τ : t ñ t1 : A “ sΓ $ τ 1 : t ñ t1 : A

Corollary

For any derivable τ, τ 1 : t ñ1: A, Γ $ τ ” τ 1 : t ñ t1 : A is derivable.

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Coherence

ù cc-Bicategories are coherent

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Coherence

ù cc-Bicategories are coherent ù “Suffices” to work in a CCC

• 1. prove result in STLC
• 2. βη-equalities ù 2-cells
• 3. axioms guaranteed

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Coherence

ù cc-Bicategories are coherent ù “Suffices” to work in a CCC

• 1. prove result in STLC
• 2. βη-equalities ù 2-cells
• 3. axioms guaranteed

Strategy

ù Coherence as a normalisation property ù Normalisation proven semantically ù If you work with universal properties enough . . . higher-categorical proof builds on categorical proof Future work: extend this to other structures

e.g. Kan extensions, monads, dependent products,. . .

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Further reading

• A type theory for cartesian closed bicategories, LICS 2019
• Relative full completeness for bicategorical cartesian closed

structure, FoSSaCS 2020

• Cartesian closed bicategories: type theory and coherence

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cc-Bicategories

1-cells evalA,B : pA “

⊲ Bq ˆ A Ñ B

Adjoint equivalences BpX, A “

⊲ Bq

BpX ˆ A, Bq

evalA,B˝p´ˆAq

% »

λ

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Rules for exponentials

explicit weakening by x free variable in context

px : Aq eval tu tincxu , xu ñ t : B u ñ λx.t : A “

⊲ B ǫ ‚ eval tp´q tincxu , xu e:px . ´q eval

f : A “

⊲ B, x : A $ evalpf , xq : B

Γ, x : A $ t : B

lam

Γ $ λx.t : A “

⊲ B

Γ, x : A $ t : B

ǫ-intro (β-rule)

Γ, x : A $ ǫt : eval tpλx.tq tincxu , xu ñ t : B Γ, x : A $ t : B Γ $ u : A “

⊲ B

Γ, x : A $ α : eval tu tincxu , xu ñ t : B

e:px . αq-intro

Γ $ e:px . αq : u ñ λx.t : A “

⊲ B

+ three equational rules ù η-rule derivable

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