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 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

3 / 25

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

3 / 25

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

4 / 25

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

4 / 25

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

4 / 25

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)

5 / 25

Coherence

Programme:

• 1. Construct a type theory Λˆ,Ñ

ps

for cartesian closed bicategories (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 Λˆ,Ñ

ps

for cartesian closed bicategories (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 Λˆ,Ñ

ps

that:

7 / 25

Desiderata

A type theory Λˆ,Ñ

ps

that:

• 1. Generalises the simply-typed lambda calculus,
• 2. Is reasonable for calculations,
• 3. Is sound and complete

7 / 25

Desiderata

A type theory Λˆ,Ñ

ps

that:

• 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 obpBq,

8 / 25

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

8 / 25

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

8 / 25

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

• Functors

1

IdX

Ý Ý Ñ BpX, Xq 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

• Functors

1

IdX

Ý Ý Ñ BpX, Xq 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

• Functors

1

IdX

Ý Ý Ñ BpX, Xq 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 biuniversal 1-cells

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Cartesian closed bicategories

Bicategories B equipped with biuniversal 1-cells (fp) πi : ΠnpA1, . . . , Anq Ñ Ai p1 ď i ď nq (cc) eval : pA ñ Bq ˆ A Ñ B

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

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Cartesian closed bicategories

Bicategories B equipped with biuniversal 1-cells (fp) πi : ΠnpA1, . . . , Anq Ñ Ai p1 ď i ď nq (cc) eval : pA ñ Bq ˆ A Ñ B inducing families of equivalences

B pX, ΠnpA1, . . . , Anqq 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 biuniversal 1-cells (fp) πi : ΠnpA1, . . . , Anq Ñ Ai p1 ď i ď nq (cc) eval : pA ñ Bq ˆ A Ñ B inducing families of equivalences

B pX, ΠnpA1, . . . , Anqq 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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Substitution and composition

In any CCC: xkru1{x1, . . . , un{xns “ uk “ πk ˝ xu1, . . . , uny

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Substitution and composition

In any CCC: xkru1{x1, . . . , un{xns “ uk “ πk ˝ xu1, . . . , uny In any cc-bicategory: xkru1{x1, . . . , un{xns “ uk – πk ˝ xu1, . . . , uny

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Substitution and composition

In any CCC: xkru1{x1, . . . , un{xns “ uk “ πk ˝ xu1, . . . , uny In any cc-bicategory: xkru1{x1, . . . , un{xns “ uk – πk ˝ xu1, . . . , uny

Question: what is bicategorical substitution?

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An algebraic theory with substitution:

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An algebraic theory with substitution:

• Sorts S,

11 / 25

An algebraic theory with substitution:

• Sorts S,
• Constants x1 : X1, . . . , xn : Xn $ tpx1, . . . , xnq : Y ,

11 / 25

An algebraic theory with substitution:

• Sorts S,
• Constants x1 : X1, . . . , xn : Xn $ tpx1, . . . , xnq : Y ,
• Variables x1 : X1, . . . , xn : Xn $ xi : Xi p1 ď i ď nq,

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An algebraic theory with substitution:

• Sorts S,
• Constants x1 : X1, . . . , xn : Xn $ tpx1, . . . , xnq : Y ,
• Variables x1 : X1, . . . , xn : Xn $ xi : Xi p1 ď i ď nq,
• A substitution rule

t, pu1, . . . , unq ÞÑ trui{xis

11 / 25

An algebraic theory with substitution:

• Sorts S,
• Constants x1 : X1, . . . , xn : Xn $ tpx1, . . . , xnq : Y ,
• Variables x1 : X1, . . . , xn : Xn $ xi : Xi p1 ď i ď nq,
• A substitution rule

t, pu1, . . . , unq ÞÑ trui{xis such that xkrui{xis “ uk p1 ď k ď nq trxi{xis “ t trui{xisrvj{yjs “ t ruirvj{yjs{xis

11 / 25

Abstract clone pS, Cq = abstract theory of substitution:

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Abstract clone pS, Cq = abstract theory of substitution:

• Sorts S,

12 / 25

Abstract clone pS, Cq = abstract theory of substitution:

• Sorts S,
• Hom-sets CpX1, . . . , Xn; Y q of operations X1, . . . , Xn

t

Ý Ñ Y ,

12 / 25

Abstract clone pS, Cq = abstract theory of substitution:

• Sorts S,
• Hom-sets CpX1, . . . , Xn; Y q of operations X1, . . . , Xn

t

Ý Ñ Y ,

• Projections X1, . . . , Xn

ppiq

X1,...,Xn

Ý Ý Ý Ý Ý Ñ Xi p1 ď i ď nq,

12 / 25

Abstract clone pS, Cq = abstract theory of substitution:

• Sorts S,
• Hom-sets CpX1, . . . , Xn; Y q of operations X1, . . . , Xn

t

Ý Ñ Y ,

• Projections X1, . . . , Xn

ppiq

X1,...,Xn

Ý Ý Ý Ý Ý Ñ Xi p1 ď i ď nq,

• Substitution mappings

CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns

12 / 25

Abstract clone pS, Cq = abstract theory of substitution:

• Sorts S,
• Hom-sets CpX1, . . . , Xn; Y q of operations X1, . . . , Xn

t

Ý Ñ Y ,

• Projections X1, . . . , Xn

ppiq

X1,...,Xn

Ý Ý Ý Ý Ý Ñ Xi p1 ď i ď nq,

• Substitution mappings

CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns such that ppkqru1, . . . , uns “ uk p1 ď k ď nq trpp1q, . . . , ppnqs “ t t ru‚s rv‚s “ trv‚ru‚ss

12 / 25

Abstract clone pS, Cq = abstract theory of substitution:

• Sorts S,
• Hom-sets CpX1, . . . , Xn; Y q of operations X1, . . . , Xn

t

Ý Ñ Y ,

• Projections X1, . . . , Xn

ppiq

X1,...,Xn

Ý Ý Ý Ý Ý Ñ Xi p1 ď i ď nq,

• Substitution mappings

CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns such that ppkqru1, . . . , uns “ uk p1 ď k ď nq trpp1q, . . . , ppnqs “ t t ru‚s rv‚s “ trv‚ru‚ss Note: every clone defines a category

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Biclone pS, Cq = abstract theory of bicategorical substitution:

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Biclone pS, Cq = abstract theory of bicategorical substitution:

• Sorts S,

13 / 25

Biclone pS, Cq = abstract theory of bicategorical substitution:

• Sorts S,
• Hom-categories

` CpX1, . . . , Xn; Y q, ‚, id ˘ ,

13 / 25

Biclone pS, Cq = abstract theory of bicategorical substitution:

• Sorts S,
• Hom-categories

` CpX1, . . . , Xn; Y q, ‚, id ˘ ,

• Projection 1-cells ppiq

X1,...,Xn : X1, . . . , Xn Ñ Xi p1 ď i ď nq,

13 / 25

Biclone pS, Cq = abstract theory of bicategorical substitution:

• Sorts S,
• Hom-categories

` CpX1, . . . , Xn; Y q, ‚, id ˘ ,

• Projection 1-cells ppiq

X1,...,Xn : X1, . . . , Xn Ñ Xi p1 ď i ď nq,

• Substitution functors

CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns τ, pσ1, . . . , σnq ÞÑ τrσ1, . . . , σns

13 / 25

Biclone pS, Cq = abstract theory of bicategorical substitution:

• Sorts S,
• Hom-categories

` CpX1, . . . , Xn; Y q, ‚, id ˘ ,

• Projection 1-cells ppiq

X1,...,Xn : X1, . . . , Xn Ñ Xi p1 ď i ď nq,

• Substitution functors

CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns τ, pσ1, . . . , σnq ÞÑ τrσ1, . . . , σns

• Structural isomorphisms

ppkqru1, . . . , uns

̺pkq

u‚

ù ù ñ uk p1 ď k ď nq trpp1q, . . . , ppnqs

ιt

ù ñ t t ru‚s rv‚s

assoct;u‚;v‚

ù ù ù ù ù ù ñ trv‚ru‚ss subject to a triangle law and pentagon law.

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Biclone pS, Cq = abstract theory of bicategorical substitution:

• Sorts S,

Note: every biclone defines a bicategory

• Hom-categories

` CpX1, . . . , Xn; Y q, ‚, id ˘ ,

• Projection 1-cells ppiq

X1,...,Xn : X1, . . . , Xn Ñ Xi p1 ď i ď nq,

• Substitution functors

CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns τ, pσ1, . . . , σnq ÞÑ τrσ1, . . . , σns

• Structural isomorphisms

ppkqru1, . . . , uns

̺pkq

u‚

ù ù ñ uk p1 ď k ď nq trpp1q, . . . , ppnqs

ιt

ù ñ t t ru‚s rv‚s

assoct;u‚;v‚

ù ù ù ù ù ù ñ trv‚ru‚ss subject to a triangle law and pentagon law.

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A type theory for biclones

14 / 25

A type theory for biclones

Hom-categories ` CpX1, . . . , Xn; Y q, ‚, id ˘

(c.f. Hilken, Seely, Hirschowitz)

14 / 25

A type theory for biclones

Hom-categories ` CpX1, . . . , Xn; Y q, ‚, id ˘

(c.f. Hilken, Seely, Hirschowitz)

Judgements:

14 / 25

A type theory for biclones

Hom-categories ` CpX1, . . . , Xn; Y q, ‚, id ˘

(c.f. Hilken, Seely, Hirschowitz)

Judgements:

• Relating terms: Γ $ t : B

14 / 25

A type theory for biclones

Hom-categories ` CpX1, . . . , Xn; Y q, ‚, id ˘

(c.f. Hilken, Seely, Hirschowitz)

Judgements:

• Relating terms: Γ $ t : B
• Relating rewrites: Γ $ τ : t ñ t1 : B

14 / 25

A type theory for biclones

Hom-categories ` CpX1, . . . , Xn; Y q, ‚, id ˘

(c.f. Hilken, Seely, Hirschowitz)

Judgements:

• Relating terms: Γ $ t : B
• Relating rewrites: Γ $ τ : t ñ t1 : B
• Equational theory Γ $ τ ” τ 1 : t ñ t1 : B

14 / 25

A type theory for biclones

Hom-categories ` CpX1, . . . , Xn; Y q, ‚, id ˘

(c.f. Hilken, Seely, Hirschowitz)

Judgements:

• Relating terms: Γ $ t : B
• Relating rewrites: Γ $ τ : t ñ t1 : B
• Equational theory Γ $ τ ” τ 1 : t ñ t1 : B

Vertical composition: Γ $ τ 1 : t1 ñ t2 : B Γ $ τ : t ñ t1 : B Γ $ τ 1 ‚ τ : t ñ t2 : B Identities: Γ $ t : B Γ $ idt : t ñ t : B

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A type theory for biclones

A substitution functor CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns τ, pσ1, . . . , σnq ÞÑ τrσ1, . . . , σns

A type theory for biclones

A substitution functor CpX1, . . . , Xn; Y q ˆ śn

i“1 CpΓ; Xiq Ñ CpΓ; Y q

t, pu1, . . . , unq ÞÑ tru1, . . . , uns τ, pσ1, . . . , σnq ÞÑ τrσ1, . . . , σns 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 txi ÞÑ u1

iu : B

ù binds the variables x1, . . . , xn

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A type theory for biclones

Structural isomorphisms ̺pkq, ι, assoc

A type theory for biclones

Structural isomorphisms ̺pkq, ι, assoc Distinguished invertible rewrites e.g.:

p∆ $ ui : Aiqi“1,...,n

p1 ď k ď nq

x1 : A1, . . . , xn : An $ ̺pkq

u‚ : xk txi ÞÑ uiu –

ù ñ uk : Ak

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The syntactic model is free

17 / 25

The syntactic model is free

SynpSq C S

syntactic model biclone signature D! strict h# @ h

17 / 25

The syntactic model is free

SynpSq ˇ ˇ

1

B S ˇ ˇ

1 syntactic model

• n unary contexts

bicategory unary signature D! strict h# @ h

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The syntactic model is free

SynpSq ˇ ˇ

1

B S ˇ ˇ

1 syntactic model

• n unary contexts

bicategory unary signature D! strict h# @ h

ù An internal language for bicategories.

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fp-Bicategories

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fp-Bicategories

1-cells πi : ΠnpA1, . . . , Anq Ñ Ai p1 ď i ď nq Adjoint equivalences B pX, ΠnpA1, . . . , Anqq śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

18 / 25

A type theory for fp-bicategories

1-cells πi : ΠnpA1, . . . , Anq Ñ Ai p1 ď i ď nq

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A type theory for fp-bicategories

1-cells πi : ΠnpA1, . . . , Anq Ñ Ai p1 ď i ď nq Projections

p1 ď i ď nq

p : ΠnpA1, . . . , Anq $ πippq : Ai

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A type theory for fp-bicategories

Equivalences

B pX, ΠnpA1, . . . , Anqq śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

̟piq ‚ πi tp´qu p:p´, . . . , “q

A type theory for fp-bicategories

Equivalences

B pX, ΠnpA1, . . . , Anqq śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

πi ˝ u ñ ti : Ai pi “ 1, . . . , nq u ñ xt1, . . . , tny : ΠnpA1, . . . , Anq

̟piq ‚ ` πi ˝ p´q ˘ p:p´, . . . , “q

for a counit p̟piq : πi ˝ xt1, . . . , tny ñ ti : Aiqi“1,...,n

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A type theory for fp-bicategories

Equivalences

B pX, ΠnpA1, . . . , Anqq śn

i“1 BpX, Aiq pπ1˝´,...,πn˝´q

% »

x´,...,“y

syntactic sugar

πi tuu ñ ti : Ai pi “ 1, . . . , nq u ñ tuppt1, . . . , tnq : ΠnpA1, . . . , Anq

̟piq ‚ πi tp´qu p:p´, . . . , “q

for a counit p̟piq : πi ttuppt1, . . . , tnqu ñ ti : ΠnpA1, . . . , Anqqi“1,...,n

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A type theory for fp-bicategories

πi tuu ñ ti : Ai pi “ 1, . . . , nq u ñ tuppt1, . . . , tnq : ΠnpA1, . . . , Anq

̟piq ‚ πi tp´qu p:p´, . . . , “q

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A type theory for fp-bicategories

πi tuu ñ ti : Ai pi “ 1, . . . , nq u ñ tuppt1, . . . , tnq : ΠnpA1, . . . , Anq

̟piq ‚ πi tp´qu p:p´, . . . , “q

Tupling map pΓ $ ti : Aiqi“1,...,n Γ $ tuppt1 . . . , tnq : ΠnpA1, . . . , Anq

21 / 25

A type theory for fp-bicategories

πi tuu ñ ti : Ai pi “ 1, . . . , nq u ñ tuppt1, . . . , tnq : ΠnpA1, . . . , Anq

̟piq ‚ πi tp´qu p:p´, . . . , “q

Tupling map pΓ $ ti : Aiqi“1,...,n Γ $ tuppt1 . . . , tnq : ΠnpA1, . . . , Anq Counit (β-law)

pΓ $ ti : Aiqi“1,...,n

p1 ď k ď nq

Γ $ ̟pkq

t‚ : πk ttuppt1 . . . , tnqu –

ù ñ tk : Ak

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A type theory for fp-bicategories

πi tuu ñ ti : Ai pi “ 1, . . . , nq u ñ tuppt1, . . . , tnq : ΠnpA1, . . . , Anq

̟piq ‚ πi tp´qu p:p´, . . . , “q

Tupling map pΓ $ ti : Aiqi“1,...,n Γ $ tuppt1 . . . , tnq : ΠnpA1, . . . , Anq Counit (β-law)

pΓ $ ti : Aiqi“1,...,n

p1 ď k ď nq

Γ $ ̟pkq

t‚ : πk ttuppt1 . . . , tnqu –

ù ñ tk : Ak

Mediating 2-cell pΓ $ αi : πi tuu ñ ti : Aiqi“1,...,n Γ $ p:pα1, . . . , αnq : u ñ tuppt1, . . . , tnq : ΠnpA1, . . . , Anq + three equational rules. ù η-law is derivable

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The syntactic model is free

pSynpSq, Πnp´qq pC, Πnp´qq S

syntactic model fp-biclone signature

(Require Π1pXq “ X)

D! strict h# @ h

22 / 25

The syntactic model is free

` SynpSq ˇ ˇ

1, Πnp´q

˘ pB, Πnp´qq S ˇ ˇ

1 syntactic model

• n unary contexts

fp-bicategory unary signature

(Require Π1pXq “ X)

D! strict h# @ h

22 / 25

The syntactic model is free

` SynpSq ˇ ˇ

1, Πnp´q

˘ pB, Πnp´qq S ˇ ˇ

1 syntactic model

• n unary contexts

fp-bicategory unary signature

(Require Π1pXq “ X)

D! strict h# @ h

ù An internal language for fp-bicategories. derived from definition of biadjoint

22 / 25

The syntactic model is free

pSynpSq, Πnp´q, “

⊲q

pC, Πnp´q, “

⊲q

S

syntactic model cc-biclone signature D! strict h# @ h

(Require Π1pXq “ X) 23 / 25

The syntactic model is free

` SynpSq ˇ ˇ

1, Πnp´q, “ ⊲˘

pB, Πnp´q, “

⊲q

S ˇ ˇ

1 syntactic model

• n unary contexts

cc-bicategory with strict products unary signature D! strict h# @ h

(Require Π1pXq “ X) 23 / 25

The syntactic model is free

` SynpSq ˇ ˇ

1, Πnp´q, “ ⊲˘

pB, Πnp´q, “

⊲q

S ˇ ˇ

1 syntactic model

• n unary contexts

cc-bicategory unary signature @ h

(Require Π1pXq “ X)

unique up to equivalence

23 / 25

The syntactic model is free

` SynpSq ˇ ˇ

1, Πnp´q, “ ⊲˘

pB, Πnp´q, “

⊲q

S ˇ ˇ

1 syntactic model

• n unary contexts

cc-bicategory unary signature @ h

(Require Π1pXq “ X)

unique up to equivalence

ù An internal language for cartesian closed bicategories.

23 / 25

STLC up to isomorphism

Embedding of STLC-terms to Λˆ,Ñ

ps

• terms:

xk ÞÑ xk πkptq ÞÑ πk t t u xt1, . . . , tny ÞÑ tupp t1 , . . . , tn q apppt, uq ÞÑ eval t t , u u λx.t ÞÑ λx. t

24 / 25

STLC up to isomorphism

Embedding of STLC-terms to Λˆ,Ñ

ps

• terms:

xk ÞÑ xk πkptq ÞÑ πk t t u xt1, . . . , tny ÞÑ tupp t1 , . . . , tn q apppt, uq ÞÑ eval t t , u u λx.t ÞÑ λx. t

pSTLC terms Γ $ t : Bq{βη – pΛˆ,Ñ

ps

• terms Γ $ t : Bq{–Γ

B

t –Γ

B t1

ô Γ $ τ : t

–

ù ñ t1 : B for some invertible τ

24 / 25

Key properties of Λˆ,Ñ

ps

:

25 / 25

Key properties of Λˆ,Ñ

ps

:

• 1. Principled development ñ few rules,
• 2. An internal language for cc-bicategories,
• 3. STLC up-to-isomorphism.

25 / 25

Key properties of Λˆ,Ñ

ps

:

• 1. Principled development ñ few rules,
• 2. An internal language for cc-bicategories,
• 3. STLC up-to-isomorphism.

A type theory for cartesian closed bicategories (LICS’19): https://arxiv.org/abs/1904.06538

25 / 25

1 / 3

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

2 / 3

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

3 / 3