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Algebraic models of dependent type theory Clive Newstead HoTT/UF Workshop 2018 Oxford, UK (in absentia) Saturday 7th July 2018 These slides: https://goo.gl/Ttacdq Natural models Polynomials Semantics End Natural models 1 2 Connection

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SLIDE 1

Algebraic models of dependent type theory

Clive Newstead

HoTT/UF Workshop 2018 Oxford, UK (in absentia)

Saturday 7th July 2018

These slides: https://goo.gl/Ttacdq

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SLIDE 2

Natural models Polynomials Semantics End

1

Natural models

2

Connection with polynomial functors

3

Natural model semantics

4

Concluding remarks

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-3
SLIDE 3

Natural models Polynomials Semantics End

1

Natural models

2

Connection with polynomial functors

3

Natural model semantics

4

Concluding remarks

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-4
SLIDE 4

Natural models Polynomials Semantics End

Representable natural transformations

A map f : Y → X in C := [Cop, Set] is representable if y(B) Y X

y y(g) y(g) f x

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-5
SLIDE 5

Natural models Polynomials Semantics End

Representable natural transformations

A map f : Y → X in C := [Cop, Set] is representable if for all A ∈ C and x ∈ X(A) y(B) Y y(A) X

y y y(g) y(g) f x

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-6
SLIDE 6

Natural models Polynomials Semantics End

Representable natural transformations

A map f : Y → X in C := [Cop, Set] is representable if for all A ∈ C and x ∈ X(A) there exist g : B → A in C and y ∈ Y(B) y(B) Y y(A) X

y y y(g) y(g) f x

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 7

Natural models Polynomials Semantics End

Representable natural transformations

A map f : Y → X in C := [Cop, Set] is representable if for all A ∈ C and x ∈ X(A) there exist g : B → A in C and y ∈ Y(B) such that the following square is a pullback: y(B) Y y(A) X

y y y(g) y(g) f x

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-8
SLIDE 8

Natural models Polynomials Semantics End

Natural models

A natural model is a representable natural transformation

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-9
SLIDE 9

Natural models Polynomials Semantics End

Natural models

A natural model is a representable natural transformation

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-10
SLIDE 10

Natural models Polynomials Semantics End

Natural models

A natural model consists of:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-11
SLIDE 11

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’);

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-12
SLIDE 12

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’); . . . with chosen terminal object ⋄ (the ‘empty context’);

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-13
SLIDE 13

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’); . . . with chosen terminal object ⋄ (the ‘empty context’); Presheaves U and

  • U (of ‘types-in-context’ and ‘terms-in-context’);

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 14

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’); . . . with chosen terminal object ⋄ (the ‘empty context’); Presheaves U and

  • U (of ‘types-in-context’ and ‘terms-in-context’);

A map of presheaves p :

  • U → U (term → its type);

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-15
SLIDE 15

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’); . . . with chosen terminal object ⋄ (the ‘empty context’); Presheaves U and

  • U (of ‘types-in-context’ and ‘terms-in-context’);

A map of presheaves p :

  • U → U (term → its type);

+ data witnessing representability of p: y(Γ • A)

  • U

U

qA qA y(pA) y(pA) p A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-16
SLIDE 16

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’); . . . with chosen terminal object ⋄ (the ‘empty context’); Presheaves U and

  • U (of ‘types-in-context’ and ‘terms-in-context’);

A map of presheaves p :

  • U → U (term → its type);

+ data witnessing representability of p: ∀Γ, A y(Γ • A)

  • U

y(Γ) U

qA qA y(pA) y(pA) p A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-17
SLIDE 17

Natural models Polynomials Semantics End

Natural models

A natural model consists of: A base category C (of ‘contexts’ and ‘substitutions’); . . . with chosen terminal object ⋄ (the ‘empty context’); Presheaves U and

  • U (of ‘types-in-context’ and ‘terms-in-context’);

A map of presheaves p :

  • U → U (term → its type);

+ data witnessing representability of p: ∀Γ, A ∃ (chosen) Γ • A, pA, qA y(Γ • A)

  • U

y(Γ) U

qA qA y(pA) y(pA)

  • p

A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 18

Natural models Polynomials Semantics End

Examples of natural models at work

Type theory: Γ ⊢ a : A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 19

Natural models Polynomials Semantics End

Examples of natural models at work

Type theory: Γ ⊢ a : A Natural model:

  • U

y(Γ) U

p a A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 20

Natural models Polynomials Semantics End

Examples of natural models at work

Type theory: Γ ⊢ A type Γ, x : A ⊢ B(x) type Γ ⊢ T(A, B) type (T -FORM)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 21

Natural models Polynomials Semantics End

Examples of natural models at work

Type theory: Γ ⊢ A type Γ, x : A ⊢ B(x) type Γ ⊢ T(A, B) type (T -FORM) Natural model: T :

  • A:U

U

  • UA → U

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 22

Natural models Polynomials Semantics End

Examples of natural models at work

Type theory: Γ ⊢ A type Γ, x : A ⊢ B(x) type Γ ⊢ T(A, B) type (T -FORM) Natural model: T :

  • A:U

U[A] → U

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-23
SLIDE 23

Natural models Polynomials Semantics End

1

Natural models

2

Connection with polynomial functors

3

Natural model semantics

4

Concluding remarks

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-24
SLIDE 24

Natural models Polynomials Semantics End

Polynomials and polynomial functors

Fix a locally cartesian closed category E.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 25

Natural models Polynomials Semantics End

Polynomials and polynomial functors

Fix a locally cartesian closed category E. f : B → A

  • Pf : E → E

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 26

Natural models Polynomials Semantics End

Polynomials and polynomial functors

Fix a locally cartesian closed category E. f : B → A

  • Pf : E → E

X →

a:A

X Ba

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-27
SLIDE 27

Natural models Polynomials Semantics End

Polynomials and polynomial functors

Fix a locally cartesian closed category E. f : B → A

  • Pf : E → E

X →

a:A

X Ba Call Pf a polynomial endofunctor and f a polynomial.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 28

Natural models Polynomials Semantics End

Polynomials and polynomial functors

Fix a locally cartesian closed category E. f : B → A

  • Pf : E → E

X →

a:A

X Ba Call Pf a polynomial endofunctor and f a polynomial. Officially, Pf is the composite E E/B E/A E

∆B→1 Πf ΣA→1

where ∆f is pullback along f and Σf ⊣ ∆f ⊣ Πf.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 29

Natural models Polynomials Semantics End

Cartesian morphisms of polynomials

  • ϕ : Pf ⇒ Pg

cartesian natural transformation

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 30

Natural models Polynomials Semantics End

Cartesian morphisms of polynomials

B D A C

ϕ1 f g ϕ0

  • ϕ : Pf ⇒ Pg

cartesian natural transformation

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 31

Natural models Polynomials Semantics End

Cartesian morphisms of polynomials

B D A C

ϕ1 f g ϕ0

  • ϕ : Pf ⇒ Pg

cartesian natural transformation

Theorem (Gambino & Kock)

Polynomials and cartesian morphisms are the 1- and 2-cells of a bicategory;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-32
SLIDE 32

Natural models Polynomials Semantics End

Cartesian morphisms of polynomials

B D A C

ϕ1 f g ϕ0

  • ϕ : Pf ⇒ Pg

cartesian natural transformation

Theorem (Gambino & Kock)

Polynomials and cartesian morphisms are the 1- and 2-cells of a bicategory; Polynomial functors and cartesian natural transformations are the 1- and 2-cells of a 2-category;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-33
SLIDE 33

Natural models Polynomials Semantics End

Cartesian morphisms of polynomials

B D A C

ϕ1 f g ϕ0

  • ϕ : Pf ⇒ Pg

cartesian natural transformation

Theorem (Gambino & Kock)

Polynomials and cartesian morphisms are the 1- and 2-cells of a bicategory; Polynomial functors and cartesian natural transformations are the 1- and 2-cells of a 2-category; These are biequivalent.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 34

Natural models Polynomials Semantics End

Admitting a unit type

Theorem (Awodey)

A natural model admits a unit type ⇔ there exist 1, ⋆ as in: y(⋄)

  • U

y(⋄) U

  • ⋆
  • ⋆
  • p
  • 1
  • 1

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-35
SLIDE 35

Natural models Polynomials Semantics End

Admitting a unit type

Theorem (Awodey)

A natural model admits a unit type ⇔ there exist 1, ⋆ as in: y(⋄)

  • U

y(⋄) U

  • ⋆
  • ⋆
  • p
  • 1
  • 1

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-36
SLIDE 36

Natural models Polynomials Semantics End

Admitting a unit type

Theorem (Awodey)

A natural model admits a unit type ⇔ there exist 1, ⋆ as in: y(⋄)

  • U

y(⋄) U

  • ⋆
  • ⋆
  • p
  • 1
  • 1

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-37
SLIDE 37

Natural models Polynomials Semantics End

Admitting a unit type

Theorem (Awodey)

A natural model admits a unit type ⇔ there exist 1, ⋆ as in: y(⋄)

  • U

y(⋄) U

  • ⋆
  • ⋆
  • p
  • 1
  • 1

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-38
SLIDE 38

Natural models Polynomials Semantics End

Admitting a unit type

Theorem (Awodey)

A natural model admits a unit type ⇔ there exist 1, ⋆ as in: y(⋄)

  • U

y(⋄) U

  • ⋆
  • ⋆
  • p
  • 1
  • 1

Corollary

interpretations

  • f unit types
  • cartesian morphisms of

polynomials 1 ⇒ p

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-39
SLIDE 39

Natural models Polynomials Semantics End

Admitting dependent sum types

Theorem (Awodey)

A natural model admits a Σ-types ⇔ there exist Σ, pair as in:

  • A:U
  • B:U[A]
  • a:[A]

[B(a)]

  • U
  • A:U U[A]

U

  • pair
  • pair

π

  • p
  • Σ
  • Σ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-40
SLIDE 40

Natural models Polynomials Semantics End

Admitting dependent sum types

Theorem (Awodey)

A natural model admits a Σ-types ⇔ there exist Σ, pair as in:

  • A:U
  • B:U[A]
  • a:[A]

[B(a)]

  • U
  • A:U

U[A] U

  • pair
  • pair

π

  • p
  • Σ
  • Σ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-41
SLIDE 41

Natural models Polynomials Semantics End

Admitting dependent sum types

Theorem (Awodey)

A natural model admits a Σ-types ⇔ there exist Σ, pair as in:

  • A:U
  • B:U[A]
  • a:[A]

[B(a)]

  • U
  • A:U

U[A] U

  • pair
  • pair

π

  • p
  • Σ
  • Σ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-42
SLIDE 42

Natural models Polynomials Semantics End

Admitting dependent sum types

Theorem (Awodey)

A natural model admits a Σ-types ⇔ there exist Σ, pair as in:

  • A:U
  • B:U[A]
  • a:[A]

[B(a)]

  • U
  • A:U

U[A] U

  • pair
  • pair

π

  • p
  • Σ
  • Σ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-43
SLIDE 43

Natural models Polynomials Semantics End

Admitting dependent sum types

Theorem (Awodey)

A natural model admits a Σ-types ⇔ there exist Σ, pair as in:

  • A:U
  • B:U[A]
  • a:[A]

[B(a)]

  • U
  • A:U

U[A] U

  • pair
  • pair

π

  • p
  • Σ
  • Σ

Note also that Pπ = Pp ◦ Pp.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-44
SLIDE 44

Natural models Polynomials Semantics End

Admitting dependent sum types

Theorem (Awodey)

A natural model admits a Σ-types ⇔ there exist Σ, pair as in:

  • A:U
  • B:U[A]
  • a:[A]

[B(a)]

  • U
  • A:U

U[A] U

  • pair
  • pair

π

  • p
  • Σ
  • Σ

Note also that Pπ = Pp ◦ Pp.

Corollary

interpretations

  • f Σ-types
  • cartesian morphisms of

polynomials p · p ⇒ p

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-45
SLIDE 45

Natural models Polynomials Semantics End

Admitting dependent product types

Theorem (Awodey)

A natural model admits a Π-types ⇔ there exist Π, λ as in:

  • A:U
  • U

[A]

  • U
  • A:U U[A]

U

  • λ
  • λ
  • A:U

p[A]

  • A:U

p[A]

  • p
  • Π
  • Π

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-46
SLIDE 46

Natural models Polynomials Semantics End

Admitting dependent product types

Theorem (Awodey)

A natural model admits a Π-types ⇔ there exist Π, λ as in:

  • A:U
  • U

[A]

  • U
  • A:U

U[A] U

  • λ
  • λ
  • A:U

p[A]

  • A:U

p[A]

  • p
  • Π
  • Π

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-47
SLIDE 47

Natural models Polynomials Semantics End

Admitting dependent product types

Theorem (Awodey)

A natural model admits a Π-types ⇔ there exist Π, λ as in:

  • A:U
  • U

[A]

  • U
  • A:U

U[A] U

  • λ
  • λ
  • A:U

p[A]

  • A:U

p[A]

  • p
  • Π
  • Π

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-48
SLIDE 48

Natural models Polynomials Semantics End

Admitting dependent product types

Theorem (Awodey)

A natural model admits a Π-types ⇔ there exist Π, λ as in:

  • A:U
  • U

[A]

  • U
  • A:U

U[A] U

  • λ
  • λ
  • A:U

p[A]

  • A:U

p[A]

  • p
  • Π
  • Π

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-49
SLIDE 49

Natural models Polynomials Semantics End

Admitting dependent product types

Theorem (Awodey)

A natural model admits a Π-types ⇔ there exist Π, λ as in:

  • A:U
  • U

[A]

  • U
  • A:U

U[A] U

  • λ
  • λ
  • A:U

p[A]

  • A:U

p[A]

  • p
  • Π
  • Π

Corollary

interpretations

  • f Π-types
  • cartesian morphisms of

polynomials Pp(p) ⇒ p

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-50
SLIDE 50

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-51
SLIDE 51

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-52
SLIDE 52

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-53
SLIDE 53

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P(p) ⇒ p

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-54
SLIDE 54

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P(p) ⇒ p This is a monad and an algebra.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-55
SLIDE 55

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P(p) ⇒ p This is almost a monad and an algebra.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-56
SLIDE 56

Natural models Polynomials Semantics End

Monad and algebra?

In summary: n.m. admits. . . ⇔ ∃ cartesian . . . 1 1 ⇒ p Σ p · p ⇒ p Π P(p) ⇒ p This is almost a monad and an algebra.

  • Goal. Find the appropriate notion of 3-cell (morphism of morphisms
  • f polynomials) allowing us to make this more precise.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-57
SLIDE 57

Natural models Polynomials Semantics End

Full internal subcategories

Given any morphism f : B → A in a locally cartesian closed category E, we can form the full internal subcategory S(f) ∈ Cat(E).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-58
SLIDE 58

Natural models Polynomials Semantics End

Full internal subcategories

Given any morphism f : B → A in a locally cartesian closed category E, we can form the full internal subcategory S(f) ∈ Cat(E). Object of objects = A;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-59
SLIDE 59

Natural models Polynomials Semantics End

Full internal subcategories

Given any morphism f : B → A in a locally cartesian closed category E, we can form the full internal subcategory S(f) ∈ Cat(E). Object of objects = A; Object of morphisms =

  • a,a′∈A

BBa

a′

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 60

Natural models Polynomials Semantics End

Full internal subcategories

Given any morphism f : B → A in a locally cartesian closed category E, we can form the full internal subcategory S(f) ∈ Cat(E). Object of objects = A; Object of morphisms =

  • a,a′∈A

BBa

a′ = (π∗ 2f)π∗

1 f over A × A. Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-61
SLIDE 61

Natural models Polynomials Semantics End

Full internal subcategories

Given any morphism f : B → A in a locally cartesian closed category E, we can form the full internal subcategory S(f) ∈ Cat(E). Object of objects = A; Object of morphisms =

  • a,a′∈A

BBa

a′ = (π∗ 2f)π∗

1 f over A × A.

Cartesian morphisms of polynomials ϕ : f ⇒ g induce full and faithful functors S(ϕ) : S(f) → S(g).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-62
SLIDE 62

Natural models Polynomials Semantics End

Full internal subcategories

Given any morphism f : B → A in a locally cartesian closed category E, we can form the full internal subcategory S(f) ∈ Cat(E). Object of objects = A; Object of morphisms =

  • a,a′∈A

BBa

a′ = (π∗ 2f)π∗

1 f over A × A.

Cartesian morphisms of polynomials ϕ : f ⇒ g induce full and faithful functors S(ϕ) : S(f) → S(g). Idea: Given cartesian morphisms ϕ, ψ : f ⇒ g, take internal natural transformations S(ϕ) ⇒ S(ψ) to be our 3-cells.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-63
SLIDE 63

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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SLIDE 64

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-65
SLIDE 65

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p-pseudoalgebra

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-66
SLIDE 66

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p-pseudoalgebra Aside: For a natural model p :

  • U → U, let U = S(p) ∈ Cat(

C).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-67
SLIDE 67

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p-pseudoalgebra Aside: For a natural model p :

  • U → U, let U = S(p) ∈ Cat(

C). Object of objects = U.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-68
SLIDE 68

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p-pseudoalgebra Aside: For a natural model p :

  • U → U, let U = S(p) ∈ Cat(

C). Object of objects = U. Object of morphisms =

A,B:U[B][A].

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-69
SLIDE 69

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p-pseudoalgebra Aside: For a natural model p :

  • U → U, let U = S(p) ∈ Cat(

C). Object of objects = U. Object of morphisms =

A,B:U[B][A].

Considered as an indexed category Cop → Cat, U is equivalent to the ‘context-indexed category of types’ of Clairambault & Dybjer (2011).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-70
SLIDE 70

Natural models Polynomials Semantics End

Full internal subcategories

Theorem

With respect to this notion of 3-cell: p admits 1, Σ ⇐ ⇒ p is a pseudomonad p also admits Π ⇐ ⇒ p is a p-pseudoalgebra Aside: For a natural model p :

  • U → U, let U = S(p) ∈ Cat(

C). Object of objects = U. Object of morphisms =

A,B:U[B][A].

Considered as an indexed category Cop → Cat, U is equivalent to the ‘context-indexed category of types’ of Clairambault & Dybjer (2011). Bonus: If (C, p) admits 1, Σ, Π, then U is cartesian closed.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-71
SLIDE 71

Natural models Polynomials Semantics End

1

Natural models

2

Connection with polynomial functors

3

Natural model semantics

4

Concluding remarks

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-72
SLIDE 72

Natural models Polynomials Semantics End

Initiality of the syntax

Idea (Initiality ‘conjecture’)

The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-73
SLIDE 73

Natural models Polynomials Semantics End

Initiality of the syntax

Idea (Initiality ‘conjecture’)

The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T. Goals:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-74
SLIDE 74

Natural models Polynomials Semantics End

Initiality of the syntax

Idea (Initiality ‘conjecture’)

The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T. Goals: Build the syntactic natural models for some basic type theories and prove that they satisfy an appropriate universal property;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-75
SLIDE 75

Natural models Polynomials Semantics End

Initiality of the syntax

Idea (Initiality ‘conjecture’)

The syntax of a dependent type theory T should itself have the structure of a natural model, which is initial amongst all natural models interpreting T. Goals: Build the syntactic natural models for some basic type theories and prove that they satisfy an appropriate universal property; Expand to more complicated type theories by (algebraically) freely adding type theoretic structure.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-76
SLIDE 76

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-77
SLIDE 77

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-78
SLIDE 78

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-79
SLIDE 79

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op Presheaf of types: UI = cod : Fin/I → Set

(A

u

− → I) → I

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-80
SLIDE 80

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op Presheaf of types: UI = cod : Fin/I → Set

(A

u

− → I) → I

Presheaf of terms:

  • UI = dom : Fin/I → Set

(A

u

− → I) → A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-81
SLIDE 81

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op Presheaf of types: UI = cod : Fin/I → Set

(A

u

− → I) → I

Presheaf of terms:

  • UI = dom : Fin/I → Set

(A

u

− → I) → A

Typing map: (pI)A

u

− →I = u : A → I

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-82
SLIDE 82

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op Presheaf of types: UI = cod : Fin/I → Set

(A

u

− → I) → I

Presheaf of terms:

  • UI = dom : Fin/I → Set

(A

u

− → I) → A

Typing map: (pI)A

u

− →I = u : A → I Representability data: given A

u

− → I and i ∈ UI(u) = I, let (A

u

− → I) • i = (A + 1

[u,i]

− − → I)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-83
SLIDE 83

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op Presheaf of types: UI = cod : Fin/I → Set

(A

u

− → I) → I

Presheaf of terms:

  • UI = dom : Fin/I → Set

(A

u

− → I) → A

Typing map: (pI)A

u

− →I = u : A → I Representability data: given A

u

− → I and i ∈ UI(u) = I, let (A

u

− → I) • i = (A + 1

[u,i]

− − → I) pi : A ֒ → A + 1

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-84
SLIDE 84

Natural models Polynomials Semantics End

Example #1: set of basic types

We’ll construct the free natural model on the theory with an I-indexed family of basic types.

Definition

Define (CI, pI :

  • UI → UI) as follows:

Category of contexts: CI = (Fin/I)op Presheaf of types: UI = cod : Fin/I → Set

(A

u

− → I) → I

Presheaf of terms:

  • UI = dom : Fin/I → Set

(A

u

− → I) → A

Typing map: (pI)A

u

− →I = u : A → I Representability data: given A

u

− → I and i ∈ UI(u) = I, let (A

u

− → I) • i = (A + 1

[u,i]

− − → I) pi : A ֒ → A + 1 qi = ⋆

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-85
SLIDE 85

Natural models Polynomials Semantics End

Example #1: set of basic types

Theorem

(CI, pI) is a natural model

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-86
SLIDE 86

Natural models Polynomials Semantics End

Example #1: set of basic types

Theorem

(CI, pI) is a natural model, and for all natural models (C, p :

  • U → U)

and all I-indexed families {Oi}i∈I ⊆ U(⋄),

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-87
SLIDE 87

Natural models Polynomials Semantics End

Example #1: set of basic types

Theorem

(CI, pI) is a natural model, and for all natural models (C, p :

  • U → U)

and all I-indexed families {Oi}i∈I ⊆ U(⋄), there is a unique F : (CI, pI) → (C, p) with F(i) = Oi for all i ∈ I.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-88
SLIDE 88

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Goal: Given a natural model (C, p), construct the ‘smallest’ natural model (CΣ, pΣ) which extends (C, p) and admits Σ-types.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-89
SLIDE 89

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-90
SLIDE 90

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

  • x,y,z:
  • x,y:

x:A B(x)

C(x,y)

 

  • w:D(x,y,z)

E(x, y, z, w)  

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-91
SLIDE 91

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

  • x,y,z:
  • x,y:

x:A B(x)

C(x,y)

 

  • w:D(x,y,z)

E(x, y, z, w)  

  • C

D E A B

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-92
SLIDE 92

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

  • x,y,z:
  • x,y:

x:A B(x)

C(x,y)

 

  • w:D(x,y,z)

E(x, y, z, w)  

  • C

D E A B

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-93
SLIDE 93

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

  • x,y,z:
  • x,y:

x:A B(x)

C(x,y)

 

  • w:D(x,y,z)

E(x, y, z, w)  

  • C

D E A B

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-94
SLIDE 94

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

  • x,y,z:
  • x,y:

x:A B(x)

C(x,y)

 

  • w:D(x,y,z)

E(x, y, z, w)  

  • C

D E A B

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-95
SLIDE 95

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can represent (iterated) Σ-types by binary trees.

  • x,y,z:
  • x,y:

x:A B(x)

C(x,y)

 

  • w:D(x,y,z)

E(x, y, z, w)  

  • C

D E A B

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-96
SLIDE 96

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Definition

Define (CΣ, pΣ :

  • UΣ → UΣ) as follows:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-97
SLIDE 97

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Definition

Define (CΣ, pΣ :

  • UΣ → UΣ) as follows:

CΣ: Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-98
SLIDE 98

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Definition

Define (CΣ, pΣ :

  • UΣ → UΣ) as follows:

CΣ: Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; UΣ is the presheaf of type trees;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-99
SLIDE 99

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Definition

Define (CΣ, pΣ :

  • UΣ → UΣ) as follows:

CΣ: Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; UΣ is the presheaf of type trees;

  • UΣ is the presheaf of term trees;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-100
SLIDE 100

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Definition

Define (CΣ, pΣ :

  • UΣ → UΣ) as follows:

CΣ: Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; UΣ is the presheaf of type trees;

  • UΣ is the presheaf of term trees;

pΣ : (tree of terms) → (tree of their types).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-101
SLIDE 101

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Definition

Define (CΣ, pΣ :

  • UΣ → UΣ) as follows:

CΣ: Objects (contexts) are the objects of C ‘formally extended’ by trees of dependent types; UΣ is the presheaf of type trees;

  • UΣ is the presheaf of term trees;

pΣ : (tree of terms) → (tree of their types). There is a morphism I : (C, p) → (CΣ, pΣ), which sends types and terms to trivial trees (one vertex, no edges).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-102
SLIDE 102

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Theorem

(CΣ, pΣ) is a natural model admitting Σ-types

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-103
SLIDE 103

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Theorem

(CΣ, pΣ) is a natural model admitting Σ-types, and for all F : (C, p) → (D, q) with (D, q) admitting Σ-types, (C, p) (D, q) (CΣ, pΣ)

F I F ♯ F ♯

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-104
SLIDE 104

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

Theorem

(CΣ, pΣ) is a natural model admitting Σ-types, and for all F : (C, p) → (D, q) with (D, q) admitting Σ-types, there is a unique Σ-type-preserving F ♯ : (CΣ, pΣ) → (D, q) extending F along I. (C, p) (D, q) (CΣ, pΣ)

F I F ♯ F ♯

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-105
SLIDE 105

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can characterise freely admitting Σ-types functorially.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-106
SLIDE 106

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can characterise freely admitting Σ-types functorially. Inspiration: Given a set S, the set of finite rooted binary trees with leaves labelled by elements of S is an initial algebra for the polynomial functor X → S + X × X.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-107
SLIDE 107

Natural models Polynomials Semantics End

Example #2: freely admitting Σ-types

We can characterise freely admitting Σ-types functorially. Inspiration: Given a set S, the set of finite rooted binary trees with leaves labelled by elements of S is an initial algebra for the polynomial functor X → S + X × X.

Theorem

pΣ is an initial algebra for the endofunctor f → p + f · f.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-108
SLIDE 108

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-109
SLIDE 109

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-110
SLIDE 110

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-111
SLIDE 111

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-112
SLIDE 112

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-113
SLIDE 113

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-114
SLIDE 114

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-115
SLIDE 115

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-116
SLIDE 116

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Let (C, p) be a natural model and O ∈ U(⋄). Idea: Freely adjoin new term x : O by slicing by O (= ⋄ • O). C

  • C

p Cx:O C/O

  • Cx:O

px:O

∆O I I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I!⊣I∗⊣I∗ I! I! ≃

Note: The objects of Cx:O look like Γ • O • A1 • . . . • An Γ • O O Γ ⋄

projections pO !•O

  • pO

!

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-117
SLIDE 117

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Theorem

(Cx:O, px:O) is a natural model

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-118
SLIDE 118

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Theorem

(Cx:O, px:O) is a natural model, and for all F : (C, p) → (D, q) and all a : FO in D, (C, p) (D, q) ∋ a : FO (Cx:O, px:O) ∋ x : O

F I F ♯ F ♯

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-119
SLIDE 119

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Theorem

(Cx:O, px:O) is a natural model, and for all F : (C, p) → (D, q) and all a : FO in D, there is a unique F ♯ : (Cx:O, px:O) → (D, q) extending F, such that F ♯(x) = a. (C, p) (D, q) ∋ a : FO (Cx:O, px:O) ∋ x : O

F I F ♯ F ♯

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-120
SLIDE 120

Natural models Polynomials Semantics End

Example #3: freely adjoining a term

Theorem

(Cx:O, px:O) is a natural model, and for all F : (C, p) → (D, q) and all a : FO in D, there is a unique F ♯ : (Cx:O, px:O) → (D, q) extending F, such that F ♯(x) = a. (C, p) (D, q) ∋ a : FO (Cx:O, px:O) ∋ x : O

F I F ♯ F ♯

Note: x is given by the diagonal map O → O × O (= ⋄ • O • O[pO]) in C/O.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-121
SLIDE 121

Natural models Polynomials Semantics End

1

Natural models

2

Connection with polynomial functors

3

Natural model semantics

4

Concluding remarks

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-122
SLIDE 122

Natural models Polynomials Semantics End

Future directions

Some areas of interest for the future:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-123
SLIDE 123

Natural models Polynomials Semantics End

Future directions

Some areas of interest for the future: Develop a formal theory of natural models in an arbitrary (suitably structured) category E, not just a presheaf topos;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-124
SLIDE 124

Natural models Polynomials Semantics End

Future directions

Some areas of interest for the future: Develop a formal theory of natural models in an arbitrary (suitably structured) category E, not just a presheaf topos; Further investigate properties of the full internal subcategory associated with a natural model;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-125
SLIDE 125

Natural models Polynomials Semantics End

Future directions

Some areas of interest for the future: Develop a formal theory of natural models in an arbitrary (suitably structured) category E, not just a presheaf topos; Further investigate properties of the full internal subcategory associated with a natural model; Translate connections between polynomial monads and operads to this setting;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-126
SLIDE 126

Natural models Polynomials Semantics End

Future directions

Some areas of interest for the future: Develop a formal theory of natural models in an arbitrary (suitably structured) category E, not just a presheaf topos; Further investigate properties of the full internal subcategory associated with a natural model; Translate connections between polynomial monads and operads to this setting; Formalise natural models in HoTT.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

slide-127
SLIDE 127

Natural models Polynomials Semantics End

Thanks for listening!

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq

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Natural models Polynomials Semantics End

References

Natural models Awodey (2016) Natural models of dependent type theory, arXiv:1406.3219 Awodey & Newstead (2018) Polynomial pseudomonads and dependent type theory, arXiv:1802.00997 Fiore (2012) Discrete generalised polynomial functors, slides from talk at ICALP 2012 Polynomials Gambino & Kock (2009) Polynomial functors and polynomial monads, arXiv:0906.4931 Related work with CwFs Clairambault & Dybjer (2011) The biequivalence of locally cartesian closed categories and Martin-L¨

  • f type theories,

arXiv:1112.3456

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Algebraic models of dependent type theory These slides: https://goo.gl/Ttacdq