Categories of natural models of type theory CT 2016 (Halifax, NS, - - PowerPoint PPT Presentation

Categories of natural models of type theory

CT 2016 (Halifax, NS, Canada) Clive Newstead

Carnegie Mellon University

Friday 12th August 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

1

Dependent type theory

2

Natural models of type theory

3

Algebraic description of homomorphisms

4

Functorial description of homomorphisms

5

Interpreting the syntax

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

1

Dependent type theory

2

Natural models of type theory

3

Algebraic description of homomorphisms

4

Functorial description of homomorphisms

5

Interpreting the syntax

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Simple type theory

Simple type theory consists of. . . Types: A, B, . . . , A × B, A → B, . . . , 0, 1, N, Z, . . .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Simple type theory

Simple type theory consists of. . . Types: A, B, . . . , A × B, A → B, . . . , 0, 1, N, Z, . . . Terms: a : A, b : B, . . . , a, b : A × B, f : A → B, . . .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Simple type theory

Simple type theory consists of. . . Types: A, B, . . . , A × B, A → B, . . . , 0, 1, N, Z, . . . Terms: a : A, b : B, . . . , a, b : A × B, f : A → B, . . . Equations: a = b : A, pr1(a, b) = a : A, . . .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Simple type theory

Simple type theory consists of. . . Types: A, B, . . . , A × B, A → B, . . . , 0, 1, N, Z, . . . Terms: a : A, b : B, . . . , a, b : A × B, f : A → B, . . . Equations: a = b : A, pr1(a, b) = a : A, . . . Rules of inference: For example, A B A × B a : A b : B a, b : A × B c : A × B pr1(c) : A · · ·

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Dependent type theory

In dependent type theory, we also have Contexts: Γ, ∆, . . . (lists of typed variables)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Dependent type theory

In dependent type theory, we also have Contexts: Γ, ∆, . . . (lists of typed variables) Dependent types: x : A ⊢ B(x), x : A, y : B(x) ⊢ C(x, y), . . .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Dependent type theory

In dependent type theory, we also have Contexts: Γ, ∆, . . . (lists of typed variables) Dependent types: x : A ⊢ B(x), x : A, y : B(x) ⊢ C(x, y), . . . e.g. n : N ⊢ List(n) “In the context n : N, List(n) is a type.”

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Dependent type theory

In dependent type theory, we also have Contexts: Γ, ∆, . . . (lists of typed variables) Dependent types: x : A ⊢ B(x), x : A, y : B(x) ⊢ C(x, y), . . . e.g. n : N ⊢ List(n) “In the context n : N, List(n) is a type.” Type-formers: If Γ, x : A ⊢ B(x), then Γ ⊢

x:A B(x).

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Dependent type theory

In dependent type theory, we also have Contexts: Γ, ∆, . . . (lists of typed variables) Dependent types: x : A ⊢ B(x), x : A, y : B(x) ⊢ C(x, y), . . . e.g. n : N ⊢ List(n) “In the context n : N, List(n) is a type.” Type-formers: If Γ, x : A ⊢ B(x), then Γ ⊢

x:A B(x).

Substitutions: ∆ ⊢ γ : Γ if Γ ⊢ a : A then ∆ ⊢ a{γ} : A{γ}

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

There are many kinds of semantics for dependent type theory, including:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

There are many kinds of semantics for dependent type theory, including: Locally cartesian closed categories (Seely) Contextual categories, categories with attributes (Cartmell) Categories with families (Dybjer) Comprehension categories (Jacobs) Display map categories (Cambridge school) Weak factorisation systems (Awodey and Warren) Algebraic weak factorisation systems (Garner; Grandis & Tholen) Homotopical categories (Garner and van den Berg) B-systems, C-systems, and universes (Voevodsky) Tribes (Joyal)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?!

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints.

They elucidate hidden structure:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints.

They elucidate hidden structure:

Connection with polynomial functors;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints.

They elucidate hidden structure:

Connection with polynomial functors; 1 + Σ + Π polynomial monad;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Yet another approach to semantics?

Why add yet another proposal?! Natural models are slick!

Straightforward interpretation of syntax; Avoids reference to (e.g.) fibrations; Also avoids heavy structure seen in (e.g.) categories with families; Flexibility—can study natural models from several viewpoints.

They elucidate hidden structure:

Connection with polynomial functors; 1 + Σ + Π polynomial monad; Natural structure of a double category.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

1

Dependent type theory

2

Natural models of type theory

3

Algebraic description of homomorphisms

4

Functorial description of homomorphisms

5

Interpreting the syntax

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Representable natural transformations

Let C be a category and let U, U : Cop → Set. A natural transformation p : U → U is representable if, for all Γ ∈ ob(C) and all A ∈ U(Γ), y(Γ · A)

• U

y(Γ) U

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

p A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Representable natural transformations

Let C be a category and let U, U : Cop → Set. A natural transformation p : U → U is representable if, for all Γ ∈ ob(C) and all A ∈ U(Γ), there exist Γ · A, pΓ

A, qΓ A making the following diagram a

pullback: y(Γ · A)

• U

y(Γ) U

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

p A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Informal semantics

Type theory Representable natural transformation

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Informal semantics

Type theory Representable natural transformation Γ context Γ ∈ ob(C)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Informal semantics

Type theory Representable natural transformation Γ context Γ ∈ ob(C) Γ ⊢ A A ∈ U(Γ)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Informal semantics

Type theory Representable natural transformation Γ context Γ ∈ ob(C) Γ ⊢ A A ∈ U(Γ) Γ ⊢ a : A

• U

y(Γ) U

p a A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Informal semantics

Type theory Representable natural transformation Γ context Γ ∈ ob(C) Γ ⊢ A A ∈ U(Γ) Γ ⊢ a : A

• U

y(Γ) U

p a A

∆ ⊢ γ : Γ γ : ∆ → Γ in C

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Informal semantics

Type theory Representable natural transformation Γ context Γ ∈ ob(C) Γ ⊢ A A ∈ U(Γ) Γ ⊢ a : A

• U

y(Γ) U

p a A

∆ ⊢ γ : Γ γ : ∆ → Γ in C ∆ ⊢ a{γ} : A{γ}

• U

y(∆) y(Γ) U

p y(γ) a{γ} A{γ} a A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Context extension ↔ representability

Type theory: Given the following: Γ ⊢ A, ∆ ⊢ γ : Γ, ∆ ⊢ a : A{γ} There is a unique substitution ∆ ⊢ γ, a : Γ · A, such that ∆ ⊢ pΓ

A ◦ γ, a = γ : Γ

and Γ ⊢ qΓ

A{γ, a} = a : A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Context extension ↔ representability

Type theory: Given the following: Γ ⊢ A, ∆ ⊢ γ : Γ, ∆ ⊢ a : A{γ} There is a unique substitution ∆ ⊢ γ, a : Γ · A, such that ∆ ⊢ pΓ

A ◦ γ, a = γ : Γ

and Γ ⊢ qΓ

A{γ, a} = a : A

Representable natural transformation: y(∆) y(Γ · A)

• U

y(Γ) U

a y(γ) y(γ,a)

• qΓ

A

y(pΓ

A)

p A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Connection with categories with families

Theorem (Awodey, 2015)

Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Connection with categories with families

Theorem (Awodey, 2015)

Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C. A natural model is a representable natural transformation.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Connection with categories with families

Theorem (Awodey, 2015)

Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C. A natural model is a representable natural transformation.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Connection with categories with families

Theorem (Awodey, 2015)

Specifying a category with families with base category C is equivalent to specifying a representable natural transformation between presheaves on C. A natural model is a representable natural transformation. We seek an essentially algebraic definition.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Representability via categories of elements

Lemma

A natural transformation p : U → U is representable if and only if the induced functor on categories of elements

• C p :
• C

U →

• C U has a

right adjoint p∗.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Representability via categories of elements

Lemma

A natural transformation p : U → U is representable if and only if the induced functor on categories of elements

• C p :
• C

U →

• C U has a

right adjoint p∗. p∗(Γ, A) = (Γ · A, qΓ

A)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Representability via categories of elements

Lemma

A natural transformation p : U → U is representable if and only if the induced functor on categories of elements

• C p :
• C

U →

• C U has a

right adjoint p∗. p∗(Γ, A) = (Γ · A, qΓ

A)

ε(Γ,A) = pΓ

A : Γ · A → A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Representability via categories of elements

Lemma

A natural transformation p : U → U is representable if and only if the induced functor on categories of elements

• C p :
• C

U →

• C U has a

right adjoint p∗. p∗(Γ, A) = (Γ · A, qΓ

A)

ε(Γ,A) = pΓ

A : Γ · A → A

γ : ∆ → Γ p∗(γ) as follows: ∆ · A{γ} Γ · A ∆ Γ

• p∗(γ)

p∆

A{γ}

pΓ

A

γ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Definition of a natural model

Definition

A natural model is an octuple C = (C, ⋄, U, U, p, p∗, η, ε) consisting of the following data:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Definition of a natural model

Definition

A natural model is an octuple C = (C, ⋄, U, U, p, p∗, η, ε) consisting of the following data: A base category C with a terminal object ⋄;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Definition of a natural model

Definition

A natural model is an octuple C = (C, ⋄, U, U, p, p∗, η, ε) consisting of the following data: A base category C with a terminal object ⋄; Presheaves U, U : Cop → Set;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Definition of a natural model

Definition

A natural model is an octuple C = (C, ⋄, U, U, p, p∗, η, ε) consisting of the following data: A base category C with a terminal object ⋄; Presheaves U, U : Cop → Set; Functors

• C

U

• C U

p p∗

such that p commutes with the projection maps to C;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Definition of a natural model

Definition

A natural model is an octuple C = (C, ⋄, U, U, p, p∗, η, ε) consisting of the following data: A base category C with a terminal object ⋄; Presheaves U, U : Cop → Set; Functors

• C

U

• C U

p p∗

such that p commutes with the projection maps to C; Natural transformations η : id → p∗ ◦ p and ε : p ◦ p∗ → id forming the unit and counit, respectively, of an adjunction p ⊣ p∗.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Outline

We’ll follow the standard pattern for functorial semantics:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Outline

We’ll follow the standard pattern for functorial semantics: Define the notion of homomorphism of natural models;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Outline

We’ll follow the standard pattern for functorial semantics: Define the notion of homomorphism of natural models; Show that the syntax for type theory on a given signature Σ presents the free natural model T on Σ;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Outline

We’ll follow the standard pattern for functorial semantics: Define the notion of homomorphism of natural models; Show that the syntax for type theory on a given signature Σ presents the free natural model T on Σ; An interpretation of Σ in a natural model C is given by a homomorphism T → C.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

1

Dependent type theory

2

Natural models of type theory

3

Algebraic description of homomorphisms

4

Functorial description of homomorphisms

5

Interpreting the syntax

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

Definition

Let C = (C, ⋄, U, U, p, p∗, η, ε) and D = (D, •, V, V, q, q∗, σ, τ) be natural models. A homomorphism from C to D is a triple (F, Φ, Φ) consisting of:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

Definition

Let C = (C, ⋄, U, U, p, p∗, η, ε) and D = (D, •, V, V, q, q∗, σ, τ) be natural models. A homomorphism from C to D is a triple (F, Φ, Φ) consisting of: A functor F : C → D;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

Definition

Let C = (C, ⋄, U, U, p, p∗, η, ε) and D = (D, •, V, V, q, q∗, σ, τ) be natural models. A homomorphism from C to D is a triple (F, Φ, Φ) consisting of: A functor F : C → D; Functors Φ :

• C

U →

• D

V and

• Φ :
• C
• U →
• D
• V

such that. . .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . the following diagrams commute (highlighted in red):

• C

U

• D

V

• C U
• D V

C D

• Φ
• Φ

p q Φ p∗ p∗ q∗ q∗ F

Action on types respects context and substitution

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . the following diagrams commute (highlighted in red):

• C

U

• D

V

• C U
• D V

C D

• Φ
• Φ

p q Φ p∗ p∗ q∗ q∗ F

Action on terms respects context and substitution

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . the following diagrams commute (highlighted in red):

• C

U

• D

V

• C U
• D V

C D

• Φ
• Φ

p q Φ p∗ p∗ q∗ q∗ F

Action on terms respects typing

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . the following diagrams commute (highlighted in red):

• C

U

• D

V

• C U
• D V

C D

• Φ
• Φ

p q Φ p∗ p∗ q∗ q∗ F

Action on contexts and substitutions respects context extension

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . and Φ, Φ respect the adjunctions (p ⊣ p∗, η, ε) and (q ⊣ q∗, σ, τ), i.e.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . and Φ, Φ respect the adjunctions (p ⊣ p∗, η, ε) and (q ⊣ q∗, σ, τ), i.e.

• C U
• C U
• D V
• D V

p◦p∗ id ε Φ Φ q◦q∗ id τ

• Counit. Φ · ε = τ · Φ
• FpΓ

A = pFΓ FA : FΓ · FA → FΓ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . and Φ, Φ respect the adjunctions (p ⊣ p∗, η, ε) and (q ⊣ q∗, σ, τ), i.e.

• C U
• C U
• D V
• D V

p◦p∗ id ε Φ Φ q◦q∗ id τ

• C

U

• C

U

• D

V

• D

V

id q∗◦q η

• Φ
• Φ

id q∗◦q σ

• Counit. Φ · ε = τ · Φ
• FpΓ

A = pFΓ FA : FΓ · FA → FΓ

Unit. Φ · η = σ · Φ

• FidΓ, qΓ

A = idFΓ, qFΓ FA : FΓ → FΓ · FA

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Algebraic description of homomorphisms

. . . and Φ, Φ respect the adjunctions (p ⊣ p∗, η, ε) and (q ⊣ q∗, σ, τ), i.e.

• C U
• C U
• D V
• D V

p◦p∗ id ε Φ Φ q◦q∗ id τ

• C

U

• C

U

• D

V

• D

V

id q∗◦q η

• Φ
• Φ

id q∗◦q σ

• Counit. Φ · ε = τ · Φ
• FpΓ

A = pFΓ FA : FΓ · FA → FΓ

Unit. Φ · η = σ · Φ

• FidΓ, qΓ

A = idFΓ, qFΓ FA : FΓ → FΓ · FA

. . . and F(⋄) = •.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Category of natural models

Theorem

There is a category NM, where: The objects of NM are natural models; The morphisms of NM are homomorphisms; The identity morphism on a natural model C is (idC, id

U, id U);

Composition is given componentwise: (G, Ψ, Ψ) ◦ (F, Φ, Φ) = (G ◦ F, Ψ ◦ Φ, Ψ ◦ Φ)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Category of natural models

Theorem

There is a category NM, where: The objects of NM are natural models; The morphisms of NM are homomorphisms; The identity morphism on a natural model C is (idC, id

U, id U);

Composition is given componentwise: (G, Ψ, Ψ) ◦ (F, Φ, Φ) = (G ◦ F, Ψ ◦ Φ, Ψ ◦ Φ) Since homomorphisms are defined diagramatically, this is extremely simple to prove.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

1

Dependent type theory

2

Natural models of type theory

3

Algebraic description of homomorphisms

4

Functorial description of homomorphisms

5

Interpreting the syntax

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Remark on Kan extension

Any functor F : C → D between small categories induces an adjunction F! ⊣ F ∗ between presheaf categories SetCop SetDop C D

F! F! ∼ = F ∗ F ∗ y F y

where F ∗ = − ◦ F is precomposition with F; and F! is left Kan extension along F.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Remark on Kan extension

Any functor F : C → D between small categories induces an adjunction F! ⊣ F ∗ between presheaf categories SetCop SetDop C D

F! F! ∼ = F ∗ F ∗ y F y

where F ∗ = − ◦ F is precomposition with F; and F! is left Kan extension along F. Moreover, F! ◦ y ∼ = y ◦ F : C → SetDop.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying: A functor F : C → D;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying: A functor F : C → D; Natural transformations ϕ : F!U → V and ϕ : F! U → V

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying: A functor F : C → D; Natural transformations ϕ : F!U → V and ϕ : F! U → V such that F(⋄) = •;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying: A functor F : C → D; Natural transformations ϕ : F!U → V and ϕ : F! U → V such that F(⋄) = •; The diagram F! U

• V

F!U V

• ϕ

F!p q ϕ

commutes;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying: A functor F : C → D; Natural transformations ϕ : F!U → V and ϕ : F! U → V such that F(⋄) = •; The diagram F! U

• V

F!U V

• ϕ

F!p q ϕ

commutes; F(Γ · A) = FΓ · FA for all Γ ∈ ob(C); and

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Functorial presentation of homomorphisms

Specifying a homomorphism (F, Φ, Φ) : C → D is equivalent to specifying: A functor F : C → D; Natural transformations ϕ : F!U → V and ϕ : F! U → V such that F(⋄) = •; The diagram F! U

• V

F!U V

• ϕ

F!p q ϕ

commutes; F(Γ · A) = FΓ · FA for all Γ ∈ ob(C); and The comparison morphisms cΓ

A : F(Γ · A) → FΓ · FA are identities.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via y(Γ) U F!y(Γ)

A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via y(Γ) U F!y(Γ)

A

• F!y(Γ)

F!U V y(Γ)

F!A F!A ϕ ∼ = ∼ = FA FA

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via y(Γ) U F!y(Γ)

A

• F!y(Γ)

F!U V y(Γ)

F!A F!A ϕ ∼ = ∼ = FA FA

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via y(Γ) U F!y(Γ)

A

• F!y(Γ)

F!U V y(FΓ)

F!A F!A ϕ ∼ = ∼ = FA FA

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via y(Γ) U F!y(Γ)

A

• F!y(Γ)

F!U V y(FΓ)

F!A F!A ϕ ∼ = ∼ = FA FA

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Action on types and terms

We obtain an action of ϕ on types and ϕ on terms as follows. Action on types. A ∈ U(Γ) FA ∈ V(FΓ) via y(Γ) U F!y(Γ)

A

• F!y(Γ)

F!U V y(FΓ)

F!A F!A ϕ ∼ = ∼ = FA FA

Action on terms. a ∈ U(Γ) Fa ∈ V(FΓ) via y(Γ)

• U

F!y(Γ)

a

• F!y(Γ)

F! U

• V

y(FΓ)

F!a

• ϕ

Fa ∼ =

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) y(Γ · A)

• U
• V

y(Γ) U V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Set-up: A type in context Γ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) y(Γ · A)

• U
• V

y(Γ) U V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

• qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Context extension of Γ by A

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

F!y(Γ) F!U V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Apply F!

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

y(FΓ) F!y(Γ) F!U V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

F! ◦ y ∼ = y(F−)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

y(FΓ) F!y(Γ) F!U V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Paste square for ϕ, ϕ

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

y(FΓ) V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Action of ϕ on types and ϕ on terms

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

y(FΓ) V

y(pFΓ

FA )

• qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Extend context FΓ by FA

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

y(FΓ) V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

Obtain cΓ

A : F(Γ · A) → FΓ · FA as shown

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Where the comparison morphisms cΓ

A come from y(FΓ · FA) y(F(Γ · A)) F!y(Γ · A) F! U

• V

y(FΓ) V

y(pFΓ

FA )

qFΓ

FA

∼ = FqΓ

A

y(FpΓ

A)

y(cΓ

A)

y(cΓ

A)

qΓ

A

qΓ

A

y(pΓ

A)

y(pΓ

A)

F!qΓ

A

F!y(pΓ

A)

p F!(p)

• ϕ

q ∼ = FA FA A F!A ϕ

cΓ

A = id

⇒ FpΓ

A = pFΓ FA

and FqΓ

A = qFΓ FA .

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

1

Dependent type theory

2

Natural models of type theory

3

Algebraic description of homomorphisms

4

Functorial description of homomorphisms

5

Interpreting the syntax

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Interpreting the syntax

We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Interpreting the syntax

We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Work in a system of type theory with four kinds of judgements Γ = Γ′ ⊢, ∆ ⊢ γ = γ′ : Γ, Γ ⊢ A = A′, Γ ⊢ a = a′ : A (We write Γ ⊢ instead of Γ = Γ ⊢, and so on.)

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Interpreting the syntax

We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Work in a system of type theory with four kinds of judgements Γ = Γ′ ⊢, ∆ ⊢ γ = γ′ : Γ, Γ ⊢ A = A′, Γ ⊢ a = a′ : A (We write Γ ⊢ instead of Γ = Γ ⊢, and so on.) From the syntax, build a natural model T = (T, [], Ty, Tm, ty, ext, sub, proj) called the term model of the system.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Interpreting the syntax

We take a similar approach to that of S. Castellan, P . Clairambault, P . Dybjer (2015). The idea is as follows: Work in a system of type theory with four kinds of judgements Γ = Γ′ ⊢, ∆ ⊢ γ = γ′ : Γ, Γ ⊢ A = A′, Γ ⊢ a = a′ : A (We write Γ ⊢ instead of Γ = Γ ⊢, and so on.) From the syntax, build a natural model T = (T, [], Ty, Tm, ty, ext, sub, proj) called the term model of the system. T will (in a suitable sense) be the free natural model supporting the derivation rules for this system.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 1: basic syntax

With no rules for type formation, the term model is very simple:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 1: basic syntax

With no rules for type formation, the term model is very simple: T has the empty context [] as its only object and the identity substitution [] ⊢ id : [] as its only morphism;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 1: basic syntax

With no rules for type formation, the term model is very simple: T has the empty context [] as its only object and the identity substitution [] ⊢ id : [] as its only morphism; Ty, Tm : Top → Set are the empty presheaves;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 1: basic syntax

With no rules for type formation, the term model is very simple: T has the empty context [] as its only object and the identity substitution [] ⊢ id : [] as its only morphism; Ty, Tm : Top → Set are the empty presheaves; ty, ext are the unique (empty) functor between empty categories;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 1: basic syntax

With no rules for type formation, the term model is very simple: T has the empty context [] as its only object and the identity substitution [] ⊢ id : [] as its only morphism; Ty, Tm : Top → Set are the empty presheaves; ty, ext are the unique (empty) functor between empty categories; sub, proj are the unique natural transformations with no components.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 1: basic syntax

With no rules for type formation, the term model is very simple: T has the empty context [] as its only object and the identity substitution [] ⊢ id : [] as its only morphism; Ty, Tm : Top → Set are the empty presheaves; ty, ext are the unique (empty) functor between empty categories; sub, proj are the unique natural transformations with no components. It is very easy to prove the following result.

Theorem

This data defines a natural model T, which is an initial object in NM.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows: The objects of T are the empty context [0] := [] and finite strings

• f the form [n] := [1 · 1 · · · 1
• n times

] for n ≥ 1;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows: The objects of T are the empty context [0] := [] and finite strings

• f the form [n] := [1 · 1 · · · 1
• n times

] for n ≥ 1; There is a unique morphism γn,m : [n] → [m] for all n, m ∈ N.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows: The objects of T are the empty context [0] := [] and finite strings

• f the form [n] := [1 · 1 · · · 1
• n times

] for n ≥ 1; There is a unique morphism γn,m : [n] → [m] for all n, m ∈ N. Ty([n]) = {[1]} and Ty(γn,m) = id{[1]};

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows: The objects of T are the empty context [0] := [] and finite strings

• f the form [n] := [1 · 1 · · · 1
• n times

] for n ≥ 1; There is a unique morphism γn,m : [n] → [m] for all n, m ∈ N. Ty([n]) = {[1]} and Ty(γn,m) = id{[1]}; Tm([n]) = {[⋆]} and Tm(γn,m) = id{[⋆]};

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows: The objects of T are the empty context [0] := [] and finite strings

• f the form [n] := [1 · 1 · · · 1
• n times

] for n ≥ 1; There is a unique morphism γn,m : [n] → [m] for all n, m ∈ N. Ty([n]) = {[1]} and Ty(γn,m) = id{[1]}; Tm([n]) = {[⋆]} and Tm(γn,m) = id{[⋆]}; ty([n], [⋆]) = ([n], [1]) and ext([n], [1]) = ([n + 1], [⋆]);

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

Consider the type theory obtained by adding a unit type 1, i.e. we add the following rules to our syntax: ⊢ ⊢ 1 ⊢ ⊢ ⋆ : 1 ⊢ a : 1 ⊢ a = ⋆ : 1 The term model T for this system is defined as follows: The objects of T are the empty context [0] := [] and finite strings

• f the form [n] := [1 · 1 · · · 1
• n times

] for n ≥ 1; There is a unique morphism γn,m : [n] → [m] for all n, m ∈ N. Ty([n]) = {[1]} and Ty(γn,m) = id{[1]}; Tm([n]) = {[⋆]} and Tm(γn,m) = id{[⋆]}; ty([n], [⋆]) = ([n], [1]) and ext([n], [1]) = ([n + 1], [⋆]); sub([n],[⋆]) = γn,n+1 and proj([n],[⋆]) = γn+1,n.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

A natural model C supports the unit type if there exist 1C ∈ U(⋄) and ⋆C ∈ U(⋄) such that p−1

⋄ ({1C}) = {⋆C}.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

A natural model C supports the unit type if there exist 1C ∈ U(⋄) and ⋆C ∈ U(⋄) such that p−1

⋄ ({1C}) = {⋆C}.

Theorem

The data T on the previous slide defines a natural model, which is the free natural model supporting the unit type, i.e.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Example 2: adding a unit type

A natural model C supports the unit type if there exist 1C ∈ U(⋄) and ⋆C ∈ U(⋄) such that p−1

⋄ ({1C}) = {⋆C}.

Theorem

The data T on the previous slide defines a natural model, which is the free natural model supporting the unit type, i.e. if C is any natural model supporting the unit type, and 1C ∈ U(⋄) and ⋆C ∈ U(⋄) are as above, then there is a unique homomorphism (F, Φ, Φ) : T → C such that F[1] = 1C and F[⋆] = ⋆C

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Future work on natural models

In the pipeline:

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Future work on natural models

In the pipeline: Uniform construction of the term model for a signature;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Future work on natural models

In the pipeline: Uniform construction of the term model for a signature; Lawvere duality for natural models;

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Future work on natural models

In the pipeline: Uniform construction of the term model for a signature; Lawvere duality for natural models; Investigation of the polynomial functor induced by p : U → U.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Acknowledgements and references

This is joint work with my PhD advisor, Steve Awodey.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Acknowledgements and references

This is joint work with my PhD advisor, Steve Awodey. Introduction and basic theory of natural models:

• S. Awodey, Natural models of homotopy type theory (2015),

arXiv:1406.3219v2.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Acknowledgements and references

This is joint work with my PhD advisor, Steve Awodey. Introduction and basic theory of natural models:

• S. Awodey, Natural models of homotopy type theory (2015),

arXiv:1406.3219v2.

Some analogous results for categories with families:

• S. Castellan, Categories with families as the initial category with families

(2014).

• S. Castellan, P

. Clairambault and P . Dybjer, Undecidability of equality in the free locally cartesian closed category (2015), arXiv:1504:03995.

• E. Palmgren, Categories with families, FOLDS and logic enriched type

theory (2016), arXiv:1605.01586.

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016

Type theory Natural models Homomorphisms (algebraic) Homomorphisms (functorial) Interpreting the syntax Closing remarks

Acknowledgements and references

This is joint work with my PhD advisor, Steve Awodey. Introduction and basic theory of natural models:

• S. Awodey, Natural models of homotopy type theory (2015),

arXiv:1406.3219v2.

Some analogous results for categories with families:

• S. Castellan, Categories with families as the initial category with families

(2014).

• S. Castellan, P

. Clairambault and P . Dybjer, Undecidability of equality in the free locally cartesian closed category (2015), arXiv:1504:03995.

• E. Palmgren, Categories with families, FOLDS and logic enriched type

theory (2016), arXiv:1605.01586.

Thank you for listening!

Clive Newstead (cnewstead@cmu.edu) Carnegie Mellon University Categories of natural models of type theory — CT 2016