Lecture 3: Homotopical models of type theory Nicola Gambino School - - PowerPoint PPT Presentation

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Lecture 3: Homotopical models of type theory

Nicola Gambino

School of Mathematics University of Leeds

Young Set Theory Copenhagen June 14th, 2016

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What happened yesterday?

Type theory

◮ the type theory T ◮ Extensional vs intensional type theories

Homotopical algebra

◮ Weak factorisation systems and model structures ◮ Groupoids ◮ Simplicial sets

Italy 2 – Belgium 0 Exercises

◮ x, y : A, u : IdA(x, y) ⊢ u−1 : IdA(y, x) ◮ x, y, z : A, u : IdA(x, y), v : IdA(y, z) ⊢ v ◦ u : IdA(x, z).

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Problems with intensionality

The axioms for identity types do not seem to capture fully what we want. Example

◮ we have

IdA×B( c, d ) ← → IdA

• π1(c), π1(d)
• × IdB
• π2(c), π2(d)
• ◮ but only

IdA→B(f , g) − → (Πx : A) IdB(fx, gx) Similar situation for Σ-types and Π-types. What about the type universe?

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Outline of Lecture 2

Part I: Models of type theory Part II: Identity types Part III: Π-types Part IV: Universes

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Part I: Models of type theory

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Models of type theories

Question: What structure on a category C do we need to have a model of T ? Idea: Look at the structure of the syntactic category of T.

◮ Objects: contexts

(x1 : A1) , (x1 : A1, x2 : A2, . . .) , . . .

◮ Maps: terms-in-context, e.g.

(a1) : Γ → (x1 : A1) if Γ ⊢ a1 : A1 (a1, a2) : Γ → (x1 : A1, x2 : A2) if Γ ⊢ a1 : A1 , Γ ⊢ a2 : A2[a1/x1] · · · Note. Γ ⊢ A : type = ⇒ (Γ, x : A)

pA

• Γ

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Axiomatization

Fix

◮ category C with a terminal object 1 ◮ a class P ⊆ Map(C)

Type Theory Syntactic category Category Theory A : type (x : A)

pA

• ( )

1.A

pA∈P

• 1

Γ ⊢ A : type (Γ, x : A)

pA

• Γ

Γ.A

pA∈P

• Γ

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Terms as sections

Type Theory Category Theory Γ ⊢ a : A Γ

a

• 1Γ
• Γ.A

pA

• Γ

a : A 1

a

A

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Substitution as pullback

Type Theory Category Theory Γ ⊢ a : A Γ, x : A ⊢ B : type Γ ⊢ B[a/x] : type Γ.B[a/x]

• Γ.A.B

pB

• Γ

a

Γ.A

Γ ⊢ a : A Γ, x : A ⊢ b : B Γ ⊢ b[a/x] : B[a/x] Γ

b[a/x]

• b◦(1Γ,a)
• 1Γ
• Γ.B[a/x]
• Γ.A.B

pB

• Γ

a

Γ.A

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Weakening

For Γ ∈ Obj(C), let P/Γ be the category with

◮ objects: P-maps A : Γ.A → Γ ◮ maps: commutative triangles

Γ.A

f

• pA
• Γ.B

pB

• Γ

Γ, x : A ⊢ f (x) : B Let pA : Γ.A → Γ be in P. Pullback Γ.A.E

• Γ.E

pE

• Γ.A

pA

Γ

Γ ⊢ E : type Γ, x : A ⊢ E : type This is the ‘weakening functor’ ∆A : P/Γ → P/Γ.A.

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General setting

Let

◮ C be a category ◮ P ⊆ Map(C)

and assume

◮ C has a terminal object. ◮ The pullback of a P-maps along any maps exists and is an P-map.

Question

◮ What additional structure on P do we need to interpret type-formers?

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Part II: Identity types

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Identity types (I)

For simplicity, let us assume Γ = 1 and work with pA : 1.A → 1. We need

• 1. a P-map

qA : IdA → A × A

• 2. a factorisation

A

reflA

• ∆A
• IdA

qA

• A × A
• 3. a diagonal filler for every commutative diagram

A

reflA

• d

IdA.E

pE ∈P

• IdA
• IdA
• Note. In fact, reflA ⋔ p for all p ∈ P.

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Identity types as path spaces

Idea p : IdA(a, b) ⇐ ⇒ p is a path in A from a to b Note

◮ This explains several aspects of the behaviour of identity types

• Theorem. The syntactic category of the type theory T admits a weak

factorisation system (L, R), where L = {i | (∀p ∈ P) i ⋔ p} R = {p | (∀i ∈ L) i ⋔ p} where P is the set of projections pA : (Γ, x : A) → Γ.

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Homotopical models of type theory

Idea

◮ Take P = Fib in some model structure (Weq, Fib, Cof).

We get a ‘dictionary’ Type Theory Homotopical algebra A : type fibrant object A x : A ⊢ B(x) : type fibration p : B → A x, y : A ⊢ IdA(x, y) path space of A (Πx : A)B(x) space of sections of p type universe U a fibrant object U x ∈ U ⊢ El(x) : type a generic fibration π : ˜ U → U

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Example: Id-types in groupoids

Given by the (Weq ∩ Cof, Fib)-factorisation of diagonal A

r

• ∆A

A × A

AJ

(s,t)

• The groupoid AJ has

◮ objects: maps α : a0 → a1 in A ◮ maps: squares

a0

• α
• b0

β

• a1

b1

Note Uniqueness of identity proofs fails in this model.

• Warning. To define a model, one needs to take care of further aspects:

◮ mere existence vs structure ◮ coherence with respect to pullback (substitution)

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Part III: Π-types

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Π-types

Let A : Γ.A → Γ be in P.

◮ Recall the ‘weakening’ functor

∆A : P/Γ → P/Γ.A Γ ⊢ E : type Γ, x : A ⊢ E : type

◮ To interpret Π-types, we require a right adjoint

ΠA : P/Γ.A → P/Γ Γ, x : A ⊢ B : type Γ ⊢ (Πx : A)B : type Idea Γ.A

b

• Γ.A.B
• Γ.A

⇐ ⇒ Γ

λ(b)

• Γ.ΠA(B)
• Γ

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Π-types in groupoids

Theorem.

◮ For p : Γ.A → Γ an isofibration, the pullback functor

∆p : Gpd/Γ → Gpd/Γ.A has a right adjoint Πp : Gpd/Γ.A → Gpd/Γ

◮ Furthermore, the right adjoint preserves isofibrations, and thus gives us

Πp : Fib/Γ.A → Fib/Γ Γ.A.B

q

Γ.A

→ Γ.ΠA(B)

Πp(q) Γ

• Example. When Γ = 1, the objects of the groupoid ΠA(B) are

A

b

• 1A
• A.B

q

• A

i.e. sections of q.

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Π-types in simplicial sets

Theorem.

◮ For any map p : B → A, pullback along p has a right adjoint

Πp : SSet/B → SSet/A

◮ If p is a Kan fibration, then Πp preserves Kan fibrations, and hence gives

Πp : Fib/B → Fib/A

• Proof. By duality, it suffices to show that

∆p : SSet/A → SSet/B preserves (Weq ∩ Cof)-maps. But

◮ Cof-maps are monomorphisms, so are always preserved. ◮ The preservation of weak equivalences by pullback along Kan fibrations is

the so-called right properness of the model structure.

• Note. Constructivity issues.

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Part IV: Universes

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Generic P-maps

We need a notion of ‘smallness’ for P-maps, e.g. fibers having cardinality < κ. Then need a P-map π : ˜ U → U that weakly classifies ‘small’ P-maps, i.e. for every such p : B → A there exists a pullback diagram B

• p
• ˜

U

π

• A

U .

• Note. Given a : 1 → U, we can form a pullback

El(a)

• ˜

U

π

• 1

a

U

We think of a as the ‘name’ in U of the object El(a).

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The type universe in groupoids and simplicial sets

Fix an inaccessible cardinal κ.

◮ In Gpd, it is not difficult to construct a universe. For example, one can

consider the groupoid of all small (discrete) groupoids.

◮ In SSet, there exists a fibration

π : ˜ U → U that weakly classifies fibrations with fibers of cardinality < κ, i.e. for every such p : B → A there exists a pullback diagram B

• p
• ˜

U

π

• A

U .

Here, Un = { p : B → ∆n | p Kan fibration }

• Problem. But U needs to be fibrant!

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The type universe in simplicial sets

Theorem.

◮ The base U of the generic Kan fibration π : ˜

U → U is a Kan complex.

• Proof. We need to show that U is a Kan complex. So show

Λn

k ∀b

• hn

k

• U

∆n

∃b′

• This reduces to the problem of extending fibrations along horn inclusions:

B

• p
• B′
• Λn

k hn

k

∆n

This can be done using the theory of minimal fibrations (AC).

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Minimal fibrations extend along (Weq ∩ Cof)-maps

Lemma 1. Let

◮ m : X → A be a minimal fibration ◮ i : A → A′ be a (Weq ∩ Cof)-map.

Then there exists m′ : M′ → A′ a minimal fibration and X

j

• m
• X ′

m′

• A

i

A′

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(Weq ∩ Fib)-maps can be extended along cofibrations

Lemma 2. Let

◮ t : B → X be a (Weq ∩ Fib)-map ◮ j : X → X ′ a cofibration

Then there exists t′ : E ′ → X ′ a (Weq ∩ Fib)-map and B

• t
• B′

t′

• X

j

X ′

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Proof of the theorem

Recall that we need to complete the diagram B

• p
• B′
• Λn

k hn

k

∆n

It suffices to

◮ factor p as a (Weq ∩ Fib)-map t followed by a minimal fibration m ◮ apply Lemma 1 and Lemma 2 so as to get

B

• t
• B′

t′

• X

j

• m
• X ′

m′

• Λn

k hn

k

∆n

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Conclusions

The homotopical models of type theory suggest:

• 1. To use type theory as a language for speaking about spaces
• 2. To develop mathematics using this language; in particular, to define

sets =def discrete spaces

• 3. To add axioms to type theory motivated by homotopy theory