Lecture 4: Univalent foundations Nicola Gambino School of - - PowerPoint PPT Presentation

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Lecture 4: Univalent foundations

Nicola Gambino

School of Mathematics University of Leeds

Young Set Theory Copenhagen June 14th, 2016

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Univalent foundations

The simplicial model of type theory suggest to:

• 1. use type theory as a language for speaking about homotopy types,
• 2. develop mathematics using this language; in particular

sets =def discrete homotopy types,

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

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Overview

Part I: Homotopy theory in type theory Part II: The univalence axiom in simplicial sets Part III: Univalent foundations

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Part I: Homotopy theory in type theory

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Contractibility

• Definition. We say that a type A is contractible if the type

iscontr(A) =def (Σx : A)(Πy : A)IdA(x, y) is inhabited. Examples

◮ The singleton type 1. ◮ For all a : A, the type

(Σx : A)IdA(a, x)

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Weak equivalences in type theory

Let f : A → B.

◮ The homotopy fiber of f at y : B is the type

hfiber(f , y) =def (Σx ∈ A) IdB(fx, y) .

◮ We say that f : A → B is a weak equivalence if the type

isweq(f ) =def (Πy : B) iscontr (hfiber(f , y)) is inhabited.

• Note. For A, B : type, there is a type

Weq(A, B) = (Σf : A → B) isweq(f )

• f weak equivalences from A to B.
• Note. The identity 1A : A → A is a weak equivalence.

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Homotopies

• Definition. Let f , g : A → B. A homotopy α : f ∼ g is an element

α : (Πx : A) IdB(fx, gx)

• Proposition. Weak equivalences are homotopy equivalences, i.e. if f : A → B is

a weak equivalence then there is g : B → A and homotopies α : g ◦ f ∼ 1A , β : f ◦ g ∼ 1B .

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Part II: The univalence axiom

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Univalent types

Let x : A ⊢ B(x) : type be a dependent type. For x, y ∈ A, we have:

◮ the type of paths from x to y, IdA(x, y) ◮ the type Weq(B(x), B(y)) of weak equivalences from B(x) to B(y).

• Note. We have

jx,y : IdA(x, y) → Weq(B(x), B(y))

• Definition. We say x : A ⊢ B(x) : type is univalent if jx,y is an equivalence for

all x, y : A.

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The univalence axiom

Recall a : U El(a) : type Univalence Axiom. The dependent type x : U ⊢ El(x) : type is univalent. Explicitly, we have equivalences jx,y : IdU(x, y) → Weq(El(x), El(y)) for all x, y : U. Slogan

◮ Isomorphism is equality

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Univalence in simplicial sets (I)

The rich structure of SSet allows us to ‘internalize’ a lot of constructions. Let p : B → A a fibration. There exists a fibration (s, t) : Weq(A) → A × A such that the fiber over (x, y) is Weq(A)x,y = {w : Bx → By | w ∈ Weq }

• Note. Given p : B → A, we have

A

• i
• Weq(B)

(s,t)

• Path(A)
• jp
• A × A

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Univalent fibrations

• Definition. A fibration p : B → A is said to be univalent if

jp : Path(A) → Weq(B) is a weak equivalence.

• Idea. Weak equivalences between fibers are ‘witnessed’ by paths in the base.
• Proposition. A fibration p : B → A is univalent if and only if for every

fibration q : D → C, the space of squares D

• q
• s

B

p

• C

t

A

such that D → C ×A B is a weak equivalence, is either empty or contractible.

• Idea. Essential uniqueness of s, t (if they exist).

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Univalence in simplicial sets (II)

• Theorem. The fibration π : ˜

U → U is univalent. We consider the diagram U

w

• ∆
• Weq(U)

(s,t)

• U × U

and show w ∈ Weq. By composing with π2 : U × U → U, we get U

w

• 1U
• Weq(U)

t

• U

Hence, it suffices to show that t ∈ Weq. Since t ∈ Fib, we show that t ∈ Weq ∩ Fib. Suffices i ⋔ t, for all i ∈ Cof.

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So, we need to prove the existence of a diagonal filler in a diagram of the form A

i

• b

Weq(U)

t

• A′

b′

U

where i ∈ Cof. By the definition of t, such a square amounts to a diagram B1

p1

• f

B2

p2

• u2

B′

2 q2

• A

i

A′

where p1 , p2 , q2 ∈ Fib, f ∈ Weq and i ∈ Cof.

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The required diagonal filler amounts to a diagram of the form B1

p1

• f
• u1

B′

1 g

• q1
• B2

p2

• u2

B′

2 q2

• A

i

A′

where q1 is a fibration, g is weak equivalence and all squares are pullbacks. This is also established via the theory of minimal fibrations.

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A relative consistency result

The definition of the simplicial model is carried out within

◮ ZFC + 2 inaccessible cardinals

• Theorem. The extension of Martin-L¨
• f type theory with the univalence axiom

is consistent relatively to ZFC + 2 inaccessible cardinals. Questions

◮ Can this result be improved? ◮ Can the simplicial model be redeveloped constructively, so as to give

relative consistency with respect to Martin-L¨

• f type theory?

Note Recent work on models in cubical sets.

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Part III: Mathematics in univalent type theories

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Homotopy levels in type theory

Definition

◮ A type A has homotopy level 0 if it is contractible.

Definition

◮ A type A has homotopy level 1 if for all x, y : A, the type IdA(x, y) has

h-level 0, i.e. it is contractible. Note: A has h-level 1 ⇔ for all x, y : A, IdA(x, y) is contractible ⇔ if A is inhabited, then it is contractible ⇔ either 0 or 1 Types of h-level 1 will be called h-propositions. Example: isweq(f ) is a h-proposition.

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Sets

Definition

◮ A type A has homotopy level 2 if for all x, y : A, the type IdA(x, y) has

h-level 1, i.e. it is an h-proposition. Note: A has h-level 2 ⇔ for all x, y : A, IdA(x, y) is a h-proposition ⇔ A is discrete Types of h-level 2 will be called h-sets.

• Idea. This hierarchy can be extended inductively:

Level 1 2 3 Types ∗ 0 , 1 sets groupoids Mathematics – “logic” “algebra” “category theory”

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Remarks on the univalence axiom

• Theorem. The univalence axiom implies Function Extensionality, i.e.

(Πx : A)IdB(fx, gx) → IdA→B(f , g)

• Theorem. Assuming the univalence axiom, the type universe U is not an h-set.
• Proof. Assume U is a h-set.

Then IdU(a, b) would be a h-proposition, i.e. either empty or contractible. By univalence, so would Weq

• El(a), El(b)
• .

But, for example, Weq

• Bool, Bool
• is neither empty nor contractible.
• Corollary. Univalence is not valid in the types-as-sets model.

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Further aspects

Other topics

◮ Synthetic homotopy theory ◮ Higher inductive types ◮ Homotopy-initial algebras in type theory ◮ Models of type theory in cubical sets ◮ Models of type theory with uniform fibrations ◮ Relation with the theory of (∞, 1)-categories

Open problems

◮ Constructivity of the simplicial model ◮ Direct definition of (∞, 1)-category in type theory

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References (I)

Type theory

◮ Martin-L¨

• f, Intuitionistic type theory

◮ Nordstr¨

• m, Petersson, Smith, Programming in Martin-L¨
• f type theory

◮ Nordstr¨

• m, Petersson, Smith, Martin-L¨
• f type theory

Type theory and set theory

◮ Aczel, The type theoretic interpretation of constructive set theory ◮ Aczel, On relating type theories and set theories ◮ Griffor and Rathjen, The strength of some Martin-L¨

• f type theories

Homotopical algebra

◮ Hovey, Model categories ◮ Joyal and Tierney, An introduction to simplicial homotopy theory

Models of type theory

◮ Pitts, Categorical logic

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References (II)

Homotopical models

◮ Awodey and Warren, Homotopy-theoretic models of identity types ◮ Bezem, Coquand, Huber, A model of type theory in cubical sets ◮ Gambino and Garner, The identity type weak factorisation system ◮ Kapulkin and Lumsdaine, The simplicial model of univalent foundations ◮ Hofmann and Streicher, The groupoid model of type theory ◮ Streicher, A Model of Type Theory in Simplicial Sets

Types and ∞-categories

◮ van den Berg and Garner, Types are weak ω-groupoids

Univalent Foundations

◮ Voevodsky, An experimental library of formalized mathematics based on

the univalent foundations

◮ Ahrens, Kapulkin, Shulman, Univalent categories and the Rezk completion