Lecture 4: Univalent Foundations

Lecture 4: Univalent foundations

Nicola Gambino

School of Mathematics University of Leeds

Young Set Theory Copenhagen June 14th, 2016

1 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

2 Overview

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

3 Part I: Homotopy theory in type theory

4 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

I The singleton type 1.

I For all a : A, the type (Σx : A)IdA(a, x)

5 Weak equivalences in type theory

Let f : A → B.

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

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

I 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 )

of weak equivalences from A to B.

Note. The identity 1A : A → A is a weak equivalence.

6 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 .

7 Part II: The univalence axiom

8 Univalent types

Let x : A ` B(x) : type be a dependent type.

For x, y ∈ A, we have:

I the type of paths from x to y, IdA(x, y)

I 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.

9 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

I Isomorphism is equality

10 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 / Weq(B) 6 j i p (s,t) Path(A) / A × A

11 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 s // B

q p C / A t

such that D → C ×A B is a weak equivalence, is either empty or contractible. Idea. Essential uniqueness of s, t (if they exist).

12 Univalence in simplicial sets (II)

Theorem. The ﬁbration π : U˜ → U is univalent. We consider the diagram

U w / Weq(U)

∆ " y (s,t) U × U

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

U w / Weq(U)

1U t { U

Hence, it suﬃces to show that t ∈ Weq.

Since t ∈ Fib, we show that t ∈ Weq ∩ Fib. Suﬃces i t t, for all i ∈ Cof.

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

A b / Weq(U)

i t A0 / U b0 where i ∈ Cof. By the deﬁnition of t, such a square amounts to a diagram

' u2 0 p1 B2 / B2

q2 p2 ~ A / A0 i where p1 , p2 , q2 ∈ Fib, f ∈ Weq and i ∈ Cof.

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

u1 0 B1 / B1

f g

' u2 ' 0 p1 B2 / B2

q1 q2 p2 ~ A / A0 i where q1 is a ﬁbration, g is weak equivalence and all squares are pullbacks. This is also established via the theory of minimal ﬁbrations.

15 A relative consistency result

The deﬁnition of the simplicial model is carried out within

I ZFC + 2 inaccessible cardinals

Theorem. The extension of Martin-L¨oftype theory with the univalence axiom is consistent relatively to ZFC + 2 inaccessible cardinals.

Questions

I Can this result be improved?

I Can the simplicial model be redeveloped constructively, so as to give relative consistency with respect to Martin-L¨oftype theory?

Note Recent work on models in cubical sets.

16 Part III: Mathematics in univalent type theories

17 Homotopy levels in type theory

Deﬁnition

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

Deﬁnition

I 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.

18 Sets

Deﬁnition

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

19 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 WeqEl(a), El(b).

But, for example, WeqBool, Bool is neither empty nor contractible.

Corollary. Univalence is not valid in the types-as-sets model.

20 Further aspects