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

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 (Σ x : A )(Π y : A )Id A ( x , y ) is inhabited. Examples ◮ The singleton type 1. ◮ For all a : A , the type (Σ x : A )Id A ( a , x ) 5

Weak equivalences in type theory Let f : A → B . ◮ The homotopy fiber of f at y : B is the type hfiber( f , y ) = def (Σ x ∈ A ) Id B ( fx , y ) . ◮ We say that f : A → B is a weak equivalence if the type isweq( f ) = def (Π y : B ) iscontr (hfiber( f , y )) is inhabited. Note. For A , B : type, there is a type Weq( A , B ) = (Σ f : A → B ) isweq( f ) of weak equivalences from A to B . Note. The identity 1 A : A → A is a weak equivalence. 6

Homotopies Definition. Let f , g : A → B . A homotopy α : f ∼ g is an element α : (Π x : A ) Id B ( 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 ∼ 1 A , β : f ◦ g ∼ 1 B . 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: ◮ the type of paths from x to y , Id A ( x , y ) ◮ the type Weq( B ( x ) , B ( y )) of weak equivalences from B ( x ) to B ( y ). Note. We have j x , y : Id A ( x , y ) → Weq( B ( x ) , B ( y )) Definition. We say x : A ⊢ B ( x ) : type is univalent if j x , 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 j x , y : Id U ( x , y ) → Weq(El( x ) , El( y )) for all x , y : U. Slogan ◮ 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 : B x → B y | w ∈ Weq } Note. Given p : B → A , we have A Weq( B ) j p ( s , t ) i Path( A ) A × A 11

� � � Univalent fibrations Definition. A fibration p : B → A is said to be univalent if j p : 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 s � B D q p � A C 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 w U Weq( U ) ∆ ( s , t ) U × U and show w ∈ Weq . By composing with π 2 : U × U → U , we get w Weq( U ) U 1 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 . 13

� � � � � So, we need to prove the existence of a diagonal filler in a diagram of the form b � Weq( U ) A t i � U A ′ b ′ where i ∈ Cof . By the definition of t , such a square amounts to a diagram B 1 f � B 2 u 2 � B ′ p 1 2 q 2 p 2 � A ′ A i where p 1 , p 2 , q 2 ∈ Fib , f ∈ Weq and i ∈ Cof . 14

� � � � � � The required diagonal filler amounts to a diagram of the form u 1 � B ′ B 1 1 g f u 2 � B ′ B 2 p 1 2 q 1 q 2 p 2 � A ′ A i where q 1 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 ◮ ZFC + 2 inaccessible cardinals Theorem. The extension of Martin-L¨ of 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¨ of type theory? Note Recent work on models in cubical sets. 16

Part III: Mathematics in univalent type theories 17

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 Id A ( x , y ) has h-level 0, i.e. it is contractible. Note: A has h-level 1 ⇔ for all x , y : A , Id A ( 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 ◮ A type A has homotopy level 2 if for all x , y : A , the type Id A ( x , y ) has h-level 1, i.e. it is an h-proposition. Note: A has h-level 2 ⇔ for all x , y : A , Id A ( 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. (Π x : A )Id B ( fx , gx ) → Id A → 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 Id U ( 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. 20

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 21

References (I) Type theory ◮ Martin-L¨ of, Intuitionistic type theory ◮ Nordstr¨ om, Petersson, Smith, Programming in Martin-L¨ of type theory ◮ Nordstr¨ om, Petersson, Smith, Martin-L¨ of 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¨ of type theories Homotopical algebra ◮ Hovey, Model categories ◮ Joyal and Tierney, An introduction to simplicial homotopy theory Models of type theory ◮ Pitts, Categorical logic 22

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 23