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 2 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 2 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 = fw : Bx ! By j w 2 Weq g 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 2 Weq. By composing with π2 : U × U ! U, we get U w / Weq(U) 1U t { U Hence, it suffices to show that t 2 Weq. Since t 2 Fib, we show that t 2 Weq \ Fib. Suffices i t t, for all i 2 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 2 Cof. By the definition of t, such a square amounts to a diagram B1 f ' u2 0 p1 B2 / B2 q2 p2 ~ A / A0 i where p1 ; p2 ; q2 2 Fib, f 2 Weq and i 2 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 Other topics I Synthetic homotopy theory I Higher inductive types I Homotopy-initial algebras in type theory I Models of type theory in cubical sets I Models of type theory with uniform fibrations I Relation with the theory of (1; 1)-categories Open problems I Constructivity of the simplicial model I Direct definition of (1; 1)-category in type theory 21 References (I) Type theory I Martin-L¨of,Intuitionistic type theory I Nordstr¨om,Petersson, Smith, Programming in Martin-L¨oftype theory I Nordstr¨om,Petersson, Smith, Martin-L¨oftype theory Type theory and set theory I Aczel, The type theoretic interpretation of constructive set theory I Aczel, On relating type theories and set theories I Griffor and Rathjen, The strength of some Martin-L¨oftype theories Homotopical algebra I Hovey, Model categories I Joyal and Tierney, An introduction to simplicial homotopy theory Models of type theory I Pitts, Categorical logic 22 References (II) Homotopical models I Awodey and Warren, Homotopy-theoretic models of identity types I Bezem, Coquand, Huber, A model of type theory in cubical sets I Gambino and Garner, The identity type weak factorisation system I Kapulkin and Lumsdaine, The simplicial model of univalent foundations I Hofmann and Streicher, The groupoid model of type theory I Streicher, A Model of Type Theory in Simplicial Sets Types and 1-categories I van den Berg and Garner, Types are weak !-groupoids Univalent Foundations I Voevodsky, An experimental library of formalized mathematics based on the univalent foundations I Ahrens, Kapulkin, Shulman, Univalent categories and the Rezk completion 23.