Univalence for Free. Matthieu Sozeau Joint work with Nicolas Tabareau - INRIA. GT Types & Réalisabilité March 13th 2013

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1 Univalence for Free Matthieu Sozeau Joint work with Nicolas Tabareau - INRIA Project Team πr 2 INRIA Rocquencourt & PPS, Paris 7 University GT Types & Réalisabilité March 13th 2013

2 Univalence for Free 1 Groupoid interpretations of type theory 1-groupoids vs weak 2-groupoids Defining weak 2-groupoids 2 The construction Propositions, proof irrelevance and propositional extensionality Functions and functional extensionality Types and homotopic equivalence Dependent products Σ types and groupoid levels The translation 3 An example Matthieu Sozeau - Univalence for Free 2

3 Weak 2 groupoids against groupoids In Hofmann and Streicher: T : GP D. GPD are 1-groupoids: ΣA : Set; Σ : A A Set; Σ equiv : x y, Equivalence (x y) eq... Morphisms representing identities are identified up to propositional equality (eq : ΠA, A A Set). Weak 2-groupoids: morphisms representing identities are identified up to another notion of equivalence. ΣA : Set; Σ 1 : A A Set; Σ 2 : {x y} (p q : x 1 y), Set; Σ 2 equiv x y, Equivalence (x 1 y) E.g: Prop := (Prop, iff, irrel,...). Matthieu Sozeau - Univalence for Free 3

4 Weak 2 groupoids agains groupoids GPD are weak 2-groupoids where the equality of morphisms is propositional equality eq (which has J but not UIP ). Not sticking to identity sets (which are at the origin of the groupoid/homotopy models), we can realize a richer model. Principle Proof-irrelevance Propositional extensionality Functional extensionality Univalence Definition of equality Irrelevant equality Logical equivalence Pointwise equality Isomorphism Matthieu Sozeau - Univalence for Free 4

5 Defining weak 2-groupoids Matthieu Sozeau - Univalence for Free 5

6 Relations Start with computational relations: Definition Hom (A : Type) := A A Type. Class Identity {A} (M : Hom A) := identity : x, M x x. Class Equivalence T (Eq : Hom T ):= { Equivalence Identity :> Identity Eq ; Equivalence Inverse :> Inverse Eq ; Equivalence Composition :> Composition Eq }. Matthieu Sozeau - Univalence for Free 6

7 Equivalence Type Notation for 1-homs, the notion of path/equality on T. Class 1Hom T := {m1 : Hom T }. Infix 1 := m1 (at level 80). 2-homs are between objects in a given 1-hom, so T will be an equality /1-Hom type itself Class EquivalenceType (T : Type) : Type := { m2: Hom T ; equiv struct :> Equivalence m2 }. Infix 2 := m2 (at level 80). We assume that m2 is an equivalence but don t assume anything about its proofs (identity, inverse, composition). Matthieu Sozeau - Univalence for Free 7

8 2-1 categories Weak 2-categories where 2-homs are iso-homs (equivalences). Class Weak2 1Category T := { M :> 1Hom T ; eq m :> x y, EquivalenceType (x 1 y) ; Weak2 1Category Identity :> Identity m1 ; Weak2 1Category Composition :> Composition m1; Matthieu Sozeau - Univalence for Free 8

9 2-1 categories id R : x y (f : x 1 y), f (identity x) 2 f ; id L : x y (f : x 1 y), (identity y) f 2 f ; assoc : x y z w (f : x 1 y) (g: y 1 z) (h: z 1 w), (h g) f 2 h (g f ); comp : x y z (f f : x 1 y) (g g : y 1 z), f 2 f g 2 g g f 2 g f }. Definition Weak2 1CatType := {T :Type & Weak2 1Category T }. Matthieu Sozeau - Univalence for Free 9

10 Weak 2-groupoids Class Weak2Groupoid T := { Weak2Groupoid Weak2 1Category :> Weak2 1Category T ; Weak2Groupoid Inverse :> Inverse m1 ; inv R : x y (f : x 1 y), f (inverse f ) 2 identity ; inv L : x y (f : x 1 y), (inverse f ) f 2 identity ; inv : x y (f f :x 1 y), f 2 f inverse f 2 inverse f }. Definition Weak2GroupoidType := {T :Type & Weak2Groupoid T }. Matthieu Sozeau - Univalence for Free 10

11 Notations Notation [ T ] := (π 1 T ). Notation ( x ; p ) := (@sigma intro x p). Notation N := ([M] N) (at level 55). Matthieu Sozeau - Univalence for Free 11

12 Univalence for Free 1 Groupoid interpretations of type theory 1-groupoids vs weak 2-groupoids Defining weak 2-groupoids 2 The construction Propositions, proof irrelevance and propositional extensionality Functions and functional extensionality Types and homotopic equivalence Dependent products Σ types and groupoid levels The translation 3 An example Matthieu Sozeau - Univalence for Free 12

13 Propositions, proof irrelevance and propositional extensionality Matthieu Sozeau - Univalence for Free 13

14 Prop extensionality and proof irrelevance Irrelevant equality: Definition Hom irr (T : Type) : Hom T := λ, unit. We define IrrRelWeak2Groupoid T m for Weak2Groupoid T when m is an equivalence and the second equality is irrelevant (all 2-homs are inhabited, all their inhabitants are equal). Class PropIrr (P : Prop) : Type := { prop irr groupoid := IrrRelWeak2Groupoid (m:= Hom irr P) }. Matthieu Sozeau - Univalence for Free 14

15 Propositional extensionality Prop forms such a degenerated 2-groupoid, with Hom-set logical equivalence of propositions and irrelevant 2-Homs representing equality of two proofs of the same proposition. Program Definition Propositions := { P : Prop & PropIrr P }. Definition eq prop (P Q : Propositions) := [P] [Q]. Program Definition Prop : Weak2GroupoidType := (Propositions ; IrrRelWeak2Groupoid (m:=eq prop) ). Matthieu Sozeau - Univalence for Free 15

16 Functions and functional extensionality Matthieu Sozeau - Univalence for Free 16

17 Functional extensionality Equality of functions requires talking about equality of 2-groupoid functors: Class Functor {T U : Weak2GroupoidType} (f : [T ] [U]) := { map : {x y}, x 1 y f x 1 f y ; map2 : x y (e e : x 1 y), e 2 e map e 2 map e ; map comp : x y z (e:x 1 y) (e :y 1 z), map (e e) 2 map e map e }. Compatibilities for identity, inverse can be derived. Matthieu Sozeau - Univalence for Free 17

18 Functors Definition Fun Type (T U : Weak2GroupoidType) := {f : [T ] [U] & Functor f }. Infix := Fun Type (at level 55). One can also define identity and composition for functors (but not inverse). We ll form the category of 2-Groupoid functors and natural transformations. Matthieu Sozeau - Univalence for Free 18

19 Functor equivalence Equivalence between functors is given by (iso-)natural transformations. Note the higher coherence condition on the functorial maps: Definition nat trans T U (f g : T U) := {α : t : [T ], f t 1 g t & t t (e : t 1 t ), (α t ) (map [f ] e) 2 (map [g] e) (α t)}. Matthieu Sozeau - Univalence for Free 19

20 Natural transformation equivalence At the higer-level: (α t : f t 1 g t) 2 β t. Definition nat trans2 T U (f g : T U) : Hom (nat trans f g) := λ α β, t : [T ], α t 2 β t. No need for naturality here as the higher equivalences are trivial (no 3). We can prove this forms an equivalence type: Program Instance nat trans eq T U (f g : T U) : EquivalenceType (nat trans f g) := {m2 := nat trans2 (f :=f ) (g:=g)}. Matthieu Sozeau - Univalence for Free 20

21 2-groupoid of the function space For two 2-groupoids T, U we have the 2-groupoid T U of functorial maps between T and U (respecting 1, natural transformations). Equivalence is natural transformation equivalence nat trans2. Program Definition fun T U : Weak2GroupoidType := sigma intro Weak2Groupoid (T U) (nat group T U). Infix := fun. Matthieu Sozeau - Univalence for Free 21

22 Types and homotopic equivalence Matthieu Sozeau - Univalence for Free 22

23 Homotopic equivalence Equivalence of 2-groupoids is homotopic equivalence: a map with its adjoint, 2 proofs that they form a section and a retraction plus 2 triangle identities relating the section and rectraction proofs. Class Equiv struct T U (f : [T U]) := { adjoint : [U T ] ; section : f adjoint 1 identity U ; retraction : identity T 1 adjoint f ; triangle : t, (section (f t)) map [ ] (retraction t) 2 identity ; triangle : u, map [ ] (section u) (retraction (adjoint u)) 2 identity }. Definition Equiv A B := {f : A B & Equiv struct f }. Infix := Equiv (at level 55). Matthieu Sozeau - Univalence for Free 23

24 3-groupoids? Again, Equiv forms an equivalence, using the identity equivalence, composition of equivalences and the obvious inverse. 2-groupoids and homotopic equivalences between them form a 3-groupoid, with equality of homotopic equivalences being given by equivalence of adjunctions. We stop at level 2, so this is not explicit here in our construction. However we can still define equivalence of adjunctions and use it. Matthieu Sozeau - Univalence for Free 24

25 Equivalence of adjunctions Two adjunctions are equivalent if their left adjoint are equivalent and they agree on their section and retraction (up-to the isomorphism). The right adjoints always agree: Definition Equiv adjoint A B (f f : Equiv A B) : [f ] 1 [f ] adjoint [f ] 1 adjoint [f ]. Record Equiv eq T U (f g : Equiv T U) : Type := {equiv : [f ] 1 [g] ; eq section : u, section [f ] u 2 section [g] (nat comp (Equiv adjoint equiv) equiv) u; eq retraction : t, (nat comp equiv (Equiv adjoint equiv)) retraction [f ] t 2 retraction [g] t }. Matthieu Sozeau - Univalence for Free 25

26 The groupoid of Types Program Definition Type : Weak2GroupoidType := (Weak2GroupoidType ; Equiv). Type is a weak 2-groupoid whose objects are types with a weak-2-groupoid structure. Their notion of equivalence is homotopic equivalence ( isomorphism). And equivalence of these homotopic equivalences is adjoint equivalence. Note that Weak2GroupoidType is used at two different universe levels here. Matthieu Sozeau - Univalence for Free 26

27 Rewriting in homotopy type theory The map function on F : [A Type] gives an (homotopic) equivalence F x 2 F y if x 1 y. This means we can rewrite [F x] into [F y] using the function part of the equivalence. Definition eq rect {A : [ Type]} {x : [A]} {F : [A Type]} {y : [A]} (e : x 1 y) (p : [F x]) : [F y] := [map [F ] e] p. Matthieu Sozeau - Univalence for Free 27

28 Rewriting theory We derive (some, not all) coherence theorems on eq rect according to the ones on map. E.g., the usual reduction for eq rect is derivable: Definition eq rect id {T :[ Type]} {F : [T Type]} (x : [T ]) (p : [F x]) : eq rect (identity x) p 1 p := (equiv (map id F x)) p. eq rect is compatible with higher equivalences as well: Definition eq rect eq {A : [ Type]} {x : [A]} {F : [A Type]} {y : [A]} {e e : x 1 y} (H : e 2 e ) (p : [F x]) : eq rect e p 1 eq rect e p := (equiv (map2 [F ] H)) p. Matthieu Sozeau - Univalence for Free 28

29 Dependent products Matthieu Sozeau - Univalence for Free 29

30 Dependent product The dependent product gives rise to dependent functors. The map component needs some adjustment by equalities due to dependencies. Class DependentFunctor (T :[ Type]) (U : [T Type]) (f : t, [U t]) : Type := { Dmap : {x y} (e: x 1 y), eq rect e (f x) 1 f y ; Dmap2 : x y (e e : x 1 y) (H: e 2 e ), Dmap e 2 Dmap e eq rect eq H (f x) ; Dmap comp : x y z (e : x 1 y) (e : y 1 z), Dmap (e e) inverse (eq rect comp e e (f x)) 2 Dmap e eq rect map e (Dmap e) }. Definition Prod Type (T :[ Type]) (U:[T Type]) := {f : t, [U t] & DependentFunctor U f }. Matthieu Sozeau - Univalence for Free 30

31 The culprit Again, identity and inverse coherence conditions can be deduced from the composition compatibility. But that requires higher-order equalities not derivable in the 2-groupoid setting, only in 3-groupoids and above. Here we show the derivability assuming an axiom. Axiom map2 id L : T (U : [T Type]) (x y : [T ]) (e:x 1 y), map2 [U] (id L e) 2 id L (map [U] e) comp (identity (map [U] e)) (map id U y) map comp [U] e (identity y). Lemma Dmap id (T :[ Type]) (U : [T Type]) (f : Prod Type U) : x, Dmap (identity x) 2 eq rect id x (f x). Matthieu Sozeau - Univalence for Free 31

32 Equality of dependent functors Dependent natural transformations. At level 2, the naturality condition is trivial again. Definition Dnat trans T (U:[T Type]) (F G : Prod Type U) := {α : t : [T ], F t 1 G t & t t (e : t 1 t ), (α t ) (Dmap e) 2 (Dmap e) eq rect map e (α t)}. Definition Dnat trans2 T U (f g : Prod Type U) : Hom (Dnat trans f g) := λ α β, t : [T ], α t 2 β t. Matthieu Sozeau - Univalence for Free 32

33 Dependent product groupoid This lets us show that dependent functors form a 2-groupoid: Program Instance Prod group T (U:[T Type]) : Weak2Groupoid (Prod Type U). Program Definition Prod T (U:[T Type]) : [ Type] := (Prod Type U ; Prod group T U). Again, universe polymorphism is needed to handle the different levels of Type here. Matthieu Sozeau - Univalence for Free 33

34 Σ types and groupoid levels Matthieu Sozeau - Univalence for Free 34

35 Dependent sums Σ types on F := λ T, Type j lives in Type (j+1). Similarly, the interpretation of a Σ type on a fibration in an n-groupoid builds an n+1-groupoid. We avoid this issue by restricting our translation to weak 1-groupoids, so we can always build the corrsponding 2-groupoid, axiom-free. -groupoids would solve this. Definition Weak1Groupoid (T : [ Type]) : Type := (x y : [T ]) (f f : x 1 y), f 2 f. Definition Weak1Fibration (T : [ Type]) : Type := { F : [T Type] & t, Weak1Groupoid (F t)}. Matthieu Sozeau - Univalence for Free 35

36 Dependent sum groupoid Definition sum type (T : [ Type]) (F : Weak1Fibration T ) := {t : [T ] & [[F ] t]}. Again, we need adjustment/transport on the second component to state equalities of dependent pairs: Definition sum eq (T : [ Type]) (F : Weak1Fibration T ) := λ m n : sum type F, {P : [m] 1 [n] & eq rect P (π 2 m) 1 π 2 n}. In the same way, 2-equality between 1-equalities is given by projections and rewriting. Definition sum eq2 T (F :Weak1Fibration T ) (m n : sum type F ) : Hom (sum eq m n) := λ e e, {P : [e] 2 [e ] & π 2 e 2 π 2 e eq rect eq P (π 2 m)}. Matthieu Sozeau - Univalence for Free 36

37 Dependent sum groupoid Program Instance sum groupoid T (F :Weak1Fibration T ) : Weak2Groupoid (sum type F ). Definition Sum T (F :Weak1Fibration T ) : [ Type] := (sum type F ; sum groupoid T F ). The proof this is a 2-groupoid uses the fact that 2 on F t is trivial. Otherwise, level 3 equalities would be needed as for the product case. Matthieu Sozeau - Univalence for Free 37

38 The translation Matthieu Sozeau - Univalence for Free 38

39 Translation Type Prop Type Prop T U T U t : T, U Prod (λ t, U ; ) λ t : T, m (λ t : T, m ; ) x : T x : [ T ] m n m n Σ t : T, U Sum (λ t, U ; ) π i m π i m Underscores represent obligations to show functoriality/naturality conditions. Matthieu Sozeau - Univalence for Free 39

40 Derived facts Lemma prop ext (P Q : [ Prop]) : [P] [Q] P 1 Q. Proof. firstorder. Qed. Lemma proof irrelevant (P : [ Prop]) (p q : [P]) : p 1 q. Proof. exact tt. Qed. Note that nat trans is not just functional extensionality, it requires a higher coherence condition. Lemma fun ext A B (f g : [A B]) : nat trans f g f 1 g. Proof. auto. Qed. Lemma fun ext dep T U (f g : [ Prod (T :=T ) U]) : Dnat trans f g f 1 g. Proof. auto. Qed. Matthieu Sozeau - Univalence for Free 40

41 Derived facts To prove equality on dependent pairs, it is enough to prove equality of the corresponding projections. Lemma sum ext T F (m n : [ Sum (T :=T ) F ]) : (P : [m] 1 [n]), eq rect P (π 2 m) 1 π 2 n m 1 n. Proof. intros. P. auto. Qed. Finally, we have the univalence principle on types. Lemma univalence (U V : [ Type]) : (Equiv U V ) U 1 V. Proof. auto. Qed. Matthieu Sozeau - Univalence for Free 41

42 Univalence for Free 1 Groupoid interpretations of type theory 1-groupoids vs weak 2-groupoids Defining weak 2-groupoids 2 The construction Propositions, proof irrelevance and propositional extensionality Functions and functional extensionality Types and homotopic equivalence Dependent products Σ types and groupoid levels The translation 3 An example Matthieu Sozeau - Univalence for Free 42

43 Church naturals and inductive naturals Define the (discrete) weak 2-groupoid of inductive natural numbers: (nat, eq, Hom irr,...). We need to use UIP on nat to show this forms a groupoid. Derive the weak 2-groupoid of church naturals cnat := ΠX : Type, X (X X) X using the function space and dependent product groupoid constructors. Prove equivalence (i.e., isomorphism) of the two groupoids. This requires parametricity at cnat (see Keller and Lasson). We can now transport the translation of any theorem on nat to cnat... This requires showing these theorems are functorial of course. Matthieu Sozeau - Univalence for Free 43

44 Conclusion Univalence should be free! Directions: Automate functoriality/naturality conditions. Internalize this, 2-dimensional type theory/ott-style. Matthieu Sozeau - Univalence for Free 44

45 The End That s all folks! Matthieu Sozeau - Univalence for Free 45