Higher Inductive Types in Homotopy Type Theory...Higher Inductive Types in Homotopy Type Theory...

Higher Inductive Types inHomotopy Type Theory

Steve Awodey

Institute for Advanced Studyand

Carnegie Mellon University

TYPES 2013Toulouse, April 2013

Introduction

A new connection has come to light between Logic and Topologyin the form of an interpretation of intensional Martin-Lof typetheory into homotopy theory.

I Homotopy can be used to construct models of systems ofconstructive logic.

I Constructive type theory can be used as a formal calculus toreason about homotopy.

I The computational implementation of type theory allowscomputer verified proofs in homotopy theory, and elsewhere.

I The homotopical interpretation suggests some new logicalconstructions and axioms.

Univalent Foundations combines these into a new program forfoundations of mathematics.

Introduction

Type theory

Martin-Lof constructive type theory consists of:

I Types: X ,Y , . . . ,A× B, A→ B, . . .

I Terms: x : A, b : B, 〈a, b〉, λx .b(x), . . .I Dependent Types: x : A ` B(x)

x :A B(x)I∏

x :A B(x)

I Equations s = t : A

Intended as a foundation for constructive mathematics.

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

x :A B(x)I∏

x :A B(x)

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

I Terms: x : A, b : B, 〈a, b〉, λx .b(x), . . .

I Dependent Types: x : A ` B(x)I∑

x :A B(x)I∏

x :A B(x)

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

x :A B(x)I∏

x :A B(x)

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

x :A B(x)

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

x :A B(x)I∏

x :A B(x)

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

x :A B(x)I∏

x :A B(x)

Type theory

I Types: X ,Y , . . . ,A× B, A→ B, . . .

x :A B(x)I∏

x :A B(x)

Propositions as Types

The system has a dual interpretation:

I once as mathematical objects: types are “sets” and theirterms are “elements”, which are being constructed,

I once as logical objects: types are “propositions” and theirterms are “proofs”, which are being derived.

This is also known as the Curry-Howard correspondence:

0 1 A + B A× B A→ B∑

x :A B(x)∏

x :A B(x)

⊥ T A ∨ B A ∧ B A⇒ B ∃x :AB(x) ∀x :AB(x)

Gives the system its constructive character.

0 1 A + B A× B A→ B∑

x :A B(x)∏

x :A B(x)

0 1 A + B A× B A→ B∑

x :A B(x)∏

x :A B(x)

0 1 A + B A× B A→ B∑

x :A B(x)∏

x :A B(x)

Identity types

According to the logical interpretation we have:

I propositional logic: A + B, A× B, A→ B,

I predicate logic: B(x),C (x , y), with quantifiers∏

and∑

So it’s natural to add a primitive relation of identity between anyterms of the same type:

x , y : A ` IdA(x , y)

This type represents the logical proposition “x is identical to y”.

On the mathematical side, the identity type admits a newlydiscovered “geometric” interpretation.

Identity types

and∑

x , y : A ` IdA(x , y)

Identity types

and∑

x , y : A ` IdA(x , y)

Rules for identity types

The introduction rule says that a : A is always identical to itself:

r(a) : IdA(a, a)

The elimination rule is a form of Leibniz’s law:

c : IdA(a, b) x : A ` d(x) : R(x , x , r(x)

)Jd(a, b, c) : R(a, b, c)

Schematically:

“ a = b & R(x , x) ⇒ R(a, b) ”

r(a) : IdA(a, a)

)Jd(a, b, c) : R(a, b, c)

Schematically:

“ a = b & R(x , x) ⇒ R(a, b) ”

r(a) : IdA(a, a)

)Jd(a, b, c) : R(a, b, c)

Schematically:

“ a = b & R(x , x) ⇒ R(a, b) ”

The homotopy interpretationSuppose we have terms of ascending identity types:

a, b : A

p, q : IdA(a, b)

α, β : IdIdA(a,b)(p, q)

. . . : IdIdId... (. . .)

Consider the following interpretation:

Types Spaces

Terms Maps

a : A Points a : 1→ A

p : IdA(a, b) Paths p : a⇒ b

α : IdIdA(a,b)(p, q) Homotopies α : p V q

The homotopy interpretationSuppose we have terms of ascending identity types:

a, b : A

p, q : IdA(a, b)

. . . : IdIdId... (. . .)

Consider the following interpretation:

Types Spaces

Terms Maps

a : A Points a : 1→ A

p : IdA(a, b) Paths p : a⇒ b

α : IdIdA(a,b)(p, q) Homotopies α : p V q

The homotopy interpretation: Type dependency

We still need to interpret dependent types x : A ` B(x).

The identity rules imply the following:

p : IdA(a, a′) b : B(a)

p∗b : B(a′)

Logically, this just says “a = a′ & B(a)⇒ B(a′)”.

But topologically, it is a familiar lifting property:

b // p∗b

A a p// a′

This is the notion of a “fibration” of spaces.

We still need to interpret dependent types x : A ` B(x).The identity rules imply the following:

p : IdA(a, a′) b : B(a)

p∗b : B(a′)

b // p∗b

A a p// a′

p : IdA(a, a′) b : B(a)

p∗b : B(a′)

b // p∗b

A a p// a′

p : IdA(a, a′) b : B(a)

p∗b : B(a′)

b // p∗b

A a p// a′

p : IdA(a, a′) b : B(a)

p∗b : B(a′)

b // p∗b

A a p// a′

p : IdA(a, a′) b : B(a)

p∗b : B(a′)

b // p∗b

A a p// a′

Thus we continue the homotopy interpretation as follows:

Dependent types x : A ` B(x) Fibrations B

The type B(a) is the fiber of B → A over the point a : A

Thus we continue the homotopy interpretation as follows:

Dependent types x : A ` B(x) Fibrations B

The type B(a) is the fiber of B → A over the point a : A

The homotopy interpretation: Identity types

To interpret the identity type x , y : A ` IdA(x , y), we thus requirea fibration over A× A.

Take the space AI of all paths in A:

Identity type x , y : A ` IdA(x , y) Path space AI

The fiber IdA(a, b) over a point (a, b) ∈ A× A is the space ofpaths from a to b in A.

IdA(a, b)

(a,b)// A× A.

To interpret the identity type x , y : A ` IdA(x , y), we thus requirea fibration over A× A.Take the space AI of all paths in A:

IdA(a, b)

(a,b)// A× A.

To interpret the identity type x , y : A ` IdA(x , y), we thus requirea fibration over A× A.Take the space AI of all paths in A:

IdA(a, b)

(a,b)// A× A.

The path space AI classifies homotopies ϑ : f ⇒ g between mapsf , g : X → A,

(f ,g)// A× A.

So given any terms x : X ` f , g : A, an identity term

x : X ` ϑ : IdA(f , g)

is interpreted as a homotopy between f and g .

The path space AI classifies homotopies ϑ : f ⇒ g between mapsf , g : X → A,

(f ,g)// A× A.

So given any terms x : X ` f , g : A, an identity term

x : X ` ϑ : IdA(f , g)

is interpreted as a homotopy between f and g .

The homotopy interpretation: Summary

1. There is a topological interpretation of the λ-calculus:

types spaces

terms continuous functions

computability continuity

2. Extend this to dependently typed λ-calculus with Id-types,using the basic idea:

p : IdX (a, b) ⇔p is a path from point a to point b in the space X

This forces dependent types to be fibrations, Id-types to bepath spaces, and terms of Id-types to be homotopies.

types spaces

2. Extend this to dependently typed λ-calculus with Id-types,

using the basic idea:

types spaces

This forces dependent types to be fibrations,

Id-types to bepath spaces, and terms of Id-types to be homotopies.

types spaces

This forces dependent types to be fibrations, Id-types to bepath spaces,

and terms of Id-types to be homotopies.

types spaces

The homotopy interpretation: First theorems

Instead of concrete spaces and homotopies, we use the axiomaticdescription provided by Quillen model categories.

I Gives a wide range of different models.

I Includes classical homotopy of spaces and simplicial sets.

I Allows the use of standard methods from categorical logic.

Theorem (Awodey & Warren 2006)

“Martin-Lof type theory has a sound interpretation into abstracthomotopy theory.”

Theorem (Gambino & Garner 2008)

“The homotopy interpretation of Martin-Lof type theory is alsocomplete.”

1. The homotopy interpretation: Conclusion

Type theory provides a “logic of homotopy”.

2. Homotopy type theory

How expressive is constructive type theory as a formal languagefor homotopy theory?

What facts, properties, and constructions from homotopy theoryare logically expressible?

One example: the fundamental group and its higher-dimensionalanalogues are logical constructions.

Fundamental groupoids

Let’s return to the system of identity terms of various orders:

a, b : A

p, q : IdA(a, b)

ϑ : IdIdId... (α, β)

These can be represented suggestively as follows:

•b//

β*4ϑ

Let’s return to the system of identity terms of various orders:

a, b : A

p, q : IdA(a, b)

ϑ : IdIdId... (α, β)

These can be represented suggestively as follows:

•b//

β*4ϑ

As in topology, the terms of order 0 and 1, (“points” and “paths”),

•b//

bear the structure of a groupoid.

The laws of identity correspond to the groupoid operations:

r : Id(a, a) reflexivity a→ a

s : Id(a, b)→ Id(b, a) symmetry a b

t : Id(a, b)× Id(b, c)→ Id(a, c) transitivity a→ b → c

This was first shown by Hofmann & Streicher (1998), who gave amodel of intensional type theory using groupoids as types.

•b//

Fundamental groupoidsBut also just as in topology, the groupoid equations ofassociativity, inverse, and unit:

p · (q · r) = (p · q) · rp−1 · p = 1 = p · p−1

1 · p = p = p · 1

do not hold strictly, but only “up to homotopy”.

This means they are witnessed by terms of the next higher order:

ϑ : IdId(p−1 · p, 1

Fundamental groupoidsBut also just as in topology, the groupoid equations ofassociativity, inverse, and unit:

p · (q · r) = (p · q) · rp−1 · p = 1 = p · p−1

1 · p = p = p · 1

do not hold strictly, but only “up to homotopy”.

This means they are witnessed by terms of the next higher order:

ϑ : IdId(p−1 · p, 1

The entire system of identity terms of all orders forms aninfinite-dimensional graph, or “globular set”:

A⇔ IdA ⇔ IdIdA ⇔ IdIdIdA ⇔ . . .

It has the structure of a (weak), infinite-dimensional, groupoid, asalready occurring homotopy theory:

Theorem (Lumsdaine, Garner & van den Berg, 2009)

The system of identity terms of all orders over any fixed type is aweak ∞-groupoid.

Every type has a fundamental weak ∞-groupoid.

Homotopy n-types

Voevodsky has discovered that the universe of all types is naturallystratified by “homotopical dimension”.

A type X is called:

contractible iff∑

∏y :X IdX (x , y) is inhabited,

A type X is called a:

proposition iff IdX (x , y) is contractible for all x , y : X ,

set iff IdX (x , y) is a proposition for all x , y : X ,

1-type iff IdX (x , y) is a set for all x , y : X ,

(n+1)-type iff IdX (x , y) is an n-type for all x , y : X .

We then let set = 0-type, and proposition = (−1)-type.This corresponds to the homotopical notion of “truncation”:the level at which the fundamental groupoid becomes trivial.

Homotopy n-types

A type X is called:

contractible iff∑

Homotopy n-types

A type X is called:

contractible iff∑

Homotopy n-types

A type X is called:

contractible iff∑

Homotopy n-types

A type X is called:

contractible iff∑

Homotopy n-types

A type X is called:

contractible iff∑

Homotopy n-types

A type X is called:

contractible iff∑

We then let set = 0-type, and proposition = (−1)-type.

This corresponds to the homotopical notion of “truncation”:the level at which the fundamental groupoid becomes trivial.

Homotopy n-types

A type X is called:

contractible iff∑

Machine implementation

Now one can combine the following:

I the representation of homotopy theory in constructive typetheory

I the well-developed implementations of type theory incomputational proof assistants like Coq.

Allows computer verified proofs in homotopy theory and relatedfields, in addition to constructive mathematics.

This aspect is being very actively pursued right now in theUnivalent Foundations Program.

A computational example

A classical result states that the higher homotopy groups of aspace are always abelian.

We can formalize this in type theory:

I the fundamental group π1(X , b) of a type X at basepointb : X consists of terms of type IdX (b, b).

I the second homotopy group π2(X , b) consists of terms of typeIdIdX (b,b)(r(b), r(b)).

I Each of these types has a group structure, and so the secondone has two group structures that are compatible.

I Now the Eckmann-Hilton argument shows that the twostructures on π2(X , b) agree, and are abelian.

This argument can be formalized in Coq and verified to be correct.In this way, we can use the homotopical interpretation to verifyproofs in homotopy theory.

A classical result states that the higher homotopy groups of aspace are always abelian.We can formalize this in type theory:

(** ** The 2-dimensional groupoid structure *)

(** Horizontal composition of 2-dimensional paths. *)

Definition concat2 A x y z : A p p’ : x = y q q’ : y = z (h : p = p’) (h’ : q = q’)

: p @ q = p’ @ q’

:= match h, h’ with idpath, idpath => 1 end.

Notation ‘‘p @@ q" := (concat2 p q)

(** 2-dimensional path inversion *)

Definition inverse2 A : Type x y : A p q : x = y (h : p = q) : p^ = q^

:= match h with idpath => 1 end.

(** *** Whiskering *)

Definition whiskerL A : Type x y z : A (p : x = y) q r : y = z (h : q = r) : p @ q = p @ r

:= 1 @@ h.

Definition whiskerR A : Type x y z : A p q : x = y (h : p = q) (r : y = z) : p @ r = q @ r

:= h @@ 1.

(** *** Unwhiskering, a.k.a. cancelling. *)

Lemma cancelL A x y z : A (p : x = y) (q r : y = z) : (p @ q = p @ r) -> (q = r).

Proof.

destruct p, r. intro a. exact ((concat_1p q)^ @ a).

Defined.

Lemma cancelR A x y z : A (p q : x = y) (r : y = z) : (p @ r = q @ r) -> (p = q).

Proof.

destruct r, p. intro a. exact (a @ concat_p1 q).

Defined.

(** Whiskering and identity paths. *)

Definition whiskerR_p1 A : Type x y : A p q : x = y (h : p = q) :

(concat_p1 p) ^ @ whiskerR h 1 @ concat_p1 q = h

match h with idpath =>

match p with idpath =>

end end.

Definition whiskerR_1p A : Type x y z : A (p : x = y) (q : y = z) :

whiskerR 1 q = 1 :> (p @ q = p @ q)

match q with idpath => 1 end.

Definition whiskerL_p1 A : Type x y z : A (p : x = y) (q : y = z) :

whiskerL p 1 = 1 :> (p @ q = p @ q)

match q with idpath => 1 end.

Definition whiskerL_1p A : Type x y : A p q : x = y (h : p = q) :

(concat_1p p) ^ @ whiskerL 1 h @ concat_1p q = h

match h with idpath =>

match p with idpath =>

end end.

Definition concat2_p1 A : Type x y : A p q : x = y (h : p = q) :

h @@ 1 = whiskerR h 1 :> (p @ 1 = q @ 1)

match h with idpath => 1 end.

Definition concat2_1p A : Type x y : A p q : x = y (h : p = q) :

1 @@ h = whiskerL 1 h :> (1 @ p = 1 @ q)

match h with idpath => 1 end.

(** The interchange law for concatenation. *)

Definition concat_concat2 A : Type x y z : A p p’ p’’ : x = y q q’ q’’ : y = z

(a : p = p’) (b : p’ = p’’) (c : q = q’) (d : q’ = q’’) :

(a @@ c) @ (b @@ d) = (a @ b) @@ (c @ d).

Proof.

case d.

case c.

case b.

case a.

reflexivity.

Defined.

(** The interchange law for whiskering. Special case of [concat_concat2]. *)

Definition concat_whisker A x y z : A (p p’ : x = y) (q q’ : y = z) (a : p = p’) (b : q = q’) :

(whiskerR a q) @ (whiskerL p’ b) = (whiskerL p b) @ (whiskerR a q’)

match b with

idpath =>

match a with idpath =>

(concat_1p _)^

(** Structure corresponding to the coherence equations of a bicategory. *)

(** The "pentagonator": the 3-cell witnessing the associativity pentagon. *)

Definition pentagon A : Type v w x y z : A (p : v = w) (q : w = x) (r : x = y) (s : y = z)

: whiskerL p (concat_p_pp q r s)

@ concat_p_pp p (q@r) s

@ whiskerR (concat_p_pp p q r) s

= concat_p_pp p q (r@s) @ concat_p_pp (p@q) r s.

Proof.

case p, q, r, s. reflexivity.

Defined.

(** The 3-cell witnessing the left unit triangle. *)

Definition triangulator A : Type x y z : A (p : x = y) (q : y = z)

: concat_p_pp p 1 q @ whiskerR (concat_p1 p) q

= whiskerL p (concat_1p q).

Proof.

case p, q. reflexivity.

Defined.

(** The Eckmann-Hilton argument *)

Definition eckmann_hilton A : Type x:A (p q : 1 = 1 :> (x = x)) : p @ q = q @ p :=

(whiskerR_p1 p @@ whiskerL_1p q)^

@ (concat_p1 _ @@ concat_p1 _)

@ (concat_1p _ @@ concat_1p _)

@ (concat_whisker _ _ _ _ p q)

@ (concat_1p _ @@ concat_1p _)^

@ (concat_p1 _ @@ concat_p1 _)^

@ (whiskerL_1p q @@ whiskerR_p1 p).

2. Homotopy type theory: Conclusion

I Type theory provides a logic of homotopy.

I Logical methods can in principle capture a lot of homotopytheory: e.g. the fundamental ∞-groupoid of a space is alogical construction, and the notion of n-type is logicallydefinable.

I Many results have already been formalized: homotopy groupsof spheres, Freudenthal suspension theorem, Postnokovtowers, Eilenberg–Mac Lane spaces, ...

I Other areas are also being developed:

I Elementary mathematics: basic algebra, real numbers, cardinalarithmetic, ...

I Foundations: quotient types, inductive types, cumulativehierarchy of sets, ...

I Some new logical ideas are suggested by the homotopyinterpretation: Higher inductive types, Univalence axiom.

3. Higher-dimensional inductive types

(Due to Lumsdaine & Shulman, but very much work in progress.)

The natural numbers N are implemented in type theory as aninductively defined structure of type:

o : Ns : N→ N

The recursion property is captured by an elimination rule:

a : X f : X → Xrec(a, f ) : N→ X

such that:

rec(a, f )(o) = a

rec(a, f )(sn) = f (rec(a, f )(n))

o : Ns : N→ N

a : X f : X → Xrec(a, f ) : N→ X

such that:

rec(a, f )(o) = a

o : Ns : N→ N

a : X f : X → Xrec(a, f ) : N→ X

such that:

rec(a, f )(o) = a

o : Ns : N→ N

a : X f : X → Xrec(a, f ) : N→ X

such that:

rec(a, f )(o) = a

Higher-dimensional inductive types

This says just that (N, o, s) is the free structure of this type:

Ns 88 rec

// X fff

The map rec(a, f ) : N→ X is unique with this property.

Higher-dimensional inductive types

This says just that (N, o, s) is the free structure of this type:

Ns 88 rec

// X fff

The map rec(a, f ) : N→ X is unique with this property.

Higher-dimensional inductive types: The circle S1

The homotopical circle S = S1 can also be given as an inductivetype, now involving a higher-dimensional generator:

base : Sloop : base base

Here we have written loop : base base for the “loop”loop : IdS(base, base).

There is an associated recursion property, captured again by anelimination rule:

a : X q : a a

rec(a, q) : S→ X

such that:

rec(a, q)(base) = a

rec(a, q)(loop) = q

The map rec(a, q) acts on paths via the Id-elimination rule.

There is an associated recursion property, captured again by anelimination rule:

a : X q : a a

rec(a, q) : S→ X

such that:

rec(a, q)(base) = a

rec(a, q)(loop) = q

The map rec(a, q) acts on paths via the Id-elimination rule.

This says that (S, base, loop) is the free structure of this (higher)type:

baseloop 88 S rec

// X a qdd

The map rec(a, q) : S→ X is then unique up to homotopy.

This says that (S, base, loop) is the free structure of this (higher)type:

baseloop 88 S rec

// X a qdd

The map rec(a, q) : S→ X is then unique up to homotopy.

Here is a sanity check:

Theorem (Shulmann 2011)

The type-theoretic circle S has the correct homotopy groups:

πn(S) =

Z, if n = 1,

0, if n 6= 1.

The proof has been formalized in Coq. It combines classicalhomotopy theory with methods from constructive type theory, anduses Voevodsky’s Univalence Axiom.

Corollary

There are homotopy 1-types.

πn(S) =

Z, if n = 1,

0, if n 6= 1.

Corollary

πn(S) =

Z, if n = 1,

0, if n 6= 1.

Corollary

Higher-dimensional inductive types: The interval I

The unit interval I = [0, 1] is also an inductive type, on the data:

0, 1 : Ip : 0 1

Again writing p : 0 1 for the path p : IdI(0, 1).

Slogan: In topology, we start with the interval and use it todefine the notion of a path.In HoTT, we start with the notion of a path, and use it to definethe interval.

0, 1 : Ip : 0 1

Slogan: In topology, we start with the interval and use it todefine the notion of a path.

In HoTT, we start with the notion of a path, and use it to definethe interval.

0, 1 : Ip : 0 1

Slogan: In topology, we start with the interval and use it todefine the notion of a path.In HoTT, we start with the notion of a path, and use it to definethe interval.

3. HITs: Impredicative construction

We can make (some) HITs by “impredicative encoding”.Recall the following examples:

p ∨ q := ∀x[(p ⇒ x) ∧ (q ⇒ x)⇒ x

]The quantifier is over all propositions x .

A + B :=∏X

[(A→ X )× (B → X )→ X

]The product is over all sets X .This really requires an additional condition saying that thetransformations

αX :((A→ X )× (B → X )

are natural in X .

p ∨ q := ∀x[(p ⇒ x) ∧ (q ⇒ x)⇒ x

A + B :=∏X

[(A→ X )× (B → X )→ X

]The product is over all sets X .

This really requires an additional condition saying that thetransformations

αX :((A→ X )× (B → X )

are natural in X .

p ∨ q := ∀x[(p ⇒ x) ∧ (q ⇒ x)⇒ x

A + B :=∏X

[(A→ X )× (B → X )→ X

]The product is over all sets X .This really requires an additional condition saying that thetransformations

αX :((A→ X )× (B → X )

are natural in X .

The same idea can be used for the interval and the circle.For a type X , define the path space I (X ) and loop space Ω(X ) by:

I (X ) :=∑x ,y :X

IdX (x , y) Ω(X ) :=∑x :X

IdX (x , x)

Morally, I (X ) = I→ X and Ω(X ) = S→ X .So we set:

I :=∏X

[I (X )→ X

[Ω(X )→ X

]Here the product is taken over all 1-types X , and the requirednaturality condition includes a further coherence condition.

I (X ) :=∑x ,y :X

IdX (x , y) Ω(X ) :=∑x :X

IdX (x , x)

Morally, I (X ) = I→ X and Ω(X ) = S→ X .

So we set:

I :=∏X

[I (X )→ X

[Ω(X )→ X

I (X ) :=∑x ,y :X

IdX (x , y) Ω(X ) :=∑x :X

IdX (x , x)

I :=∏X

[I (X )→ X

[Ω(X )→ X

Here the product is taken over all 1-types X , and the requirednaturality condition includes a further coherence condition.

I (X ) :=∑x ,y :X

IdX (x , y) Ω(X ) :=∑x :X

IdX (x , x)

I :=∏X

[I (X )→ X

[Ω(X )→ X

3. Higher-dimensional inductive types: Conclusion

Many basic spaces and constructions can be introduced as HITs:

I higher spheres Sn, cylinders, tori, cell complexes, . . . ,

I suspensions ΣA,

I homotopy pullbacks, pushouts, etc.,

I truncations, such as connected components π0(A) and“bracket” types,

I quotients by equivalence relations,

I higher homotopy groups,

I Quillen model structure.

Some of these have impredicative constructions, some do not.A general theory is the topic of current research.

3. Higher-dimensional inductive types: Conclusion

Many basic spaces and constructions can be introduced as HITs:

I higher spheres Sn, cylinders, tori, cell complexes, . . . ,

I suspensions ΣA,

I homotopy pullbacks, pushouts, etc.,

I truncations, such as connected components π0(A) and“bracket” types,

I quotients by equivalence relations,

I higher homotopy groups,

I Quillen model structure.

Some of these have impredicative constructions, some do not.A general theory is the topic of current research.

4. Univalence

Voevodsky has proposed a new foundational axiom to be added toHoTT: the Univalence Axiom.

I It captures the informal mathematical practice of identifyingisomorphic objects.

I It is formally incompatible with conventional foundations.

I It is formally consistent with homotopy type theory.

I It has has powerful consequences, especially when combinedwith HITs.

4. Univalence

Isomorphism

In type theory, the notion of type isomorphism A ∼= B is definableas usual:

A ∼= B ⇔ there are f : A→ B and g : B → Asuch that gfx = x and fgy = y .

Formally, there is a type of isomorphisms:

Iso(A,B) :=∑

f :A→B

∑g :B→A

(∏x :A

IdA(gfx , x)×∏y :B

IdB(fgy , y))

We say that A ∼= B if this type is “inhabited” by a closed term –which is then an isomorphism between A and B.

Isomorphism

Iso(A,B) :=∑

f :A→B

∑g :B→A

(∏x :A

IdB(fgy , y))

Isomorphism

Iso(A,B) :=∑

f :A→B

∑g :B→A

(∏x :A

IdB(fgy , y))

Isomorphism: Remarks

Iso(A,B) :=∑

f :A→B

∑g :B→A

(∏x :A

IdB(fgy , y))

I It is convenient to add a “coherence” condition relating theproofs of gfx = x and fgy = y .

I Under the homotopy interpretation, this then becomes thetype of homotopy equivalences.

I The same notion also subsumes logical equivalence,isomorphism of sets, and categorical equivalence.

Iso(A,B) :=∑

f :A→B

∑g :B→A

(∏x :A

IdB(fgy , y))

Iso(A,B) :=∑

f :A→B

∑g :B→A

(∏x :A

IdB(fgy , y))

Invariance

One can show that in type theory without universes, all definableproperties P(X ) of types X respect type isomorphism, in the sensethat the following inference holds:

A ∼= B P(A)

In this sense, all definable properties are invariant.

So type isomorphism A ∼= B is equivalent to indiscernability:

P(A)⇒ P(B), for all P

How are these conditions related to identity of types A and B ina system with universes?

Invariance

A ∼= B P(A)

Invariance

A ∼= B P(A)

Invariance

A ∼= B P(A)

Universes

In order to reason directly about identity of types, we need auniverse U , which then has an identity type:

IdU (A,B)

One can then easily construct a “comparison map” of types:

IdU (A,B)→ Iso(A,B).

So, of course, identity implies isomorphism.

Universes

IdU (A,B)

Universes

IdU (A,B)

The Univalence Axiom

Voevodsky’s Univalence Axiom asserts that this comparison map isitself an isomorphism:

IdU (A,B)∼−→ Iso(A,B) (UA)

So UA can be stated: “Identity is isomorphic to isomorphism.”

The Univalence Axiom

Voevodsky’s Univalence Axiom asserts that this comparison map isitself an isomorphism:

IdU (A,B)∼−→ Iso(A,B) (UA)

So UA can be stated: “Identity is isomorphic to isomorphism.”

The Univalence Axiom: How it works

Definition (Universal Cover of S1)

Define a dependent type cov : S // U by circle-recursion, with

cov(base) := Zcov(loop) := ua(succ).

The Univalence Axiom: How it worksTo define a dependent type on S1,

cov : S1 −→ U

we need the following data:

I a point A : UI a loop p : A A

For the point A we take the integers Z.By Univalence, to give a loop p : Z Z in U , it suffices to give anisomorphism Z ∼= Z, for which we choose the successor function

succ : Z ∼= Z.

Now we can set:

cov : S1 −→ U

For the point A we take the integers Z.

By Univalence, to give a loop p : Z Z in U , it suffices to give anisomorphism Z ∼= Z, for which we choose the successor function

succ : Z ∼= Z.

Now we can set:

cov : S1 −→ U

succ : Z ∼= Z.

Now we can set:

cov : S1 −→ U

succ : Z ∼= Z.

Now we can set:

The Univalence Axiom: Remarks

I Since UA is an iso, there is a map coming back:

IdU (A,B)←− Iso(A,B)

So isomorphic objects are identical.

I The system with UA still has the invariance property:

A ∼= B P(A)

In the presence of a universe, UA is equivalent to invariance.

I Incompatible with conventional foundations in set theory, butconsistent with HoTT: Voevodsky has a model in SSets.

I Conjecture (Voevodsky): UA preserves the computationalcharacter of the total system.

A ∼= B P(A)

Conclusion

Homotopy Type Theory is a topological interpretation ofconstructive type theory that allows purely formal reasoning inhomotopy theory.

Univalent Foundations is a new approach to the foundations ofmathematics based on Homotopy Type Theory, with both intrinsicgeometric content and a computational implementation.

The Univalence Axiom is a powerful new principle of reasoningthat is incompatible with conventional foundations, and yet(conjecture!) computationally admissible.

Conclusion

References and Further Information

General information:

www.HomotopyTypeTheory.org

Current state of the Univalent Foundations Program:

uf-ias-2012.wikispaces.com

Forthcoming book:

Homotopy type theory:Univalent foundations of mathematics

Homotopy Type TheoryUNIVALENT FOUNDATIONS OF MATHEMATICS

The Univalent Foundations Program

Institute for Advanced Study

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