A naive view of Homotopy Type Theoryand its relation to the Calculus of Constructions

Yves BertotInria

2014

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Overview

I Conjunction between type theory and homotopy theoryI Intensional type theory

I Proofs as first-class objectsI Inductive presentation of equality

I Paths in homotopy theoryI A precise structure as higher categoryI Abstract view: synthetic homotopy theoryI Need for more axioms: not the last word?

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Elements of context

I Growing complexity in mathematicsI The odd-order theorem: 250 pagesI discontent among leading mathematicians, V. Voevodsky, T.

Hales

I Computer verification of proofsI One attractive approach: Curry-Howard Isomorphism

I A proposal for a new foundation of mathematics

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Formalized mathematics in a few words

I Computer language to represent objects of mathematicsI numbers, figuresI logical statementsI proofs

I Verification algorithmsI Verifying that formulas are well-formedI Verifying that proofs respect the rules of logic

I Variety of choicesI LCF style: proofs are not in the same worldI Intensional Type Theory: proofs are algorithms

New unusual questions

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Logical reading of types

I Thanks to de Bruijn, Curry, Howard, Martin-LofI Implication behaves pretty much like a function type

I a proof of A⇒ B constructs proofs of B when given proofs ofA

I Use the function type A -> B to represent implicationI Requires that all functions are total

I Data constructors as logical connectivesI The statement proved by the pair of two proofs of A and B

that means A ∧ B

I A logic inherent to functional programmingI One type-checker to rule them all

I Remember: Propositions are types

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Uses of proofs as first-class citizens

I With dependent types and pairs: an approach to subsetsI Using proof as certificatesI E.g. : (n,P) is a ‘certified prime number’

if n is a number and P a proof that it is prime

I Link to computationI The proof language is itself a programming languageI Possibility to “extract” algorithms

I Raising new questionsI if (a, p) and (a, p′) are two “certified values”, are they equal?I Are proofs relevant?I Important question when equality proofs

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What is this thing called Equality

I A family of equality types: for every x y : A, x = y is a type

I Described as an inductive type with a single constructornamed refl

I Induction principle illuminatingeq rect:

∀A : Type.∀x : A.∀P : A→ Prop.P(x)⇒ ∀y : A. x = y ⇒ P(y)

I If x = y then every property satisfied by x is also satisfied by yI x and y are undistinguishableI Not reallyI Remember: the induction principle is named eq rectI Computing behavior: eq rect refl u computes to u

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Not really interchangeable

I In context

x, y : A ; B : A -> Type; a : B y; H : x = y

if u has type B y, then u = a is well typedif u has type B x, then u = a is not

I but eq rect H u = a is well typedI Do we know eq rect H does not modify u?

I Solution proposed long ago: add an axiom Unicity of IdentityProofs (Hoffmann&Streicher98)

I The story could end here!I If proofs are unique then all are equal to reflexivity proofsI eq rect computes nicely on reflexivity proofs

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Questioning “undistinguishable”

I In the induction principle for equalityeq rect:

∀A : Type.∀x : A.∀P : A→ Prop.P(x)⇒ ∀y : A.x = y ⇒ P(y)

I the range of P is limited to functions definable in the language

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Homotopic perspective

I Consider the model where:I Types are (equivalence classes of) topological spaces

collections of points with a notion of proximity or continuityI Functions are equivalence classes of continuous functionsI Two points are equal if there is a continuous path between

them

I equality types are types, they have their own equality types,etc.

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Paths of several dimensions

I Paths between points are lines

I Paths between lines are surfaces, etc.

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contracting paths

The space of all paths with one given extremity can be reduced toa singleton.

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Contractibility

I A topological space is contractible whenI There exists a point a in this space (center of contraction)I There exists a continuous function mapping any other point x

to a path from x to a

I A circle is not contractible

I For any point the space of paths from this point is contractible

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Introduction to Groupoids

I A special kind of category where every morphism is anisomorphism

I The abstract description of equivalence relationsI for p, p1, p2, p3 proofs of equivalence:

I p, p1 proof that x ∼ y , p2 proof that y ∼ z , p3 proof thatz ∼ t

I The equivalence relation’s transitivity is noted •I p1 • p2 is a proof that x ∼ z

I The equivalence relation’s symmetry is noted −1

I p−1 is a proof that y ∼ x

I The equivalence relation’s reflexivity is an identity element forevery x , noted 1x or 1

I With a few more coherence laws

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Groupoid structure and ∞-groupoid

I Coherence lawsI associativity • : (p1 • p2) • p3 p1 • (p2 • p3) are the same (1)I neutral property of 1 : 1 • p, p • 1, and p are the same (2,3)I The symetry property maps every arrow to its inverse p • p−1

is the same as 1 (4)I A bit like a group, but • is partialI A groupoid is a category where every morphism is an

isomorphism

I A higher groupoid: when the equalities of (1,2,3,4) are statedas proofs of equivalence in another groupoid

I ∞-groupoid when proofs of equivalence are used all the wayup

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Unification: homotopy type theory

I In homotopy theory, paths form an ∞-groupoid

I In type theory, proofs of equality form an ∞-groupoid

I New correspondence: types are topological spaces

I Many proofs of homotopy theory can be modeled directly intype theory

I Topological structures can be described by adding new pathsaround points

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Examples of topological spaces

I The interval: two points and a path between them,I The circle

I one point and a path from it to itself (distinct from thereflexivity path),

I two points and two paths between them,

I The sphereI two points (north and south) and, for every point in the circle

(equator), a path (meridian) from north to south,I one base point and a path between the reflexivity path at the

base and itself

I types of paths are themselves topological spaces

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Generic topological operations

I Suspension of A: two points (north and south) and for everyelement in A a path from north to south

I bool is the suspension of the empty set,I the circle is the suspension of boolI the sphere is the suspension of the circleI the 3-sphere is the suspension of the sphereI the n + 1-sphere is the suspension of the n sphere

I Truncation operation: force all path at a certain level to beequal

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Homotopy equivalence

I In a structure, reduce any path to a constant pathI Only if the two extremities are distinct

I Example: 2 presentations of the circle

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Should you be afraid?

I Homotopy type theory looks at the microscopical levelI Continuity in real numbers lives at another level

I It distinguishes types where all points have trivial path spacesI Called hsetsI For them UIP holds

I Hedberg theorem: datatypes with decidable equality are hsetsI For instance: nat

I Also distinguishes types with at most one elementI For them proof-irrelevance holds

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What are the gains?

I Synthetic homotopy theoryI Basic homotopy theory directly in type theoryI Already done proofs about loop spaces, e.g.

I Understanding of quotientsI A systematic approach to adding equalities

I Advantages of hsets already exploited in ssreflectI Homotopy type theory generalizes beyond decidable equality

I The Univalence axiom (isomorphic types are equal)I Functional extensionalityI Propositional extensionality

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Questions not discussed in this talk

I universe polymorphism: resizing rulesI Higher inductive types

I Adding explicit paths between elements in types (higherconstructors)

I Avoid inconsistenciesI Consider higher constructors in computations

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