Categorical Homotopy Type Theory · The emergence of Homotopy Type Theory Gestation: IRussell: Mathematical logic based on the theory of types (1908) IChurch: A formulation of the

Categorical Homotopy Type Theory

Andre Joyal

UQAM

MIT Topology Seminar, March 17, 2014

Warning

The present slides include corrections and modifications that weremade during the week following my talk. Thanks to Steve Awodey,David Spivak, Thierry Coquand, Nicola Gambino, and MichaelShulman.

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The emergence of Homotopy Type Theory

Gestation:

I Russell: Mathematical logic based on the theory of types(1908)

I Church: A formulation of the simple theory of types (1940)

I Lawvere: Equality in hyperdoctrines and comprehensionschema as an adjoint functor (1968)

I Martin-Lof: Intuitionistic theory of types (1971, 1975, 1984)

I Hofmann, Streicher: The groupoid interpretation of typetheory (1995)

Birth:

I Awodey, Warren: Homotopy theoretic models of identitytypes (2006∼2007)

I Voevodsky: Notes on type systems (2006∼2009)

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The emergence of Homotopy Type Theory

Gestation:

I Russell: Mathematical logic based on the theory of types(1908)

I Church: A formulation of the simple theory of types (1940)

I Lawvere: Equality in hyperdoctrines and comprehensionschema as an adjoint functor (1968)

I Martin-Lof: Intuitionistic theory of types (1971, 1975, 1984)

I Hofmann, Streicher: The groupoid interpretation of typetheory (1995)

Birth:

I Awodey, Warren: Homotopy theoretic models of identitytypes (2006∼2007)

I Voevodsky: Notes on type systems (2006∼2009)

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Suggested readings

Recent work in homotopy type theorySlides of a talk by Steve Awodey at the AMS meeting January 2014

Notes on homotopy λ-calculusVladimir Voevodsky

Homotopy Type TheoryA book by the participants to the Univalent Foundation Programheld at the IAS in 2012-13

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Axiomatic Homotopy Theory

Henry Whitehead (1950):The ultimate aim of algebraic homotopy is to construct a purelyalgebraic theory, which is equivalent to homotopy theory in thesame sort of way that analytic is equivalent to pure projectivegeometry.

Examples of axiomatic systems

I Triangulated categories (Verdier 1963);

I Homotopical algebra (Quillen 1967);

I Homotopy theories (Heller 1988)

I Theory of derivators (Grothendieck 198?)

I Homotopy type theory

I Elementary higher topos?

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Axiomatic Homotopy Theory

Henry Whitehead (1950):The ultimate aim of algebraic homotopy is to construct a purelyalgebraic theory, which is equivalent to homotopy theory in thesame sort of way that analytic is equivalent to pure projectivegeometry.

Examples of axiomatic systems

I Triangulated categories (Verdier 1963);

I Homotopical algebra (Quillen 1967);

I Homotopy theories (Heller 1988)

I Theory of derivators (Grothendieck 198?)

I Homotopy type theory

I Elementary higher topos?

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Some features of Hott

Hott replaces

I sets by spaces,

I isomorphisms by equivalences,

I proofs of equality x = y by paths x y ,

I the relation x = y by the homotopy relation x ∼ y ,

I equivalences X ' Y by paths X Y .

The formal system of Hott is decidable in a precise way.

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Potential applications

I to constructive mathematics,

I to proof verification and proof assistant,

I to homotopy theory.

A wish list:

I to higher topos theory,

I higher category theory,

I derived algebraic geometry.

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Category theory as a bridge

Category theory

pp ,,Type theory

22

$$

Homotopy theory

mm

yyHott

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Overview of the talk

tribe

**ttπ − tribe

**

h − tribe

ttMartin-Lof tribe

Voevodsky tribe

Elementary higher topos?

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Quadrable objects and maps

An object X of a category C is quadrable if the cartesian productA× X exists for every object A ∈ C.

A map p : X → B is quadrable if the object (X , p) of thecategory C/B is quadrable. This means that the pullback square

A×B X

p1

p2 // X

p

A

f // B

exists for every map f : A→ B.

The projection p1 is called the base change of p : X → B alongf : A→ B.

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Tribes

Let C be a category with terminal object ?.

DefinitionA tribe structure on C is a class of maps F ⊆ C satisfying thefollowing conditions:

I F contains the isomorphisms and is closed under composition;

I every map in F is quadrable and F is closed under basechanges;

I the map X → ? belongs to F for every object X ∈ C.

A tribe is a category C with terminal object equipped with a tribestructure F . A map in F is called a fibration.

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Examples of tribes

I A category with finite products, if the fibrations are theprojections;

I The category of small groupoids Grpd if the fibrations are theiso-fibrations;

I The category of Kan complexes Kan if the fibrations are theKan fibrations;

I The category of fibrant objects of a Quillen model category.

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Types and terms

An object E of a tribe C is called a type. Notation:

` E : Type

A map t : ?→ E in C is called a term of type E . Notation:

` t : E

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Fibrations and families

The fiber E (a) of a fibration p : E → A at a point a : A is definedby the pullback square

E (a)

// E

p

?

a // A.

A fibration p : E → A is a family (E (x) : x ∈ A) of objects of Cparametrized by a variable element x ∈ A.

A tribe is a collection of families closed under certain operations.

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The local tribe C(A)

For an object A of a tribe C.

The local tribe C(A) is the full sub-category of C/A whose objects(E , p) are the fibrations p : E → A with codomain A.

A map f : (E , p)→ (F , q) in C(A) is a fibration if the mapf : E → F is a fibration in C.

An object (E , p) of C(A) is a dependent type in context x : A.

x : A ` E (x) : Type

A section t of p : E → A is called a dependent term t(x) : E (x)

x : A ` t(x) : E (x)

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General contexts

Type declarations can be iterated:

A : Type

x : A ` B(x) : Type

x : A, y : B(x) ` C (x , y) : Type

x : A, y : B(x), z : C (x , y) ` E (x , y , z) : Type

E

A Boo Coo

Γ = (x : A, y : B(x), z : C (x , y)) is an example of general context.

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The syntactic category

An object of the syntactic category is a formal expression [Γ] whereΓ is a (general) context.

A map f : [x : A]→ [y : B] is a term

x : A ` f (x) : B

Two maps f , g : [x : A]→ [y : B] are equal if f (x) = g(x) can beproved in context x : A,

x : A ` f (x) = g(x) : B

Composition of maps is defined by substituting:

x : A ` f (x) : B, y : B ` g(y) : C

x : A ` g(f (x)) : C

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Homomorphism of tribes

A homomorphism of tribes is a functor F : C → D which

I takes fibrations to fibrations;

I preserves base changes of fibrations;

I preserves terminal objects.

Remark: The category of tribes is a 2-category, where a 1-cell is ahomomorphism and 2-cell is a natural transformation.

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Base change=change of parameters

If f : A→ B is a map in a tribe C, then the base change functor

f ? : C(B)→ C(A)

is a homomorphism of tribes.

In type theory, it is expressed by the following deduction rule

y : B ` E (y) : Type

x : A ` E (f (x)) : Type.

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Restriction of context

Let A be an object of a tribe C.

The base change functor iA : C → C(A) along the map A→ ? is ahomomorphism of tribes.

By definition iA(E ) = (E × A, p2).

The functor iA : C → C(A) is expressed in type theory by adeduction rule called context weakening:

` E : Type

x : A ` E : Type.

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Free extension

The extension iA : C → C(A) is freely generated by a term xA oftype A.

An analogy:

Recall that if R is a commutative ring, then the polynomialextension i : R → R[x ] is freely generated by the element x . Thefreeness means that for every homomorphism f : R → S and everyelement s ∈ S , there exists a unique homomorphism h : R[x ]→ Ssuch that hi = f and h(x) = s,

Ri //

f''

R[x ]

h

S

The element x ∈ R[x ] can be assigned any value. It is generic.

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Generic terms

The functor i = iA : C → C(A) takes the object A to the objecti(A) = (A× A, p2).

The diagonal δA : A→ A× A is a map δA : ?A → i(A) in C(A); itis thus a term δA : i(A).

TheoremThe extension i : C → C(A) is freely generated by the termδA : i(A). Thus, C(A) = C[xA] with xA = δA.

Hence the diagonal δA : i(A) is a generic term.

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Total space and summation

The forgetful functor C(A)→ C associates to a fibration p : E → Aits total space E =

∑x :A E (x). It is thus a summation operation,

ΣA : C(A)→ C.

It leads to the Σ-formation rule,


`∑x :A

E (x) : Type

A term t :∑

x :A E (x) is a pair t = (a, u), where a : A and u : E (a).

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Display maps

The projection

pr1 :∑x :A

E (x)→ A

is called a display map.

The syntactic category of type theory is a tribe, where a fibrationis a map isomorphic to a display map

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Push-forward

If f : A→ B is a fibration in a tribe C, then the push-forwardfunctor

f! : C(A)→ C(B)

is defined by putting f!(E , p) = (E , fp).

The functor f! is left adjoint to the pullback functorf ? : C(B)→ C(A).

Formally, we have

f!(E )(y) =∑

f (x)=y

E (x).

for a term y : B.

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Function space [A,B]

Our goal is now to introduce the notion of π-tribe.

Let A be a quadrable object in a category C.

Recall that the exponential of an object B ∈ C by A is an object[A,B] equipped with a map ε : [A,B]× A→ B called theevaluation such that for every object C ∈ C and every mapu : C × A→ B, there exists a unique map v : C → [A,B] suchthat ε(v × A) = u.

[A,B]× A

ε

C × A

v×A88

u // B

We write v = λA(u).

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Space of sections

Let A be a quadrable object in a category C.

The space of sections of an object E = (E , p) ∈ C/A is an objectΠA(E ) ∈ C equipped with a map ε : ΠA(E )× A→ E called theevaluation such that:

I pε = p2

I for every object C ∈ C and every map u : C × A→ E in C/Athere exists a unique map v : C → ΠA(E ) such thatε(v × A) = u.

ΠA(E )× A

ε

C × A

v×A88

u // E

We write v = λA(u).

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Products along a map

Let f : A→ B be a quadrable map in a category C.

The product Πf (E ) of an object E = (E , p) ∈ C/A along a mapf : A→ B is the space of sections of the map (E , fp)→ (A, f ) inthe category C/B,

E

p

Πf (E )

A

f //// B

For every y : B we have

Πf (E )(y) =∏

f (x)=y

E (x)

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π-tribes

DefinitionWe say that a tribe C is π-closed, and that it is a π-tribe, if everyfibration E → A has a product along any fibration f : A→ B andif the structure map Πf (E )→ B is a fibration,

The functor Πf : C(A)→ C(B) is right adjoint to the functorf ? : C(B)→ C(A).

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Examples of π-tribes

I A cartesian closed category, where a fibration is a projection;

I A locally cartesian category is a Π-tribe in which every map isa fibration;

I The category of small groupoids Grpd, where a fibration is aniso-fibration (Hofmann, Streicher);

I The category of Kan complexes Kan, where a fibrations is aKan fibration (Streicher, Voevodsky);

If C is a π-tribe, then so is the tribe C(A) for every object A ∈ C.

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Π-formation rule

In a Π-tribe, we have the following Π-formation rule:


`∏x :A

E (x) : Type.

There is also a rule for the introduction of λ-terms:

x : A ` t(x) : E (x)

` (λx)t(x) :∏x :A

E (x)

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Homotopical tribes

DefinitionWe say that a map u : A→ B in a tribe C is anodyne if it has theleft lifting property with respect to every fibration f : X → Y .

This means that every commutative square

A

u

a // X

f

Bb // Y

has a diagonal filler d : B → X ( du = a and fd = b).

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Homotopical tribes

DefinitionWe say that a tribe C is homotopical, or a h-tribe, if thefollowing two conditions are satisfied

I every map f : A→ B admits a factorization f = pu with u ananodyne map and p a fibration;

I the base change of an anodyne map along a fibration isanodyne.

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Examples of h-tribes

I The category of groupoids Grpd, where a functor is anodyneif it is a monic equivalence (Hofmann, Streicher);

I The category of Kan complexes Kan, where a map is anodyneif it is a monic homotopy equivalence (Streicher, Awodey andWarren, Voevodsky);

I The syntactic category of Martin-Lof type theory, where afibration is a map isomorphic to a display map (Gambino andGarner).

If C is a h-tribe, then so is the tribe C(A) for every object A ∈ C.

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Path object

A path object for an object A ∈ C is a factorisation of thediagonal ∆ : A→ A× A as an anodyne map r : A→ PA followedby a fibration (s, t) : PA→ A× A,

PA

(s,t)

A

r

==

∆ // A× A.

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Identity type

In Martin-Lof type theory, there is a type constructor whichassociates to every type A a dependent type

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

called the identity type of A,

A term p : IdA(x , y) is regarded as a proof that x = y .

There is a termx :A ` r(x) : IdA(x , x)

called the reflexivity term. It is a proof that x = x .

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The J-ruleThe identity type IdA is defined by putting

IdA =∑

(x ,y):A×A

IdA(x , y).

In type theory, there is an operation J which takes a commutativesquare

A

r

u // E

p

IdA IdA

with p a fibration, to a diagonal filler d = J(u, p)

A

r

u // E

p

IdA

d

==

IdA

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Identity type as a path object

Awodey and Warren: The J-rule shows that the reflexivity termr : A→ IdA is anodyne! Hence the identity type

IdA =∑

(x ,y):A×A

IdA(x , y)

is a path object for A,

IdA

〈s,t〉

A

r

==

∆ // A× A

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Mapping path space

The mapping path space P(f ) of a map f : A→ B is defined bythe pullback square

P(f )

p1

p2 // PB

s

Af // B.

This gives a factorization f = pu : A→ P(f )→ B withu = 〈1A, rf 〉 an anodyne map and p = tp2 a fibration.

The homotopy fiber of a map f : A→ B at a point y : B is thefiber of the fibration p : P(f )→ B at the same point,

fibf (y) =∑x :A

IdB(f (x), y).

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Homotopic mapsLet C be a h-tribe.

A homotopy h : f g between two maps f , g : A→ B in Cis a map h : A→ PB

B

A

g //

f//

h // PB

s

==

t

!!B

such that sh = f and th = g .

In type theory, h is regarded as a proof that f = g ,

x : A ` h(x) : IdB(f (x), g(x)).

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The homotopy category

Let C be a h-tribe.

TheoremThe homotopy relation f ∼ g is a congruence on the arrows of C.

The homotopy category Ho(C) is the quotient category C/ ∼.

A map f : X → Y in C is called a homotopy equivalence if it isinvertible in Ho(C).

Every anodyne map is a homotopy equivalence.

An object X is contractible if the map X → ? is a homotopyequivalence.

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Local homotopy categories

A map f : (E , p)→ (F , q) in C/A is called a weak equivalence ifthe map f : E → F is a homotopy equivalence in C.

The local homotopy category Ho(C/A) is defined to be thecategory of fraction

Ho(C/A) = W−1A (C/A)

where WA is the class of weak equivalences in C/A.

The inclusion C(A)→ C/A induces an equivalence of categories:

Ho(C(A)) = Ho(C/A)

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Homotopy pullback

Recall that a squareA //

C

B // D

is called a homotopy pullback if the canonical map A→ B ×hD C is

a homotopy equivalence, where B ×hD C = (f × g)?(PD)

B ×hD C //

PD

B × C

f×g // D × D

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h-propositions

A map u : A→ B is homotopy monic if the square

A1A //

1A

A

u

Au // B

is homotopy pullback.

DefinitionAn object A ∈ C is a h-proposition if the map A→ ? is homotopymonic.

An object A is a h-proposition if and only if the diagonalA→ A× A is a homotopy equivalence.

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n-types

The fibration 〈s, t〉 : PA→ A× A defines an object P(A) of thelocal tribe C(A× A).

An object A is

I a 0-type if P(A) is a h-proposition in C(A× A);

I a (n + 1)-type if P(A) is a n-type in C(A× A).

A 0-type is also called a h-set.

An object A is a h-set if the diagonal A→ A× A is homotopymonic.

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Homotopy initial objects

Let C be a h-tribe.

An object ⊥ ∈ C is homotopy initial if every fibration p : E → ⊥has a section σ : ⊥ → E ,

E

p⊥.

σ

]]

A homotopy initial object remains initial in the homotopy categoryHo(C).

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Homotopy coproducts

An object A t B equipped with a pair of maps i , j : A,B → A t B

such that for every fibration p : E → A t B and every pair of mapsf , g : A,B → E such that pf = i and pg = j ,

E

p

Ai//

f

<<

A t B Bj

oo

gbb

there exists a section σ : AtB → E such that σi = f and σj = g .

A homotopy coproduct remains a coproduct in the homotopycategory Ho(C).

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Homotopy natural number object

It is a homotopy initial object (N, s, 0) in the category of triples(X , f , a), for X ∈ C, f : X → X and a : X .

For every fibration p : X → N, such that pf = sp and p(a) = 0

?a // X

f //

p

X

p

?0 // N s // N

there exists a section σ : N→ X such that σs = f σ and σ(0) = a.

A homotopy natural number object (N, s, 0) is not necessarily anatural number object in the homotopy category Ho(C).

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Martin-Lof tribes

DefinitionA tribe is a πh-tribe if it is both a π-tribe and a h-tribe.

A πh-tribe C satisfies the axiom of function extensionality if theproduct functor

Πf : C(A)→ C(B)

preserves the homotopy relation for every fibration f : A→ B.

DefinitionA ML-tribe is a πh-tribe which satisfies the axiom of functionextensionality.

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Examples of ML-tribes

I The category of groupoids Grpd (Hofmann and Streicher);

I The category of Kan complexes Kan (Awodey and Warren,Voevodsky);

I The syntactic category of type theory with functionextensionality (Gambino and Garner).

If C is a ML-tribe, then so is the tribe C(A) for every A ∈ C.

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Elementary toposes

Let E be a category with finite limits

Recall that a monomorphism t : 1→ Ω in E is said to be universalif for every monomorphism S → A there exists a unique mapf : A→ Ω, such that f −1(t) = S ,

S //

1

t

Af // Ω

The pair (Ω, t) is called a sub-object classifier.

Lawvere and Tierney: An elementary topos is a locally cartesiancategory with a sub-object classifier (Ω, t).

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Small fibrations and universes

A class of small fibrations in a tribe C = (C,F) is a class of mapsF ′ ⊆ F which contains the isomorphisms and is closed undercomposition and base changes.

A small fibration q : U ′ → U is universal if for every smallfibration p : E → A there exists a cartesian square:

E //

p

U ′

q

A // U.

A universe is the codomain of a universal small fibration U ′ → U.

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Martin-Lof universes

A universe U ′ → U in a π-tribe C is π-closed if the product of asmall fibration along a small fibration is small.

A universe U ′ → U in a h-tribe C is h-closed if the path fibrationPA→ A× A can be chosen small for each object A.

A universe U ′ → U in πh-tribe C is a πh-closed if it is bothπ-closed and h-closed.

We may also say that πh-closed universe is a ML-universe.

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Decidability

A set S is decidable if the relations x ∈ S and the equality relationx = y for x , y ∈ S can be decided recursively.

I The set of natural numbers N is decidable;

I Not every finitely presented group is decidable (Post).

Martin-Lof’s theorem :The relations ` t : A and ` s = t : A aredecidable in type theory without function extensionality, but with aML-universe, with finite (homotopy) coproducts and (homotopy)natural numbers. Moreover, every globally defined term ` t : N isdefinitionaly equal to a numeral sn(0) : N.

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Homotopical pre-sheaves

Let C be a ML-tribe.

DefinitionA presheaf F : Cop → Set homotopical if it respects thehomotopy relation: f ∼ g ⇒ F (f ) = F (g).

A homotopical presheaf is the same thing as a functorF : Ho(C)op → Set.

A homotopical presheaf F is representable if the functorF : Ho(C)op → Set is representable.

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IsContr(X )


If E ∈ C, then the presheaf F : Cop → Set defined by putting

F (A) =

1, if EA is contractible in C(A)

∅ otherwise

is homotopical.

It is represented by the h-proposition

IsContr(E ) =def

∑x :E

∏y :E

IdE (x , y)

Compare with(∃x ∈ E ) (∀y ∈ E ) x = y

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IsEq(f )


If f : X → Y is a map in C, then the presheaf F : Cop → Setdefined by putting

F (A) =

1, if fA : XA → XA is an equivalence

∅ otherwise

is homotopical.

It is represented by the h-proposition

IsEq(f ) =def

∏y :Y

IsCont(fibf (y)),

where fibf (y) is the homotopy fiber of f at y : Y .

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Eq(X ,Y )


If X ,Y ∈ C, let us put

Eq(X ,Y ) =def

∑f :X→Y

IsEq(f )

For every object A ∈ C, there is a bijection between the maps

A→ Eq(X ,Y )

in Ho(C) and the isomorphism XA ' YA in Ho(C(A))

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EqA(E )


For every fibration p : E → A let us put

EqA(E ) =∑x :A

∑y :A

Eq(E (x),E (y))

This defines a fibration EqA(E )→ A× A.

The identity of E (x) is represented by a term

x : A ` u(x) : Eq(E (x),E (x))

which defines the unit map u : A→ EqA(E ),

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Univalent fibrations

Voevodsky:

DefinitionA fibration E → A is univalent if the unit map u : A→ EqA(E ) isa homotopy equivalence.

In which case the fibration EqA(E )→ A× A is equivalent to thepath fibration PA→ A× A.

PA

〈s,t〉 ""

' // EqA(E )

(s,t)yyA× A

Remark: The notion of univalent fibration can be defined in anyπh-tribe.

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Uncompressible fibrations

A Kan fibration is univalent if and only if it is uncompressible.

To compress a Kan fibration p : X → A is to find a homotopypullback square

X //

Y

A

f // B

in which f is homotopy surjective but not homotopy monic.

Every Kan fibration X → A is the pullback of an uncompressiblefibration X ′ → A′ along a homotopy surjection A→ A′. Moreover,the fibration X ′ → A′ is homotopy unique.

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Voevodsky tribes

Voevodsky: The tribe of Kan complexes Kan admits a univalentML-universe U ′ → U.

DefinitionA V-tribe is a πh-tribe C equipped with a univalent ML-universeU ′ → U.

Voevodsky’s theorem: A V -tribe satisfies function extensionality, itis thus a ML-tribe.

Voevodsky’s conjecture : The relations ` t : A and ` s = t : A aredecidable in V-type theory. Moreover, every globally defined term` t : N is definitionaly equal to a numeral sn(0) : N.

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What is an elementary higher topos?

Grothendieck topos Elementary topos

Higher topos EH-topos?

Rezk and Lurie:

DefinitionA higher topos is a locally presentable (∞, 1)-category with aclassifying universe U ′k → Uk for k-compact morphisms for eachregular cardinal k .

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What is an elementary higher topos?

We hope that the notion of elementary higher topos will emergeafter a period of experimentations with the axioms.

In principle, the notion could be formalized with any notion of(∞, 1)-category:

I a quasi-category;

I a complete Segal space;

I a Segal category;

I a simplicial category;

I a model category;

I a relative category.

A formalization may emerge from homotopy type theory.

Here we propose an axiomatization using the notion of generalizedmodel category (to be defined next).

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Generalised model categories

Let E be a category with terminal object > and initial object ⊥.

DefinitionA generalized model structure on E is a triple (C,W,F) of classesmaps in E such that

I every map in F is quadrable and every map in C isco-quadrable.

I W satisfies 3-for-2;

I the pairs (C ∩W,F) and (C,W ∩F) are weak factorizationsystems;

A generalized model category is a category E equipped with ageneralised model structure (C,W,F).

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A map in C is called a cofibration, a map in W is acyclic and a mapin F a fibration. An object A is cofibrant if the map ⊥ → A is acofibration, an object X is fibrant if the map X → > is a fibration.

A generalized model structure is right proper (resp. left proper) ifthe base (resp. cobase) change of a weak equivalence along afibration (resp. a cofibation) is a weak equivalence. A generalizedmodel structure is proper if it is both left and right proper.

A generalized model structure is smooth if every object iscofibrant, if it is right proper, if the base change of a cofibrationalong a fibration is a cofibration and if the product of a fibrationalong a fibration exists.

Remark: A smooth generalized model structure is proper and theproduct of a fibration along a fibration is a fibration.

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EH-topos?

Definition (β-version):An EH-topos is a smooth generalised model category E equippeda univalent ML-universe U ′ → U.

Examples:

I The category of simplicial sets sSet (Voevodsky);

I The category of simplicial presheaves over any elegant Reedycategory (Shulman).

I The category of symmetric cubical sets (Coquand).

I The category of presheaves over any elegant (local) testcategory (Cisinski).

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Critics

Critic 1: We may want a hierarchy of universes U0 : U1 : U2 : · · · .

Critic 2: We may want a fibrant-cofibrant natural number object N.

Critic 3: Every fibration should factor as a homotopy surjectionfollowed by a monic fibration.

Critic 4: The initial object should be strict.

Critic 5: The inclusions i1 : X → X t Y and i2 : Y → X t Yshould be fibration for every pair of objects (X ,Y ).

Critic 5’: The functor (i?1 , i?2 ) : E/(X t Y )→ E/X × E/Y should

be an equivalence of generalized model categories.

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More critics

Critic 6: If u : A→ B is a cofibration between fibrant objects andp : E → B is a fibration, then the map

u? : ΠB(E )→ ΠA(u?(E ))

induced by u should be a fibration. Moreover, u? should be acyclicwhen u is acyclic.

Critic 6’: Condition 6 should be true in every slice category E/C .

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Epilogue

What is mathematics?

Georg Cantor:”The essence of mathematics lies in its freedom”

Bertrand Russell:”Mathematics is the subject in which we never know what we aretalking about, nor whether what we are saying is true”

Godfrey H. Hardy:”Beauty is the first test; there is no permanent place in the worldfor ugly mathematics”

John von Neumann:”In mathematics you don’t understand things. You just get used tothem”

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Epilogue


Georg Cantor:

”The essence of mathematics lies in its freedom”




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Epilogue






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Epilogue



Bertrand Russell:

”Mathematics is the subject in which we never know what we aretalking about, nor whether what we are saying is true”



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Epilogue






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Epilogue




Godfrey H. Hardy:

”Beauty is the first test; there is no permanent place in the worldfor ugly mathematics”


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Epilogue






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Epilogue





John von Neumann:

”In mathematics you don’t understand things. You just get used tothem”

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Epilogue






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THANK YOU FOR YOUR ATTENTION!

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Documents

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