categorical homotopy type theory · the emergence of homotopy type theory gestation: irussell:...
TRANSCRIPT
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Categorical Homotopy Type Theory
Andre Joyal
UQAM
MIT Topology Seminar, March 17, 2014
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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
`∑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
`∏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 )
Let C be a ML-tribe.
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 )
Let C be a ML-tribe.
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 )
Let C be a ML-tribe.
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 )
Let C be a ML-tribe.
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
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
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
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
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
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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![Page 78: 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](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d0ef77e708231d4387916/html5/thumbnails/78.jpg)
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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![Page 79: 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](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d0ef77e708231d4387916/html5/thumbnails/79.jpg)
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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![Page 80: 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](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d0ef77e708231d4387916/html5/thumbnails/80.jpg)
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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![Page 81: 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](https://reader033.vdocument.in/reader033/viewer/2022060320/5f0d0ef77e708231d4387916/html5/thumbnails/81.jpg)
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