Sets in HoTT Egbert Rijke Bas Spitters Radboud University Nijmegen - - PowerPoint PPT Presentation

Introduction Regularity Quotients The object classifier

Sets in HoTT

Egbert Rijke Bas Spitters

Radboud University Nijmegen

April 23rd, 2013

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Challenges of current type theories

For finitary mathematics (Coq) type theory works very well. The extension to infinitary mathematics is challenging. No: quotients, functional extensionality, subset types... Univalence Axiom to the rescue!

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Setting

◮ Univalent foundations of mathematics.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Setting

◮ Univalent foundations of mathematics.

◮ We work with a univalent universe U. Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Setting

◮ Univalent foundations of mathematics.

◮ We work with a univalent universe U.

◮ Types in U have the structure of weak ∞-groupoids.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Setting

◮ Univalent foundations of mathematics.

◮ We work with a univalent universe U.

◮ Types in U have the structure of weak ∞-groupoids. ◮ Sets (Set) in U are types for which UIP is valid.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Setting

◮ Univalent foundations of mathematics.

◮ We work with a univalent universe U.

◮ Types in U have the structure of weak ∞-groupoids. ◮ Sets (Set) in U are types for which UIP is valid. ◮ Mere propositions (Prop) in U are types with proof irrelevance.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos? Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom? Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom?

◮ We will use ideas from

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom?

◮ We will use ideas from

◮ Algebraic Set Theory (Joyal, Moerdijk). Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Aim of the current work

◮ To understand the category Set in a univalent universe.

◮ Initial question: Is Set a predicative topos?

Do we have

◮ (Stable) image factorization, ◮ quotients, ◮ a (large) subobject classifier, ◮ an object classifier, ◮ the collection axiom?

◮ We will use ideas from

◮ Algebraic Set Theory (Joyal, Moerdijk). ◮ Higher category theory (Lurie, Rezk) Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Squash

◮ The inclusion of Prop in U has a left adjoint called

(−1)-truncation.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Squash

◮ The inclusion of Prop in U has a left adjoint called

(−1)-truncation. − −1 : U → Prop.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Squash

◮ The inclusion of Prop in U has a left adjoint called

(−1)-truncation. − −1 : U → Prop. With unit | − |−1 : A → A for every A : U.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Squash

◮ The inclusion of Prop in U has a left adjoint called

(−1)-truncation. − −1 : U → Prop. With unit | − |−1 : A → A for every A : U.

◮ The universal property of (−1)-truncation is that

◮ For every type A : U, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Squash

◮ The inclusion of Prop in U has a left adjoint called

(−1)-truncation. − −1 : U → Prop. With unit | − |−1 : A → A for every A : U.

◮ The universal property of (−1)-truncation is that

◮ For every type A : U, ◮ For every family P : A−1 → Prop of mere propositions over

A−1,

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Squash

◮ The inclusion of Prop in U has a left adjoint called

(−1)-truncation. − −1 : U → Prop. With unit | − |−1 : A → A for every A : U.

◮ The universal property of (−1)-truncation is that

◮ For every type A : U, ◮ For every family P : A−1 → Prop of mere propositions over

A−1,

◮ The pre-composition function

λs. λa. s(|a|−1) :

• x:A−1

P(x) →

• a:A

P(|a|−1) is an equivalence.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

0-truncation

◮ The inclusion of Set in U has a left adjoint called 0-truncation.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

0-truncation

◮ The inclusion of Set in U has a left adjoint called 0-truncation.

− 0 : U → Set.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

0-truncation

◮ The inclusion of Set in U has a left adjoint called 0-truncation.

− 0 : U → Set. With unit | − |0 : A → A0 for every A : U.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

0-truncation

◮ The inclusion of Set in U has a left adjoint called 0-truncation.

− 0 : U → Set. With unit | − |0 : A → A0 for every A : U.

◮ The universal property of 0-truncation is that

◮ For every type A : U, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

0-truncation

◮ The inclusion of Set in U has a left adjoint called 0-truncation.

− 0 : U → Set. With unit | − |0 : A → A0 for every A : U.

◮ The universal property of 0-truncation is that

◮ For every type A : U, ◮ For every family P : A0 → Set of sets over A0, Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

0-truncation

◮ The inclusion of Set in U has a left adjoint called 0-truncation.

− 0 : U → Set. With unit | − |0 : A → A0 for every A : U.

◮ The universal property of 0-truncation is that

◮ For every type A : U, ◮ For every family P : A0 → Set of sets over A0, ◮ The pre-composition function

λs. λa. s(|a|0) :

• x:A0

P(x) →

• a:A

P(|a|0) is an equivalence.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

◮ We use higher inductive types to implement colimits.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

◮ We use higher inductive types to implement colimits. ◮ In particular we will get:

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

◮ We use higher inductive types to implement colimits. ◮ In particular we will get:

◮ Coequalizers Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

◮ We use higher inductive types to implement colimits. ◮ In particular we will get:

◮ Coequalizers ◮ Pushouts Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

◮ We use higher inductive types to implement colimits. ◮ In particular we will get:

◮ Coequalizers ◮ Pushouts ◮ Quotients Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Higher inductive types

◮ The truncations are implemented as higher inductive types

(Brunerie).

◮ We use higher inductive types to implement colimits. ◮ In particular we will get:

◮ Coequalizers ◮ Pushouts ◮ Quotients (only when we work with sets) Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Quotients as HIT’s

Suppose A is a type and R : A → A → Prop is an equivalence relation over A.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Quotients as HIT’s

Suppose A is a type and R : A → A → Prop is an equivalence relation over A. We define A/R as a Higher Inductive Type with basic constructors:

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Quotients as HIT’s

Suppose A is a type and R : A → A → Prop is an equivalence relation over A. We define A/R as a Higher Inductive Type with basic constructors: π : A → A/R paths :

(x,y:A)R(x, y) → (π(x) = π(y))

and isSet :

(x,y:A/R)

• (p,q:x=y)(p = q)

Introduction Regularity Quotients The object classifier

Quotients as HIT’s

Suppose A is a type and R : A → A → Prop is an equivalence relation over A. We define A/R as a Higher Inductive Type with basic constructors: π : A → A/R paths :

(x,y:A)R(x, y) → (π(x) = π(y))

and isSet :

(x,y:A/R)

• (p,q:x=y)(p = q)

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

To find a map out of A/R into a set X, the induction principle tells us that it suffices to find f : A → X H :

(x,y:A)R(x, y) → (f (x) = f (y)).

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Regularity

Definition

For a function f : A → B, we define im(f ) :≡

• b:B

fibf (b)

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Regularity

Definition

For a function f : A → B, we define im(f ) :≡

• b:B

fibf (b)

Theorem

For any function f : A → B between sets, im(f ) is the coequalizer

• f

π1, π2 :

• x,y:A

(f (x) = f (y)) → A.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

surjective.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective. ◮ Hence it suffices to show that

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective. ◮ Hence it suffices to show that

for any function g : A → C

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective. ◮ Hence it suffices to show that

for any function g : A → C and any homotopy g ◦ π1 ∼ g ◦ π2

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective. ◮ Hence it suffices to show that

for any function g : A → C and any homotopy g ◦ π1 ∼ g ◦ π2 there is a function h : im(f ) → C

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective. ◮ Hence it suffices to show that

for any function g : A → C and any homotopy g ◦ π1 ∼ g ◦ π2 there is a function h : im(f ) → C and a homotopy h ◦ f ∼ g.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ First prove the Principle of Unique Choice. ◮ Also show that a function is an epimorphism if and only if it is

• surjective. (predicative, using univalence)

◮ The function A → im(f ) is surjective. ◮ Hence it suffices to show that

for any function g : A → C and any homotopy g ◦ π1 ∼ g ◦ π2 there is a function h : im(f ) → C and a homotopy h ◦ f ∼ g.

◮ Define the function h with the Principle of Unique Choice.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Quotients

Definition

Suppose R : A → A → Prop is a mere relation on A. We define cR : A → A/R to be the coequalizer (in Set) of the pair π1, π2 :

• x,y:A

R(x, y) → A.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Quotients

Definition

Suppose R : A → A → Prop is a mere relation on A. We define cR : A → A/R to be the coequalizer (in Set) of the pair π1, π2 :

• x,y:A

R(x, y) → A.

Definition

A mere relation R : A → A → Prop is said to be effective if

• (x,y:A) R(x, y)

A A A/R

π1 π2 cR cR

is a pullback square.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Theorem

The mere effective relations are precisely the mere equivalence relations.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Theorem

The mere effective relations are precisely the mere equivalence relations.

Theorem

Set is a ΠW-pretopos.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

(Sub)object classifier

Without any further assumptions, we have no proof that Set is a predicative topos. However, we can prove that

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

(Sub)object classifier

Without any further assumptions, we have no proof that Set is a predicative topos. However, we can prove that

Theorem

For any type B : U, there is an equivalence (B → U) ≃

• A:U

(A → B)

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ Define ϕ : (B → U) → (A:U)(A → B) by

ϕ :≡ λP.

(b:B)P(b), pr1

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ Define ϕ : (B → U) → (A:U)(A → B) by

ϕ :≡ λP.

(b:B)P(b), pr1 ◮ Define the function ψ : (A:U)(A → B) → (B → U) by

ψ :≡ λA, f . λb. fibf (b)

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Sketch of the proof

◮ Define ϕ : (B → U) → (A:U)(A → B) by

ϕ :≡ λP.

(b:B)P(b), pr1 ◮ Define the function ψ : (A:U)(A → B) → (B → U) by

ψ :≡ λA, f . λb. fibf (b)

◮ Show that these functions are each other’s inverses (using

Univalence)

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

In fact we can show:

Theorem

The function pr1 :

(X:U) X → U is the object classifier for U.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

In fact we can show:

Theorem

The function pr1 :

(X:U) X → U is the object classifier for U.

Theorem

The function pr1 :

(X:n-TypeU) X → n-TypeU is the object

classifier for n-TypeU.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

In fact we can show:

Theorem

The function pr1 :

(X:U) X → U is the object classifier for U.

Theorem

The function pr1 :

(X:n-TypeU) X → n-TypeU is the object

classifier for n-TypeU. And in particular

Theorem

The function pr1 :

(X:Set) X → Set is the object classifier for Set.

The function pr1 :

(X:Prop) X → Prop is the subobject classifier

for Set.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

◮ The pullback of a map in S is again in S.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

◮ The pullback of a map in S is again in S. ◮ When g is the pullback of f along a surjective map, then

S(g) → S(f ).

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

◮ The pullback of a map in S is again in S. ◮ When g is the pullback of f along a surjective map, then

S(g) → S(f ).

◮ S(f ) → S(g) → S(f + g).

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

◮ The pullback of a map in S is again in S. ◮ When g is the pullback of f along a surjective map, then

S(g) → S(f ).

◮ S(f ) → S(g) → S(f + g). ◮ S is locally full.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

◮ The pullback of a map in S is again in S. ◮ When g is the pullback of f along a surjective map, then

S(g) → S(f ).

◮ S(f ) → S(g) → S(f + g). ◮ S is locally full.

However, we have no proof of the collection axiom for S.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Using pr1 :

(X:Set) X → Set, we obtain a class S of small maps,

by the methods of algebraic set theory. Writing S(f ) if f is in S, the class S satisfies:

◮ The pullback of a map in S is again in S. ◮ When g is the pullback of f along a surjective map, then

S(g) → S(f ).

◮ S(f ) → S(g) → S(f + g). ◮ S is locally full.

However, we have no proof of the collection axiom for S. Conjecture: higher inductive types can be used instead. Theorem: We do have collection in the Aczel encoding of sets.

Egbert Rijke, Bas Spitters Sets in HoTT

Introduction Regularity Quotients The object classifier

Conclusion

Sets form a ΠW-pretopos with an object classifier: well-behaved:

◮ class of proof irrelevant types (Prop) ◮ squash types ◮ quotient types ◮ equivalence between Fam and Pow

Egbert Rijke, Bas Spitters Sets in HoTT