Towards a constructive simplicial model of univalent foundations - - PowerPoint PPT Presentation

1

Towards a constructive simplicial model

• f univalent foundations

Nicola Gambino1 Simon Henry2

1University of Leeds 2University of Ottawa

Homotopy Type Theory 2019 Carnegie Mellon University August 15th, 2019

2

Goal

To define a model of Univalent Foundations that is (1) definable constructively, i.e. without EM and AC (2) defined in a category homotopically-equivalent to Top.

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Goal

To define a model of Univalent Foundations that is (1) definable constructively, i.e. without EM and AC (2) defined in a category homotopically-equivalent to Top. Univalent Foundations = ML + UA , where

◮ ML = Martin-L¨

• f type theory with one universe type

◮ UA = Voevodsky’s Univalence Axiom

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Related work

Cubical approach:

◮ [BCH], [BCHM], [OP], . . . do (1) but not (2). ◮ Recent [ACCRS] does (1) and (2) using equivariant fibrations.

5

Related work

Cubical approach:

◮ [BCH], [BCHM], [OP], . . . do (1) but not (2). ◮ Recent [ACCRS] does (1) and (2) using equivariant fibrations.

Simplicial approach has some advantages:

◮ more familiar ◮ uses standard notion of Kan fibration ◮ straightforward equivalence with Top.

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Main result

Theorem (Gambino and Henry). Constructively, there exists a comprehension category Fib

χ

• SSet→

cof cod

• SSetcof

with

◮ all the type constructors of ML ◮ univalence of the universe ◮ Π-types are weakly stable, other type constructors are pseudo-stable.

SSetcof = full subcategory of cofibrant simplicial sets SSet

7

References

[H1]

• S. Henry

Weak model structures in classical and constructive mathematics ArXiv, 2018 [H2]

• S. Henry

A constructive account of the Kan-Quillen model structure and of Kan’s Ex∞ functor ArXiv, 2019 [GSS]

• N. Gambino and K. Szumi

lo and C. Sattler The constructive Kan-Quillen model structure: two new proofs ArXiv, 2019 [GH]

• N. Gambino and S. Henry

Towards a constructive simplicial model of Univalent Foundations ArXiv, 2019

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Outline of the talk

◮ Review of the classical simplicial model ◮ Constructive simplicial homotopy theory

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Voevodsky’s classical simplicial model

Idea

◮ contexts = simplicial sets ◮ dependent types = Kan fibrations.

10

Voevodsky’s classical simplicial model

Idea

◮ contexts = simplicial sets ◮ dependent types = Kan fibrations.

⇒ The comprehension category Fib

χ

• SSet→

cod

• SSet

11

Voevodsky’s classical simplicial model

Idea

◮ contexts = simplicial sets ◮ dependent types = Kan fibrations.

⇒ The comprehension category Fib

χ

• SSet→

cod

• SSet

It supports

◮ all the type constructors of ML ◮ a univalent universe

satisfying stability conditions.

12

Voevodsky’s classical simplicial model

Idea

◮ contexts = simplicial sets ◮ dependent types = Kan fibrations.

⇒ The comprehension category Fib

χ

• SSet→

cod

• SSet

It supports

◮ all the type constructors of ML ◮ a univalent universe

satisfying stability conditions. It gives rise to a strict model via a splitting process.

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Key facts

(0) Existence of the Kan-Quillen model structure on SSet.

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Key facts

(0) Existence of the Kan-Quillen model structure on SSet. (1) A, B ∈ SSet, B Kan complex ⇒ BA Kan complex.

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Key facts

(0) Existence of the Kan-Quillen model structure on SSet. (1) A, B ∈ SSet, B Kan complex ⇒ BA Kan complex. (2) p : A → X Kan fibration ⇒ the right adjoint to pullback Πp : SSet/A → SSet/X preserves Kan fibrations.

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Key facts

(0) Existence of the Kan-Quillen model structure on SSet. (1) A, B ∈ SSet, B Kan complex ⇒ BA Kan complex. (2) p : A → X Kan fibration ⇒ the right adjoint to pullback Πp : SSet/A → SSet/X preserves Kan fibrations. (3) There is a Kan fibration π : ˜ U → U, with U Kan complex, that classifies small Kan fibrations, i.e. A

• ∀
• ˜

U

π

• X

∃

U

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Key facts

(0) Existence of the Kan-Quillen model structure on SSet. (1) A, B ∈ SSet, B Kan complex ⇒ BA Kan complex. (2) p : A → X Kan fibration ⇒ the right adjoint to pullback Πp : SSet/A → SSet/X preserves Kan fibrations. (3) There is a Kan fibration π : ˜ U → U, with U Kan complex, that classifies small Kan fibrations, i.e. A

• ∀
• ˜

U

π

• X

∃

U (4) The Kan fibration π : ˜ U → U is univalent.

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Constructivity problems

◮ Kan-Quillen model structure has classical proofs.

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Constructivity problems

◮ Kan-Quillen model structure has classical proofs. ◮ [BCP] shows that (1), (2) require classical logic.

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Constructivity problems

◮ Kan-Quillen model structure has classical proofs. ◮ [BCP] shows that (1), (2) require classical logic. ◮ [GS] fixed (1), (2) by introducing uniform Kan fibrations in SSet,

but this creates problems for (3), (4).

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Constructive simplicial homotopy theory

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Constructive simplicial homotopy theory

We start with I =

• ∂∆n → ∆n | n ≥ 0
• J =
• Λk

n → ∆n | 0 ≤ k ≤ n

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Constructive simplicial homotopy theory

We start with I =

• ∂∆n → ∆n | n ≥ 0
• J =
• Λk

n → ∆n | 0 ≤ k ≤ n

• and generate wfs’s

(Sat(I) , I ⋔) , (Sat(J) , J⋔)

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Constructive simplicial homotopy theory

We start with I =

• ∂∆n → ∆n | n ≥ 0
• J =
• Λk

n → ∆n | 0 ≤ k ≤ n

• and generate wfs’s

(Sat(I) , I ⋔) , (Sat(J) , J⋔) We wish to have a model structure (W, C, F) such that C = Sat(I) , W ∩ F = I ⋔ W ∩ C = Sat(J) , F = J⋔

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Constructive simplicial homotopy theory

We start with I =

• ∂∆n → ∆n | n ≥ 0
• J =
• Λk

n → ∆n | 0 ≤ k ≤ n

• and generate wfs’s

(Sat(I) , I ⋔) , (Sat(J) , J⋔) We wish to have a model structure (W, C, F) such that C = Sat(I) , W ∩ F = I ⋔ W ∩ C = Sat(J) , F = J⋔ In particular, F = Kan fibrations. This helps with (3).

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Constructive cofibrations

Let C = Sat(I).

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Constructive cofibrations

Let C = Sat(I). Classically, for i : A → B in SSet, TFAE

◮ i ∈ C ◮ i is a monomorphism

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Constructive cofibrations

Let C = Sat(I). Classically, for i : A → B in SSet, TFAE

◮ i ∈ C ◮ i is a monomorphism

Constructively, for i : A → B in SSet, TFAE

◮ i ∈ C ◮ i is a monomorphism s.t. ∀n, in : An → Bn is complemented, i.e.

∀y ∈ Bn

• y ∈ An ∨ y /

∈ An

• ,

and degeneracy of simplices in Bn \ An is decidable.

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Constructive cofibrations

Let C = Sat(I). Classically, for i : A → B in SSet, TFAE

◮ i ∈ C ◮ i is a monomorphism

Constructively, for i : A → B in SSet, TFAE

◮ i ∈ C ◮ i is a monomorphism s.t. ∀n, in : An → Bn is complemented, i.e.

∀y ∈ Bn

• y ∈ An ∨ y /

∈ An

• ,

and degeneracy of simplices in Bn \ An is decidable.

• Note. C = cofibrations in Reedy wfs generated by the wfs

(Complemented mono, Split epi)

• n Set.

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The constructive Kan-Quillen model structure

Theorem [H2]. Constructively, the category SSet admits a model structure (W, C, F) such that C = Sat(I) , F = Kan fibrations .

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The constructive Kan-Quillen model structure

Theorem [H2]. Constructively, the category SSet admits a model structure (W, C, F) such that C = Sat(I) , F = Kan fibrations . Two other proofs in [GSS].

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The constructive Kan-Quillen model structure

Theorem [H2]. Constructively, the category SSet admits a model structure (W, C, F) such that C = Sat(I) , F = Kan fibrations . Two other proofs in [GSS]. Note

◮ Constructively, not every object is cofibrant: X is cofibrant if and

• nly if degeneracy of simplices in X is decidable.

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The constructive Kan-Quillen model structure

Theorem [H2]. Constructively, the category SSet admits a model structure (W, C, F) such that C = Sat(I) , F = Kan fibrations . Two other proofs in [GSS]. Note

◮ Constructively, not every object is cofibrant: X is cofibrant if and

• nly if degeneracy of simplices in X is decidable.

◮ Every object X has a cofibrant replacement, given by L(X) cofibrant

and t : L(X) → X in W ∩ C.

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Towards a constructive simplicial model

Idea

◮ use cofibrancy to solve constructivity issues,

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Towards a constructive simplicial model

Idea

◮ use cofibrancy to solve constructivity issues, ◮ contexts are cofibrant simplicial sets, ◮ types are Kan fibrations between cofibrant simplicial sets.

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Towards a constructive simplicial model

Idea

◮ use cofibrancy to solve constructivity issues, ◮ contexts are cofibrant simplicial sets, ◮ types are Kan fibrations between cofibrant simplicial sets.

⇒ The comprehension category Fibcof

χ

• SSet→

cof cod

• SSetcof

37

Towards a constructive simplicial model

Idea

◮ use cofibrancy to solve constructivity issues, ◮ contexts are cofibrant simplicial sets, ◮ types are Kan fibrations between cofibrant simplicial sets.

⇒ The comprehension category Fibcof

χ

• SSet→

cof cod

• SSetcof

Challenge

◮ stay within the cofibrant fragment.

38

Key facts

• 0. Existence of the constructive Kan-Quillen model structure.

39

Key facts

• 0. Existence of the constructive Kan-Quillen model structure.
• 1. A, B ∈ SSet, A cofibrant, B Kan ⇒ BA Kan.

40

Key facts

• 0. Existence of the constructive Kan-Quillen model structure.
• 1. A, B ∈ SSet, A cofibrant, B Kan ⇒ BA Kan.
• 2. p : A → X Kan fibration, A cofibrant ⇒ the right adjoint to pullback

Πp : SSet/A → SSet/X preserves Kan fibrations.

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Key facts

• 0. Existence of the constructive Kan-Quillen model structure.
• 1. A, B ∈ SSet, A cofibrant, B Kan ⇒ BA Kan.
• 2. p : A → X Kan fibration, A cofibrant ⇒ the right adjoint to pullback

Πp : SSet/A → SSet/X preserves Kan fibrations.

• 3. There is a Kan fibration π : ˜

Uc → Uc, with Uc cofibrant Kan complex, that weakly classifies small Kan fibrations between cofibrant simplicial sets A

• ∀
• ˜

U

π

• X

∃

U

42

Key facts

• 0. Existence of the constructive Kan-Quillen model structure.
• 1. A, B ∈ SSet, A cofibrant, B Kan ⇒ BA Kan.
• 2. p : A → X Kan fibration, A cofibrant ⇒ the right adjoint to pullback

Πp : SSet/A → SSet/X preserves Kan fibrations.

• 3. There is a Kan fibration π : ˜

Uc → Uc, with Uc cofibrant Kan complex, that weakly classifies small Kan fibrations between cofibrant simplicial sets A

• ∀
• ˜

U

π

• X

∃

U

• 4. The fibration π : ˜

Uc → Uc is univalent.

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Function types

Let A, B be cofibrant Kan complexes.

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Function types

Let A, B be cofibrant Kan complexes. Step 1. Consider BA, which is a Kan complex by (1). We have app : BA × A → B universal,

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Function types

Let A, B be cofibrant Kan complexes. Step 1. Consider BA, which is a Kan complex by (1). We have app : BA × A → B universal, i.e. such that X

f

BA X × A

f ×1A

BA × A

app

B is a bijection.

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Function types

Let A, B be cofibrant Kan complexes. Step 1. Consider BA, which is a Kan complex by (1). We have app : BA × A → B universal, i.e. such that X

f

BA X × A

f ×1A

BA × A

app

B is a bijection. Its inverse is written X × A

f

B X

λ(f )

BA In general, BA is not cofibrant.

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Step 2. Let L(BA) be a cofibrant replacement of BA, with t : L(BA) → BA in W ∩ F Now L(BA) is cofibrant Kan complex.

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Step 2. Let L(BA) be a cofibrant replacement of BA, with t : L(BA) → BA in W ∩ F Now L(BA) is cofibrant Kan complex. We have

• app : L(BA) × A

t×1A BA × A app B

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Step 2. Let L(BA) be a cofibrant replacement of BA, with t : L(BA) → BA in W ∩ F Now L(BA) is cofibrant Kan complex. We have

• app : L(BA) × A

t×1A BA × A app B

For f : X × A → B, with X cofibrant Kan complex, we get X × A

f

B X

λ(f ) BA

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Step 2. Let L(BA) be a cofibrant replacement of BA, with t : L(BA) → BA in W ∩ F Now L(BA) is cofibrant Kan complex. We have

• app : L(BA) × A

t×1A BA × A app B

For f : X × A → B, with X cofibrant Kan complex, we get X × A

f

B X

λ(f ) BA

X

• λ(f ) L(BA)

where

• L(BA)

t

• X

λ(f )

• λ(f )
• BA

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Step 2. Let L(BA) be a cofibrant replacement of BA, with t : L(BA) → BA in W ∩ F Now L(BA) is cofibrant Kan complex. We have

• app : L(BA) × A

t×1A BA × A app B

For f : X × A → B, with X cofibrant Kan complex, we get X × A

f

B X

λ(f ) BA

X

• λ(f ) L(BA)

where

• L(BA)

t

• X

λ(f )

• λ(f )
• BA

Note

◮ β-rule holds judgementally, η-rule holds propositionally. ◮ This extends to Π-types.

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The universe (I)

Step 1. Construct a Kan fibration π : ˜ U → U which classifies small Kan fibrations with cofibrant fibers. Un = {p : A → ∆[n] | p small fibration , A cofibrant}

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The universe (I)

Step 1. Construct a Kan fibration π : ˜ U → U which classifies small Kan fibrations with cofibrant fibers. Un = {p : A → ∆[n] | p small fibration , A cofibrant} Step 2.

◮ Let Uc be a cofibrant replacement of U, with t : Uc → U in W ∩ F ◮ Pullback

˜ Uc

• πc
• ˜

U

π

• Uc

t

U

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The universe (II)

• Proposition. The map πc : ˜

Uc → Uc classifies small Kan fibrations between cofibrant objects.

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The universe (II)

• Proposition. The map πc : ˜

Uc → Uc classifies small Kan fibrations between cofibrant objects.

• Proof. Let p : A → X be such a map.

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The universe (II)

• Proposition. The map πc : ˜

Uc → Uc classifies small Kan fibrations between cofibrant objects.

• Proof. Let p : A → X be such a map. Since p has cofibrant fibers, we

have A

• p
• ˜

U

π

• X

a

U

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The universe (II)

• Proposition. The map πc : ˜

Uc → Uc classifies small Kan fibrations between cofibrant objects.

• Proof. Let p : A → X be such a map. Since p has cofibrant fibers, we

have A

• p
• ˜

U

π

• X

a

U But Uc

t

• X

a

• ac
• U

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The universe (II)

• Proposition. The map πc : ˜

Uc → Uc classifies small Kan fibrations between cofibrant objects.

• Proof. Let p : A → X be such a map. Since p has cofibrant fibers, we

have A

• p
• ˜

U

π

• X

a

U But Uc

t

• X

a

• ac
• U

and so A

• p
• ˜

Uc

πc

• ˜

U

π

• X

ac

Uc

t

U .

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Fibrancy and univalence of the universe

Step 1. Prove equivalence extension property.

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Fibrancy and univalence of the universe

Step 1. Prove equivalence extension property.

◮ Key Lemma. Let f : Y → X be a cofibration between cofibrant

• bjects. If q : B → Y has cofibrant domain, then so does

Πf (q) : ΠY (B) → X.

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Fibrancy and univalence of the universe

Step 1. Prove equivalence extension property.

◮ Key Lemma. Let f : Y → X be a cofibration between cofibrant

• bjects. If q : B → Y has cofibrant domain, then so does

Πf (q) : ΠY (B) → X. Step 2. Prove U Kan complex, so that Uc is a cofibrant Kan complex.

62

Fibrancy and univalence of the universe

Step 1. Prove equivalence extension property.

◮ Key Lemma. Let f : Y → X be a cofibration between cofibrant

• bjects. If q : B → Y has cofibrant domain, then so does

Πf (q) : ΠY (B) → X. Step 2. Prove U Kan complex, so that Uc is a cofibrant Kan complex.

◮ Familiar argument, via instance of equivalence extensional property.

63

Fibrancy and univalence of the universe

Step 1. Prove equivalence extension property.

◮ Key Lemma. Let f : Y → X be a cofibration between cofibrant

• bjects. If q : B → Y has cofibrant domain, then so does

Πf (q) : ΠY (B) → X. Step 2. Prove U Kan complex, so that Uc is a cofibrant Kan complex.

◮ Familiar argument, via instance of equivalence extensional property.

Step 3. Prove π univalent, so that πc univalent.

64

Fibrancy and univalence of the universe

Step 1. Prove equivalence extension property.

◮ Key Lemma. Let f : Y → X be a cofibration between cofibrant

• bjects. If q : B → Y has cofibrant domain, then so does

Πf (q) : ΠY (B) → X. Step 2. Prove U Kan complex, so that Uc is a cofibrant Kan complex.

◮ Familiar argument, via instance of equivalence extensional property.

Step 3. Prove π univalent, so that πc univalent.

◮ Equivalence extension property ◮ Diagram-chasing, using 3-for-2 for W.

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Coherence issues

The comprehension category Fibcof

χ

• SSet→

cof cod

• SSetcof

66

Coherence issues

The comprehension category Fibcof

χ

• SSet→

cof cod

• SSetcof

It is not split and satisfies only weak versions of stability conditions.

67

Coherence issues

The comprehension category Fibcof

χ

• SSet→

cof cod

• SSetcof

It is not split and satisfies only weak versions of stability conditions. Open problem. Can we construct a strict model from this?

68

Coherence issues

The comprehension category Fibcof

χ

• SSet→

cof cod

• SSetcof

It is not split and satisfies only weak versions of stability conditions. Open problem. Can we construct a strict model from this? None of the known strictification methods seems to apply constructively.

69

Future work

◮ Solve coherence problem.

70

Future work

◮ Solve coherence problem. ◮ Generalise from Set to a Grothendieck topos E

◮ Model structure on simplicial sheaves [∆op, E] ◮ Connections to higher topos theory

71

Future work

◮ Solve coherence problem. ◮ Generalise from Set to a Grothendieck topos E

◮ Model structure on simplicial sheaves [∆op, E] ◮ Connections to higher topos theory

◮ A simplicial type theory extracted from the comprehension category,

in which univalence axiom is provable.

72

Future work

◮ Solve coherence problem. ◮ Generalise from Set to a Grothendieck topos E

◮ Model structure on simplicial sheaves [∆op, E] ◮ Connections to higher topos theory

◮ A simplicial type theory extracted from the comprehension category,

in which univalence axiom is provable.

73

References

[H1]

• S. Henry

Weak model structures in classical and constructive mathematics ArXiv, 2018. [H2]

• S. Henry

A constructive account of the Kan-Quillen model structure and of Kan’s Ex∞ functor ArXiv, 2019 [GSS]

• N. Gambino and K. Szumi

lo and C. Sattler The constructive Kan-Quillen model structure: two new proofs ArXiv, 2019 [GH]

• N. Gambino and S. Henry

Towards a constructive simplicial model of Univalent Foundations ArXiv, 2019