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

Towards a constructive simplicial model of univalent foundations Nicola Gambino 1 Simon Henry 2 1 University of Leeds 2 University of Ottawa Homotopy Type Theory 2019 Carnegie Mellon University August 15th, 2019 1

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 . 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 . Univalent Foundations = ML + UA , where ◮ ML = Martin-L¨ of type theory with one universe type ◮ UA = Voevodsky’s Univalence Axiom 3

Related work Cubical approach: ◮ [BCH], [BCHM], [OP], . . . do (1) but not (2). ◮ Recent [ACCRS] does (1) and (2) using equivariant fibrations. 4

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 . 5

� � � Main result Theorem (Gambino and Henry) . Constructively, there exists a comprehension category χ SSet → Fib cof cod SSet cof with ◮ all the type constructors of ML ◮ univalence of the universe ◮ Π-types are weakly stable, other type constructors are pseudo-stable. SSet cof = full subcategory of cofibrant simplicial sets � SSet 6

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 7

Outline of the talk ◮ Review of the classical simplicial model ◮ Constructive simplicial homotopy theory 8

Voevodsky’s classical simplicial model Idea ◮ contexts = simplicial sets ◮ dependent types = Kan fibrations. 9

� � � Voevodsky’s classical simplicial model Idea ◮ contexts = simplicial sets ◮ dependent types = Kan fibrations. ⇒ The comprehension category χ SSet → Fib cod SSet 10

� � � Voevodsky’s classical simplicial model Idea ◮ contexts = simplicial sets ◮ dependent types = Kan fibrations. ⇒ The comprehension category χ SSet → Fib cod SSet It supports ◮ all the type constructors of ML ◮ a univalent universe satisfying stability conditions. 11

� � � Voevodsky’s classical simplicial model Idea ◮ contexts = simplicial sets ◮ dependent types = Kan fibrations. ⇒ The comprehension category χ SSet → Fib 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. 12

Key facts (0) Existence of the Kan-Quillen model structure on SSet . 13

Key facts (0) Existence of the Kan-Quillen model structure on SSet . (1) A , B ∈ SSet , B Kan complex ⇒ B A Kan complex. 14

Key facts (0) Existence of the Kan-Quillen model structure on SSet . (1) A , B ∈ SSet , B Kan complex ⇒ B A Kan complex. (2) p : A → X Kan fibration ⇒ the right adjoint to pullback Π p : SSet / A → SSet / X preserves Kan fibrations. 15

� � � Key facts (0) Existence of the Kan-Quillen model structure on SSet . (1) A , B ∈ SSet , B Kan complex ⇒ B A Kan complex. (2) p : A → X Kan fibration ⇒ the right adjoint to pullback Π p : 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 ∀ � U X ∃ 16

� � � Key facts (0) Existence of the Kan-Quillen model structure on SSet . (1) A , B ∈ SSet , B Kan complex ⇒ B A Kan complex. (2) p : A → X Kan fibration ⇒ the right adjoint to pullback Π p : 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 ∀ � U X ∃ (4) The Kan fibration π : ˜ U → U is univalent. 17

Constructivity problems ◮ Kan-Quillen model structure has classical proofs. 18

Constructivity problems ◮ Kan-Quillen model structure has classical proofs. ◮ [BCP] shows that (1), (2) require classical logic. 19

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). 20

Constructive simplicial homotopy theory 21

Constructive simplicial homotopy theory We start with � � I = ∂ ∆ n → ∆ n | n ≥ 0 � � Λ k J = n → ∆ n | 0 ≤ k ≤ n 22

Constructive simplicial homotopy theory We start with � � I = ∂ ∆ n → ∆ n | n ≥ 0 � � Λ k J = n → ∆ n | 0 ≤ k ≤ n and generate wfs’s ( Sat ( I ) , I ⋔ ) , ( Sat ( J ) , J ⋔ ) 23

Constructive simplicial homotopy theory We start with � � I = ∂ ∆ n → ∆ n | n ≥ 0 � � Λ k J = 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 W ∩ F = I ⋔ C = Sat ( I ) , F = J ⋔ W ∩ C = Sat ( J ) , 24

Constructive simplicial homotopy theory We start with � � I = ∂ ∆ n → ∆ n | n ≥ 0 � � Λ k J = 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 W ∩ F = I ⋔ C = Sat ( I ) , F = J ⋔ W ∩ C = Sat ( J ) , In particular, F = Kan fibrations. This helps with (3). 25

Constructive cofibrations Let C = Sat ( I ). 26

Constructive cofibrations Let C = Sat ( I ). Classically, for i : A → B in SSet , TFAE ◮ i ∈ C ◮ i is a monomorphism 27

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 , i n : A n → B n is complemented, i.e. � � ∀ y ∈ B n y ∈ A n ∨ y / ∈ A n , and degeneracy of simplices in B n \ A n is decidable. 28

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 , i n : A n → B n is complemented, i.e. � � ∀ y ∈ B n y ∈ A n ∨ y / ∈ A n , and degeneracy of simplices in B n \ A n is decidable. Note. C = cofibrations in Reedy wfs generated by the wfs (Complemented mono, Split epi) on Set . 29

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 . 30

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]. 31

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 only if degeneracy of simplices in X is decidable. 32

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 only 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 . 33

Towards a constructive simplicial model Idea ◮ use cofibrancy to solve constructivity issues, 34

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. 35

� � � 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 χ SSet → Fib cof cof cod SSet cof 36

� � � 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 χ SSet → Fib cof cof cod SSet cof Challenge ◮ stay within the cofibrant fragment. 37

Key facts 0. Existence of the constructive Kan-Quillen model structure. 38

Key facts 0. Existence of the constructive Kan-Quillen model structure. 1. A , B ∈ SSet , A cofibrant, B Kan ⇒ B A Kan. 39