Lecture 3: Homotopical models of type theory Nicola Gambino School - PowerPoint PPT Presentation

Lecture 3: Homotopical models of type theory Nicola Gambino School of Mathematics University of Leeds Young Set Theory Copenhagen June 14th, 2016 1

What happened yesterday? Type theory ◮ the type theory T ◮ Extensional vs intensional type theories Homotopical algebra ◮ Weak factorisation systems and model structures ◮ Groupoids ◮ Simplicial sets Italy 2 – Belgium 0 Exercises ◮ x , y : A , u : Id A ( x , y ) ⊢ u − 1 : Id A ( y , x ) ◮ x , y , z : A , u : Id A ( x , y ) , v : Id A ( y , z ) ⊢ v ◦ u : Id A ( x , z ). 2

Problems with intensionality The axioms for identity types do not seem to capture fully what we want. Example ◮ we have � � � � Id A × B ( c , d ) ← → Id A π 1 ( c ) , π 1 ( d ) × Id B π 2 ( c ) , π 2 ( d ) ◮ but only Id A → B ( f , g ) − → (Π x : A ) Id B ( fx , gx ) Similar situation for Σ-types and Π-types. What about the type universe? 3

Outline of Lecture 2 Part I: Models of type theory Part II: Identity types Part III: Π-types Part IV: Universes 4

Part I: Models of type theory 5

� Models of type theories Question: What structure on a category C do we need to have a model of T ? Idea: Look at the structure of the syntactic category of T . ◮ Objects: contexts ( x 1 : A 1 ) , ( x 1 : A 1 , x 2 : A 2 , . . . ) , . . . ◮ Maps: terms-in-context, e.g. ( a 1 ) : Γ → ( x 1 : A 1 ) if Γ ⊢ a 1 : A 1 ( a 1 , a 2 ) : Γ → ( x 1 : A 1 , x 2 : A 2 ) if Γ ⊢ a 1 : A 1 , Γ ⊢ a 2 : A 2 [ a 1 / x 1 ] · · · Note. (Γ , x : A ) Γ ⊢ A : type = ⇒ p A Γ 6

� � � � Axiomatization Fix ◮ category C with a terminal object 1 ◮ a class P ⊆ Map ( C ) Type Theory Syntactic category Category Theory ( x : A ) 1 . A A : type p A p A ∈ P 1 ( ) (Γ , x : A ) Γ . A Γ ⊢ A : type p A p A ∈ P Γ Γ 7

� � � Terms as sections Type Theory Category Theory a Γ Γ . A Γ ⊢ a : A 1 Γ p A Γ a � A a : A 1 8

� � � � � � � Substitution as pullback Type Theory Category Theory � Γ . A . B Γ . B [ a / x ] Γ ⊢ a : A Γ , x : A ⊢ B : type p B Γ ⊢ B [ a / x ] : type � Γ . A Γ a b ◦ (1 Γ , a ) Γ b [ a / x ] Γ ⊢ a : A Γ , x : A ⊢ b : B � Γ . A . B Γ . B [ a / x ] Γ ⊢ b [ a / x ] : B [ a / x ] 1 Γ p B � Γ . A Γ a 9

� � � � � � Weakening For Γ ∈ Obj ( C ), let P / Γ be the category with ◮ objects: P -maps A : Γ . A → Γ ◮ maps: commutative triangles f Γ . A Γ . B Γ , x : A ⊢ f ( x ) : B p A p B Γ Let p A : Γ . A → Γ be in P . Pullback Γ . A . E Γ . E Γ ⊢ E : type p E Γ , x : A ⊢ E : type � Γ Γ . A p A This is the ‘weakening functor’ ∆ A : P / Γ → P / Γ . A . 10

General setting Let ◮ C be a category ◮ P ⊆ Map( C ) and assume ◮ C has a terminal object. ◮ The pullback of a P -maps along any maps exists and is an P -map. Question ◮ What additional structure on P do we need to interpret type-formers? 11

Part II: Identity types 12

� � � � � � Identity types (I) For simplicity, let us assume Γ = 1 and work with p A : 1 . A → 1. We need 1. a P -map q A : Id A → A × A 2. a factorisation refl A Id A A q A ∆ A A × A 3. a diagonal filler for every commutative diagram d � Id A . E A refl A p E ∈ P Id A Id A Note. In fact, refl A ⋔ p for all p ∈ P . 13

Identity types as path spaces Idea p : Id A ( a , b ) ⇐ ⇒ p is a path in A from a to b Note ◮ This explains several aspects of the behaviour of identity types Theorem. The syntactic category of the type theory T admits a weak factorisation system ( L , R ), where L = { i | ( ∀ p ∈ P ) i ⋔ p } R = { p | ( ∀ i ∈ L ) i ⋔ p } where P is the set of projections p A : (Γ , x : A ) → Γ. 14

Homotopical models of type theory Idea ◮ Take P = Fib in some model structure ( Weq , Fib , Cof ). We get a ‘dictionary’ Type Theory Homotopical algebra A : type fibrant object A x : A ⊢ B ( x ) : type fibration p : B → A x , y : A ⊢ Id A ( x , y ) path space of A (Π x : A ) B ( x ) space of sections of p type universe U a fibrant object U a generic fibration π : ˜ x ∈ U ⊢ El( x ) : type U → U 15

� � � � � Example: Id-types in groupoids Given by the ( Weq ∩ Cof , Fib )-factorisation of diagonal ∆ A � A × A A r ( s , t ) A J The groupoid A J has ◮ objects: maps α : a 0 → a 1 in A ◮ maps: squares a 0 b 0 α β � b 1 a 1 Note Uniqueness of identity proofs fails in this model. Warning. To define a model, one needs to take care of further aspects: ◮ mere existence vs structure ◮ coherence with respect to pullback (substitution) 16

Part III: Π -types 17

� � � � � � Π-types Let A : Γ . A → Γ be in P . ◮ Recall the ‘weakening’ functor Γ ⊢ E : type ∆ A : P / Γ → P / Γ . A Γ , x : A ⊢ E : type ◮ To interpret Π-types, we require a right adjoint Γ , x : A ⊢ B : type Π A : P / Γ . A → P / Γ Γ ⊢ (Π x : A ) B : type Idea λ ( b ) b Γ . A Γ . A . B Γ Γ . Π A ( B ) ⇐ ⇒ Γ . A Γ 18

� � � Π-types in groupoids Theorem. ◮ For p : Γ . A → Γ an isofibration, the pullback functor ∆ p : Gpd / Γ → Gpd / Γ . A has a right adjoint Π p : Gpd / Γ . A → Gpd / Γ ◮ Furthermore, the right adjoint preserves isofibrations, and thus gives us Π p : Fib / Γ . A → Fib / Γ Π p ( q ) � Γ q � Γ . A Γ . A . B �→ Γ . Π A ( B ) Example. When Γ = 1, the objects of the groupoid Π A ( B ) are b A A . B q 1 A A i.e. sections of q . 19

Π-types in simplicial sets Theorem. ◮ For any map p : B → A , pullback along p has a right adjoint Π p : SSet / B → SSet / A ◮ If p is a Kan fibration, then Π p preserves Kan fibrations, and hence gives Π p : Fib / B → Fib / A Proof. By duality, it suffices to show that ∆ p : SSet / A → SSet / B preserves ( Weq ∩ Cof )-maps. But ◮ Cof -maps are monomorphisms, so are always preserved. ◮ The preservation of weak equivalences by pullback along Kan fibrations is the so-called right properness of the model structure. Note. Constructivity issues. 20

Part IV: Universes 21

� � � � � � Generic P -maps We need a notion of ‘smallness’ for P -maps, e.g. fibers having cardinality < κ . Then need a P -map π : ˜ U → U that weakly classifies ‘small’ P -maps, i.e. for every such p : B → A there exists a pullback diagram ˜ B U p π � U . A Note. Given a : 1 → U , we can form a pullback ˜ El( a ) U π � U 1 a We think of a as the ‘name’ in U of the object El( a ). 22

� � � The type universe in groupoids and simplicial sets Fix an inaccessible cardinal κ . ◮ In Gpd , it is not difficult to construct a universe. For example, one can consider the groupoid of all small (discrete) groupoids. ◮ In SSet , there exists a fibration π : ˜ U → U that weakly classifies fibrations with fibers of cardinality < κ , i.e. for every such p : B → A there exists a pullback diagram ˜ B U p π � U . A Here, U n = { p : B → ∆ n | p Kan fibration } Problem. But U needs to be fibrant! 23

� � � � � � The type universe in simplicial sets Theorem. ◮ The base U of the generic Kan fibration π : ˜ U → U is a Kan complex. Proof. We need to show that U is a Kan complex. So show ∀ b Λ n U k h n k ∃ b ′ ∆ n This reduces to the problem of extending fibrations along horn inclusions: B ′ B p � ∆ n Λ n k h n k This can be done using the theory of minimal fibrations (AC). 24

� � � Minimal fibrations extend along ( Weq ∩ Cof )-maps Lemma 1. Let ◮ m : X → A be a minimal fibration ◮ i : A → A ′ be a ( Weq ∩ Cof )-map. Then there exists m ′ : M ′ → A ′ a minimal fibration and j X ′ X m m ′ � A ′ A i 25

� � � ( Weq ∩ Fib )-maps can be extended along cofibrations Lemma 2. Let ◮ t : B → X be a ( Weq ∩ Fib )-map ◮ j : X → X ′ a cofibration Then there exists t ′ : E ′ → X ′ a ( Weq ∩ Fib )-map and B ′ B t t ′ � X ′ X j 26

� � � � � � � � � Proof of the theorem Recall that we need to complete the diagram B ′ B p � ∆ n Λ n k h n k It suffices to ◮ factor p as a ( Weq ∩ Fib )-map t followed by a minimal fibration m ◮ apply Lemma 1 and Lemma 2 so as to get B ′ B t t ′ X ′ X j m m ′ � ∆ n Λ n k h n k 27

Conclusions The homotopical models of type theory suggest: 1. To use type theory as a language for speaking about spaces 2. To develop mathematics using this language; in particular, to define sets = def discrete spaces 3. To add axioms to type theory motivated by homotopy theory 28