Homotopy-theoretic aspects of Martin-L of type theory Nicola - PowerPoint PPT Presentation

Homotopy-theoretic aspects of Martin-L¨ of type theory Nicola Gambino University of Palermo visiting The University of Manchester Logic Colloquium Paris – July 30th, 2010

Background Identity types: for a type A and a , b ∈ A , we have a new type Id A ( a , b ) Idea: p ∈ Id A ( a , b ) ⇔ “ p is a proof that a equals b ” Key discovery (Hofmann and Streicher, 1995) : p , q ∈ Id A ( a , b ) � p = q Question: ◮ What is the combinatorics of identity types?

Recent advances Models ◮ Awodey and Warren (2007), Warren (2008) ◮ van den Berg and Garner (2010) Identity types and homotopy theory ◮ Gambino and Garner (2008) ◮ Awodey, Hofstra and Warren (2009) Identity types and higher-dimensional categories ◮ van den Berg and Garner (2008) ◮ Lumsdaine (2008) Voevodsky’s work ◮ Homotopy λ -calculus (2006) ◮ Univalent models (2010)

Overview Part I ◮ Identity types Part II ◮ The identity type weak factorization system Part III ◮ Weak ω -groupoids

Part I Identity types

Martin-L¨ of type theory (I) Dependent types: x ∈ A ⊢ B ( x ) ∈ Type Key ideas: ◮ Propositions-as-types ◮ Theory of inductive definitions ◮ Computer implementation Forms of type: 0 , 1 , A × B , A ⇒ B , A + B , N , � � Id A ( a , b ) , x ∈ A B ( x ) , x ∈ A B ( x ) , . . . We will only need the rules for identity types.

Martin-L¨ of type theory (II) Judgements A ∈ Type , a ∈ A , A = B ∈ Type , a = b ∈ A . Hypothetical judgements Γ ⊢ J where Γ = ( x 1 ∈ A 1 , . . . , x n ∈ A n ) . Deduction rules Γ 1 ⊢ J 1 · · · Γ n ⊢ J n Γ ⊢ J

Identity types Formation rule A ∈ Type a ∈ A b ∈ A Id A ( a , b ) ∈ Type For example, if a ∈ A then Id A ( a , a ) ∈ Type Introduction rule a ∈ A r ( a ) ∈ Id A ( a , a )

Elimination rule p ∈ Id A ( a , b ) x ∈ A , y ∈ A , u ∈ Id A ( x , y ) ⊢ C ( x , y , u ) ∈ Type x ∈ A ⊢ c ( x ) ∈ C ( x , x , r ( x )) J ( a , b , p , c ) ∈ C ( a , b , p ) Idea: [ x ∈ A ] · · · · a = b C ( x , x ) C ( a , b ) Cf. Lawvere’s treatment of equality in categorical logic.

Computation rule a ∈ A x ∈ A , y ∈ A , u ∈ Id A ( x , y ) ⊢ C ( x , y , u ) ∈ Type x ∈ A ⊢ c ( x ) ∈ C ( x , x , r ( x )) J ( a , a , r ( a ) , c ) = c ( a ) ∈ C ( a , a , r ( a )) Idea: a ∈ A [ x ∈ A ] · · · · a ∈ A · · · · − → · a = a C ( x , x ) · · C ( a , a ) C ( a , a )

Definitional equality vs. propositional equality Definition. We say that a , b ∈ A are propositionally equal if there exists p ∈ Id A ( a , b ) . Theorem (Hofmann and Streicher) . ⇒ a = b ∈ A There exists p ∈ Id A ( a , b ) � Proposition (Hofmann) . Adding the rules p ∈ Id A ( a , b ) p ∈ Id A ( a , b ) p = r ( a ) ∈ Id A ( a , b ) a = b ∈ A makes type-checking undecidable.

Weakness of propositional equality We have p ∈ Id A ( a , b ) q ∈ Id A ( b , c ) q ◦ p ∈ Id A ( a , c ) The rule p ∈ Id A ( a , b ) q ∈ Id A ( b , c ) r ∈ Id A ( c , d ) ( r ◦ q ) ◦ p = r ◦ ( q ◦ p ) ∈ Id A ( a , d ) does not seem derivable, but only p ∈ Id A ( a , b ) q ∈ Id A ( b , c ) r ∈ Id A ( c , d ) � � α ∈ Id Id A ( a , d ) ( r ◦ q ) ◦ p , r ◦ ( q ◦ p )

Part II The identity type weak factorisation system

Types as spaces Idea ◮ Elements a ∈ A as points ◮ Elements p ∈ Id A ( a , b ) as paths from a to b ◮ Elements α ∈ Id Id A ( a , b ) ( p , q ) as homotopies from p to q ◮ . . . Examples a ∈ A p ∈ Id A ( a , b ) q ∈ Id A ( b , c ) r ∈ Id A ( c , d ) � � r ( a ) ∈ Id A ( a , a ) α ∈ Id Id A ( a , d ) ( r ◦ q ) ◦ p , r ◦ ( q ◦ p )

The syntactic category ML ◮ Objects. Types A , B , C , . . . . ◮ Maps. Terms-in-context, i.e. f : X → A is x ∈ X ⊢ f ( x ) ∈ A . Examples. For A ∈ Type, let Id ( A ) = � x , y ∈ A Id A ( x , y ) We have maps r A p A � Id ( A ) , � A × A Id ( A ) A � ( x , x , r ( x )) � ( x , y ) x � ( x , y , u ) �

� � � � � Identity types as path spaces ◮ ML r A Id ( A ) p A A × A A ∆ A ◮ Top X [ 0 , 1 ] � X × X X ∆ X

� � � � � � Propositional equality as homotopy ◮ Terms x ∈ X ⊢ f ( x ) , g ( x ) ∈ A are propositionally equal iff Id ( A ) ∃ p A X A × A ( f , g ) ◮ Maps f : X → A and g : X → A in Top are homotopic iff A [ 0 , 1 ] ∃ A × A X ( f , g ) Still missing: elimination and computation rules.

� � � � � � Lifting properties Let C be a category. Definition. Let i and p be maps in C . We say that i has the left lifting property with respect to p if � • � • • • � � ⇒ � p p i i � � � � � • • • • Notation: i ⋔ p . Let S be a class of maps in C . Define S ⋔ = { p | ( ∀ i ∈ S ) i ⋔ p } . ⋔ S = { i | ( ∀ p ∈ S ) i ⋔ p }

� � Example: fibrations Definition. A continuous map p : B → A is a fibration if it has the homotopy lifting property, i.e. every diagram � B X × { 0 } p i X � A X × [ 0 , 1 ] has a diagonal filler. { Fibrations } = { i X | X ∈ Top } ⋔ p : A [ 0 , 1 ] → A × A . Example.

� � � Weak factorization systems Definition (Bousfield 1977) . A weak factorization system on C is a pair ( L , R ) of classes of maps such that: (1) Every map f in C factors as f • • � � � with i ∈ L , p ∈ R . � � � � � � p � � i � � � • (2) L = ⋔ R , R = L ⋔ . Example. The category Top has a w.f.s. ( L , R ) where R = { Fibrations } , L ⊆ { Homotopy equivalences } . Examples. Quillen model structures.

Projections in ML Definition. A map in ML is a projection if it has the form p : � x ∈ A B ( x ) → A ( x , y ) �→ x where x ∈ A ⊢ B ( x ) ∈ Type. Example. Recall � Id ( A ) = Id A ( x , y ) . x , y ∈ A We have the projection p A : Id ( A ) − → A × A ( x , y , u ) �− → ( x , y )

The identity type weak factorisation system Theorem (Gambino and Garner) . The syntactic category ML has a weak factorisation system ( L , R ) given by R = L ⋔ . L = ⋔ P , where P = { Projections } . Note. L -maps and R -maps can be characterized explicitly. Example. The diagonal ∆ A : A → A × A factors as p A r A � Id ( A ) � A × A A To show: { r A } ⋔ P .

� � � � It suffices to consider � ( x , y , u ) ∈ Id ( A ) C ( x , y , u ) A r A p Id ( A ) Id ( A ) Top horizontal arrow gives x ∈ A ⊢ c ( x ) ∈ C ( x , x , r ( x )) So, we can apply the elimination rule: x ∈ A ⊢ c ( x ) ∈ C ( x , x , r ( x )) x ∈ A , y ∈ A , u ∈ Id A ( x , y ) ⊢ J ( x , y , u , c ) ∈ C ( x , y , u ) Top triangle commutes by computation rule.

� � � Homotopy-theoretic models Theorem (Awodey and Warren) . The rules for identity types admit an interpretation in every category C with a w.f.s. ( L , R ) . Idea. ◮ Dependent types as R -maps � B � x ∈ A ⊢ B ( x ) ∈ Type = ⇒ � A � ◮ Terms as sections � b � � A � � B � � � ������ � x ∈ A ⊢ b ( x ) ∈ B ( x ) = ⇒ � � � � 1 � A �

� � ◮ Identity types as path objects  x ∈ A , y ∈ A ⊢ Id A ( x , y ) ∈ Type   = ⇒ x ∈ A ⊢ r ( x ) ∈ Id A ( x , x ) � r � � A � � Id A � � � � ������������� � � � � � � � ∆ � A � � � � � � A � × � A � Elimination terms given by diagonal fillers. Note. Coherence issues (Warren, van den Berg and Garner).

Part III The fundamental weak ω -groupoid of a type

The fundamental groupoid π 1 ( A ) of a type A ◮ Objects. Elements a , b , . . . ∈ A ◮ Maps. Equivalence classes [ p ] : a → b , where p ∈ Id A ( a , b ) and p ∼ q ⇔ there exists α ∈ Id Id A ( a , b ) ( p , q ) . ≈ Fundamental groupoid of a space. Question ◮ What happens if we do not quotient identity proofs?

� � � � � � � � � � � � � � � � � The globular set π ( A ) of a type A π ( A ) X X 0 a , b , . . . A i s t p X 1 Id A ( a , b ) a b s i t p X 2 Id Id A ( a , b ) ( p , q ) a b α q s i t p Φ � Id Id Id A ( a , b ) ( p , q ) ( α, β ) X 3 a α β b q . . . . . . . . .

� � � � � � � Weak ω -groupoids Definition (Batanin 1998, Leinster 2004) . = Weak ω -category Globular set + action by a contractible operad = Globular set + ‘composition operations’ Example. p 1 q ◦ p 1 q q ◦ α a c �− → a c α b b p 2 q ◦ p 2 We have a weak ω -groupoid if all n -cells have weak inverses.

The weak ω -groupoid of a type Theorem (Garner and van den Berg, Lumsdaine) . For every type A , the globular set π ( A ) is a weak ω -groupoid. Examples. α ∈ Id Id A ( a , b ) ( p 1 , p 2 ) q ∈ Id A ( b , c ) q ◦ α ∈ Id Id A ( a , c ) ( q ◦ p 1 , q ◦ p 2 ) p ∈ Id A ( a , b ) p ∈ Id A ( a , b ) θ p ∈ Id Id A ( a , a ) ( p − 1 ◦ p , r ( a )) p − 1 ∈ Id A ( b , a )

Open problems Models ◮ Models in weak ω -groupoids Relationship with homotopy theory ◮ Simplicial identity types Relationship with higher categories ◮ Free higher categories from syntax