Two-dimensional models of type theory

Transcription

1 Two-dimensional models of type theory Richard Garner Uppsala University Logic Colloquium 2008, Bern 1/33

2 Outline Motivation Two-dimensional type theory: syntax Two-dimensional type theory: semantics Further directions (NB: Talk notes available athttp:// rhgg2) 2/33

3 Motivation Robert Seely showed in 1984 that we have: Extensional Martin-Löf type theories semantics syntax Locally cartesian closed categories. 3/33

4 Motivation Robert Seely showed in 1984 that we have: Extensional Martin-Löf type theories semantics syntax Locally cartesian closed categories. What is the analogous result for intensional type theory? Intensional Martin-Löf type theories semantics? syntax 3/33

5 We want to retain two key aspects of Seely s semantics: Σ-type formation is left adjoint to substitution and Π-type formation is right adjoint to substitution. 4/33

6 We want to retain two key aspects of Seely s semantics: Σ-type formation is left adjoint to substitution and Π-type formation is right adjoint to substitution. So we can t use categories with families, categories with attributes, type-categories, D-categories, categories with display maps, etc. 4/33

7 We want to retain two key aspects of Seely s semantics: Σ-type formation is left adjoint to substitution and Π-type formation is right adjoint to substitution. So we can t use categories with families, categories with attributes, type-categories, D-categories, categories with display maps, etc. Instead look at models in higher dimensional categories. 4/33

8 These models should have: Objects A, modelling types; 1-cells f: A B, modelling terms x : A f(x) : B; 2-cells f A g α B modelling judgements x : A α(x) : Id B (f(x), g(x)); 3-cells f A α Γ β B g modelling judgements x : A Γ(x) : Id IdB (f(x),g(x))(α(x), β(x)); And so on... 5/33

9 In such a way that Σ-type formation is a higher-dimensional left adjoint to substitution and Π-type formation is a higher-dimensional right adjoint to substitution. 6/33

10 In such a way that Σ-type formation is a higher-dimensional left adjoint to substitution and Π-type formation is a higher-dimensional right adjoint to substitution. I.e., should have Intensional Martin-Löf type theories semantics syntax Higher-dimensional locally cartesian closed categories. 6/33

11 The two-dimensional case Extensional Martin-Löf type theories semantics syntax Locally cartesian closed categories inclusion inclusion Intensional Martin-Löf type theories semantics syntax Higher-dimensional locally cartesian closed categories 7/33

12 The two-dimensional case Extensional Martin-Löf type theories inclusion Two-dimensional Martin-Löf type theories inclusion Intensional Martin-Löf type theories semantics syntax semantics syntax semantics syntax Locally cartesian closed categories inclusion Two-dimensional locally cartesian closed categories inclusion Higher-dimensional locally cartesian closed categories 8/33

13 The two-dimensional case Two-dimensional Martin-Löf type theories semantics syntax Two-dimensional locally cartesian closed categories 9/33

14 Coherence problems Seely 1984 said that: Extensional Martin-Löf type theories semantics syntax Locally cartesian closed categories Split comprehension categories with products, strong sums and strong equality 10/33

15 Coherence problems Hofmann 1994 corrected this picture: Extensional Martin-Löf type theories syntax semantics Split comprehension categories with products, strong sums and strong equality inclusion coherence Locally cartesian closed categories fibre over1 codomain fibration Comprehension categories with products, strong sums and strong equality. 11/33

16 Coherence problems So in two dimensions we are looking for Two-dimensional Martin-Löf type theories Two-dimensional locally cartesian closed categories syntax semantics fibre over1 codomain fibration Split comprehension 2-categories with products, strong sums and strong equality inclusion coherence Comprehension 2-categories with products, strong sums and strong equality. 12/33

17 Coherence problems And we ll concentrate on looking Two-dimensional Martin-Löf type theories Two-dimensional locally cartesian closed categories syntax semantics Split comprehension 2-categories with products, strong sums and strong equality Comprehension 2-categories with products, strong sums and strong equality. 13/33

18 Intensional Martin-Löf type theory Sequent calculus with four forms of judgement: Atype ( A is a type ); a : A ( a is a term of type A ) A = Btype ( A and B are definitionally equal types ); a = b : A ( a and b are definitionally equal terms of type A ). Can also have judgements under hypotheses; so if Atype, we can have x : A B(x)type; x : A f(x) : B(x); x : A B(x) = C(x) type; x : A f(x) = g(x) : B(x). And so on; in general can have things like x : A, y : B(x), z : C(x, y) f(x, y, z) : D(x, y, z). 14/33

19 These come with inference rules for: Congruence of definitional equality; Weakening, contraction and exchange; Substitution; for example: x : A f(x) : B y : B C(y)type x : A C(f(x)) type or x : A f(x) : B y : B g(y) : C(y) type; x : A g(f(x)) : C(f(x)) Logical operations: Σ-types,Π-types and identity types. 15/33

20 Identity types Atype a, b : A Id A (a, b)type Id-FORM; a : A r(a) : Id A (a, a) Id-INTRO; x, y : A, z : Id A (x, y) C(x, y, z) type x : A d(x) : C(x, x,r(x)) a, b : A p : Id A (a, b) J C (d, a, b, p) : C(a, b, p) Id-ELIM; x, y : A, z : Id A (x, y) C(x, y, z) type x : A d(x) : C(x, x,r(x)) a : A J C (d, a, a,r(a)) = d(a) : C(a, a,r(a)) Id-COMP. 16/33

21 Extensional Martin-Löf type theory Obtained by adding the equality reflection rules: Atype a, b : A p : Id A (a, b) a = b : A Atype a, b : A p : Id A (a, b) p = r(a) : Id A (a, b) Id-REFL-1; Id-REFL-2; causing definitional equality andid-equality to coincide. 17/33

22 Extensional Martin-Löf type theory Obtained by adding the equality reflection rules: Atype a, b : A p : Id A (a, b) a = b : A Atype a, b : A p : Id A (a, b) p = r(a) : Id A (a, b) Id-REFL-1; Id-REFL-2; causing definitional equality andid-equality to coincide. Good from a category-theoretic perspective; Bad from a proof-theoretic one. 17/33

23 Two-dimensional Martin-Löf type theory We call a type A is discrete just when the Id-reflection rules Atype a, b : A p : Id A (a, b) a = b : A Atype a, b : A p : Id A (a, b) p = r(a) : Id A (a, b) Id-REFL-1; Id-REFL-2; obtain at the type A. 18/33

24 Two-dimensional Martin-Löf type theory We call a type A is discrete just when the Id-reflection rules Atype a, b : A p : Id A (a, b) a = b : A Atype a, b : A p : Id A (a, b) p = r(a) : Id A (a, b) Id-REFL-1; Id-REFL-2; obtain at the type A. So the intensional theory says that no types need be discrete; The extensional theory says that all types are discrete; Our two-dimensional theory will say that all identity types are discrete. 18/33

25 Two-dimensional Martin-Löf type theory Explicitly, obtained by adding to intensional type theory the discrete identity rules: Atype a, b : A p, q : Id A (a, b) s : Id IdA (a,b)(p, q) p = q : Id A (a, b) Atype a, b : A p, q : Id A (a, b) s : Id IdA (a,b)(p, q) s = r(p) : Id IdA (a,b)(p, q) Id-DISC-1; Id-DISC-2. Resultant type theory is intermediate between the intensional and the extensional ones. 19/33

26 Two-dimensional syntactic model We build a2-category Ctxt from the syntax of two-dimensional type theory. Objects are contexts Γ = ( x 1 : C 1, x 2 : C 2 (x 1 ),..., x n : C n (x 1,..., x n 1 ) ) ; Morphisms f: Γ are context morphisms x : Γ f 1 (x) : D 1 x : Γ f 2 (x) : D 2 (f 1 (x))... x : Γ f n (x) : D n (f 1 (x),..., f n 1 (x)) (or x : Γ f(x) : for short) (all modulo definitional equality). 20/33

27 What about 2-cells of Ctxt? Easy to describe into contexts of length1: Given f, g: Γ (y : B) in Ctxt, a 2-cell α: f g is given by x : Γ α(x) : Id B (f(x), g(x)). (modulo definitional equality). 21/33

28 What about 2-cells of Ctxt? Easy to describe into contexts of length1: Given f, g: Γ (y : B) in Ctxt, a 2-cell α: f g is given by x : Γ α(x) : Id B (f(x), g(x)). (modulo definitional equality). To describe2-cells into longer contexts, we need the Leibnitz rule: a, b : A c : C(a) p : Id A (a, b) p (c) : C(b) obtained byid-elimination on p. 21/33

29 Using Leibnitz: Given f, g: Γ (y : B, z : C(y)) in Ctxt looking like x : Γ f 1 (x) : B, x : Γ g 1 (x) : B, x : Γ f 2 (x) : C(f 1 (x)) x : Γ g 2 (x) : C(g 1 (x)) a 2-cell α: f g will be given by x : Γ α 1 (x) : Id B (f 1 (x), g 1 (x)) x : Γ α 2 (x) : Id C(f1 (x))( α1 (x) (f 2 (x)), g 2 (x) ). And so on. 22/33

30 Composition in Ctxt is defined using substitution (for1-cell composition) andid-elimination (for2-cell composition). 23/33

31 Composition in Ctxt is defined using substitution (for1-cell composition) andid-elimination (for2-cell composition). Example Composite of2-cells f (x : A) α g β h (y : B) in Ctxt given by the transitivity proof x : A trans(α(x), β(x)) : Id B (f(x), h(x)). 23/33

32 A two-dimensional indexed category For each contextγ Ctxt, we have a 2-category Type(Γ) with: Objects being judgements x : Γ A(x)type; Morphisms f: A B being judgements x : Γ, y : A(x) f(x, y) : B(x); 2-cells α: f g being judgements x : Γ, y : A(x) α(x, y) : Id B(x) (f(x, y), g(x, y)). 24/33

33 A two-dimensional indexed category Each f: Γ in Ctxt induces a substitution 2-functor f : Type( ) Type( ); 25/33

34 A two-dimensional indexed category Each f: Γ in Ctxt induces a substitution 2-functor f : Type( ) Type( ); each α: f g: Γ induces a pseudo-natural transformation α : g f : Type( ) Type( ); 25/33

35 A two-dimensional indexed category Each f: Γ in Ctxt induces a substitution 2-functor f : Type( ) Type( ); each α: f g: Γ induces a pseudo-natural transformation α : g f : Type( ) Type( ); so we obtain a split indexed2-category Type( ): Ctxt coop 2-Cat ps. 25/33

36 Sum types Each type x : Γ A(x)type induces a dependent projection π A : Γ, A Γ in Ctxt; so obtain a weakening2-functor A := π A : Type(Γ) Type(Γ, A). 26/33

37 Sum types Each type x : Γ A(x)type induces a dependent projection π A : Γ, A Γ in Ctxt; so obtain a weakening2-functor A := π A : Type(Γ) Type(Γ, A). Proposition A has a left biadjoint Σ A : Type(Γ, A) Type(Γ) B Σx : A. B(x) 26/33

38 Product types However, for the following to be true... Proposition A has a right biadjoint Π A : Type(Γ, A) Type(Γ) B Πx : A. B(x) 27/33

39 Product types However, for the following to be true... Proposition A has a right biadjoint Π A : Type(Γ, A) Type(Γ) B Πx : A. B(x)... we need to add function extensionality. 27/33

40 Function extensionality M, N : Π(A, B) K : Πx : A.Id Π(A,B) (M x, N x) ext(m, N, K) : Id(M, N) EXT; x : A f(x) : B(x) ext ( λ(f), λ(f), λx.r(f(x)) ) = r(λ(f)) : Id Π(A,B) ( λ(f), λ(f) ) EXT-COMP; M, N : Π(A, B) K : Πx : A.Id(M x, N x) a : A µ(m, N, K, a) : Id IdB(a) (M a,n a)(ext(m, N, K) a, K a) EXT-COMP 2 ; x : A f(x) : B(x) a : A µ ( λ(f), λ(f), λx.r(f(x)), a) = r(r(f(a))) : Id ( r(f(a)), r(f(a)) ) EXT-COMP 3. 28/33

41 Two-dimensional semantics We define a model of two-dimensional type theory to be: A2-category C; A split indexed2-category T( ): C coop 2-Cat ps ; Left and right biadjoints for each weakening2-functor A ; Plus extra data (Beck-Chevalley, strong sums,... ) 29/33

42 Soundness and completeness Every two-dimensional type theory gives rise to a two-dimensional syntactic model. 30/33

43 Soundness and completeness Every two-dimensional type theory gives rise to a two-dimensional syntactic model. Every two-dimensional model admits a sound interpretation of the syntax. 30/33

44 Soundness and completeness Every two-dimensional type theory gives rise to a two-dimensional syntactic model. Every two-dimensional model admits a sound interpretation of the syntax. Can also associate a type theory to every two-dimensional model: the internal language. 30/33

45 Further directions Need to extend this: Two-dimensional Martin-Löf type theories Two-dimensional locally cartesian closed categories syntax semantics Split comprehension 2-categories with products, strong sums and strong equality Comprehension 2-categories with products, strong sums and strong equality. 31/33

46 Further directions To this: Two-dimensional Martin-Löf type theories Two-dimensional locally cartesian closed categories syntax semantics fibre over1 codomain fibration Split comprehension 2-categories with products, strong sums and strong equality inclusion coherence Comprehension 2-categories with products, strong sums and strong equality. 32/33

47 Further directions Construct examples of two-dimensional models: Hofmann and Streicher s groupoid model; 2-category of groupoid-valued fibrations on a category C; 2-category of stacks on a site(c, J). 33/33

48 Further directions Construct examples of two-dimensional models: Hofmann and Streicher s groupoid model; 2-category of groupoid-valued fibrations on a category C; 2-category of stacks on a site(c, J). Higher dimensional models... 33/33