A Type Theory for Formalising Category Theory

Transcription

1 A Type Theory for Formalising Category Theory Robin Adams 24 July 2012 Robin Adams Type Theory for Category Theory 24 July / 28

2 1 Introduction The Problem of Notation Category Theory in Five Minutes 2 The Type Theory Syntax 3 Conclusion Robin Adams Type Theory for Category Theory 24 July / 28

3 Notation is Important Euclid IX.36 If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. Robin Adams Type Theory for Category Theory 24 July / 28

4 Notation is Important Euclid IX.36 If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect. If k = p is prime, then p2 k is perfect. Robin Adams Type Theory for Category Theory 24 July / 28

5 From a Category Theory Textbook (We have defined y : Nat(D(r, ), K) Kr.) The Yoneda map y [...] is natural in K and r. To state this fact formally, we must consider K as an object in the functor category Set D, regard both domain and codomain of the map y as functors of the pair K, r, and consider this pair as an object in the category Set D D. The codomain for y is then the evaluation functor E, which maps each pair K, r to the value Kr of the functor K at the object r; the domain is the functor N which maps the object K, r to the set Nat(D(r, ), K) of all natural transformations and which maps a pair of arrows F : K K, f : r r to Nat(D(f, ), F ). Robin Adams Type Theory for Category Theory 24 July / 28

6 In my Notation Let N = λk.λr.nat(λx.d(r, x), K) and E = λk.λr.kr. There is a natural isomorphism between N and E, namely λk.λr.λα.α r 1 r Robin Adams Type Theory for Category Theory 24 July / 28

7 In my Notation Let N = λk.λr.nat(λx.d(r, x), K) and E = λk.λr.kr. There is a natural isomorphism between N and E, namely λk.λr.λα.α r 1 r Problem: How can we guarantee that a lambda-expression of the appropriate type defines a functor or a natural transformation? Robin Adams Type Theory for Category Theory 24 July / 28

8 Categories A category A consists of: a class A of objects; given any two objects A, B, a set A[A, B] of arrows from A to B. We write f : A B iff f is an arrow from A to B; for any object A, an arrow 1 A : A A, the identity on A; given arrows f : A B, g : B C, an arrow g f : A C, the composite of g with f such that, for all f : A B, g : B C, h : C D, 1 B f = f f 1 A = f h (g f ) = (h g) f Robin Adams Type Theory for Category Theory 24 July / 28

9 Functors Given categories A, B, a functor F : A B consists of: for every object A on A, an object FA of B; for every arrow f : A A in A, an arrow Ff : FA FA such that F 1 A = 1 FA F (g f ) = Fg Ff Robin Adams Type Theory for Category Theory 24 July / 28

10 Natural Transformations Given functors F, G : A B, a natural transformation τ : F G consists of: such that for every object A of A, an arrow τ A : FA GA in B for every arrow f : A B in A, Gf τ A = τ B Ff Robin Adams Type Theory for Category Theory 24 July / 28

11 The Setup We have six syntactic categories: categories A, B, C,... objects A, B, C,..., X, Y, Z sets S, T,... elements a, b, c,..., x, y, z propositions φ, ψ,... proofs δ,..., p, q A context consists of declarations of the form X : A, x : S and p : φ. Robin Adams Type Theory for Category Theory 24 July / 28

12 The Setup We have six judgement forms: Γ A cat Γ A : A Γ S set Γ a : S Γ φ prop Γ δ : φ Robin Adams Type Theory for Category Theory 24 July / 28

13 Reduction We are going to introduce a reduction relation. We define convertibility as usual, and have the rules Γ A : A Γ B cat (A B) Γ A : B Γ a : S Γ T set (S T ) Γ a : T Γ δ : φ Γ ψ prop (φ ψ) Γ δ : ψ Robin Adams Type Theory for Category Theory 24 July / 28

14 Propositions Let there be propositions. Robin Adams Type Theory for Category Theory 24 July / 28

15 Propositions Let there be propositions. Let propositions have proofs. Robin Adams Type Theory for Category Theory 24 July / 28

16 Propositions Let there be propositions. Let propositions have proofs. There is no notion of equality between proofs. Robin Adams Type Theory for Category Theory 24 July / 28

17 Sets Let there be sets. Robin Adams Type Theory for Category Theory 24 July / 28

18 Sets Let there be sets. Let sets have elements. Robin Adams Type Theory for Category Theory 24 July / 28

19 Sets Let there be sets. Let sets have elements. Given elements a and b of the set S, let there be a proposition a = b : S. Robin Adams Type Theory for Category Theory 24 July / 28

20 Sets Let there be sets. Let sets have elements. Given elements a and b of the set S, let there be a proposition a = b : S. Γ a : S Γ ref (a) : a = a : S Γ, x : S φ[x] prop Γ δ : a = b : S Γ δ : φ[a] Γ sub ([x : S]φ[x], δ, δ ) : φ[b] Robin Adams Type Theory for Category Theory 24 July / 28

21 Categories Let there be categories. Robin Adams Type Theory for Category Theory 24 July / 28

22 Categories Let there be categories. Let categories have objects. Robin Adams Type Theory for Category Theory 24 July / 28

23 Categories Let there be categories. Let categories have objects. Given objects A and B of C, let there be a set A B : C. Robin Adams Type Theory for Category Theory 24 July / 28

24 Categories Let there be categories. Let categories have objects. Given objects A and B of C, let there be a set A B : C. Γ A : C Γ 1 A : A A : C Γ f : A B : C Γ g : B C : C Γ g f : A C : C Γ f : A B : C Γ unitl (f ) : 1 B f = f : A B : C Γ f : A B : C Γ unitr (f ) : f 1 A = f : A B : C Γ f : A B : C Γ g : B C : C Γ h : C D : C Γ assoc (f, g, h) : h (g f ) = (h g) f : A D : C Robin Adams Type Theory for Category Theory 24 July / 28

25 Implication Given propositions φ and ψ, let there be a proposition φ ψ. Robin Adams Type Theory for Category Theory 24 July / 28

26 Implication Given propositions φ and ψ, let there be a proposition φ ψ. Γ, p : φ δ : ψ Γ impi([p : φ]δ) : φ ψ Γ δ : φ ψ Γ δ : φ Γ impe(δ, δ ) : ψ Robin Adams Type Theory for Category Theory 24 July / 28

27 Functions Given sets S and T, let there be a set S T, whose elements are called functions. Robin Adams Type Theory for Category Theory 24 July / 28

28 Functions Given sets S and T, let there be a set S T, whose elements are called functions. Γ, x : S b : T Γ λx : S.b : S T S Γ f : S T Γ a : S Γ f (a) : T Γ, x : S b : T Γ a : S Γ β([x : S]b, a) : (λx : S.b)(a) = [a/x]b : T Γ, x : S δ : f (x) = g(x) : T Γ ext ([x : S]δ) : f = g : S T Robin Adams Type Theory for Category Theory 24 July / 28

29 Γ F : A B Γ A : AΓ FA : B Γ F : A B Γ f : A A Γ Ff : FA FA : B Γ τ : F G : A B Γ A : AΓ τ A : FA GA : B Robin Adams Type Theory for Category Theory 24 July / 28

30 Given categories A and B, let there be a category A B, whose objects are called functors and whose arrows are called natural transformations. Γ, X : A B : BΓ ΛX : A.B : A B Γ, X : A f : FX GX : BΓ λx : A.f : F G : A B Robin Adams Type Theory for Category Theory 24 July / 28

31 Γ τ : F G : A B Γ f : A B : AΓ nat (τ, f ) : Gf τ A = τ B Ff : FA GB : B We introduce rules to guarantee the desired properties of functors and natural transformations: Γ F : A B Γ A : AΓ funcid (F, A) : F 1 A = 1 FA : FA FA : B Γ F : A B Γ f : A B : A Γ g : B C : AΓ funcomp (F, f, g) : F (g f ) = Fg Ff : FA FC : B Robin Adams Type Theory for Category Theory 24 July / 28

32 Introduce the reduction rule (ΛX : A.B)A [A/X ]B. Γ, X : A f : FX GX : B Γ A : AΓ β nat ([X : A]f, A) : (λx : A.f ) A = [A/X ]f : FA GA : B Robin Adams Type Theory for Category Theory 24 July / 28

33 What should (ΛX : A.B)f mean? Introduce the reduction rule (ΛX : A.B)A [A/X ]B. Γ, X : A f : FX GX : B Γ A : AΓ β nat ([X : A]f, A) : (λx : A.f ) A = [A/X ]f : FA GA : B Robin Adams Type Theory for Category Theory 24 July / 28

34 The definition is: {f /X }X f {f /X }Y 1 Y {f /X }(ΛY : C.D) λy : C.{f /X }D {f : A A /X }(FB) [A /X ]F {f /X }B ({f /X }F ) [A/X ]B Define a new notion of substitution. {f : A A /X }B (f an element, X an object variable, A an object, result an element) We want the following rule to be admissible: Γ, X : A B : B Γ f : A A : AΓ {f : A A /X }B : [A/X ]B [A /X ]B : B Robin Adams Type Theory for Category Theory 24 July / 28

35 Now introduce the rule: Γ, X : A B : B Γ f : A A : AΓ β func ([X : A]B, f ) : (ΛX : A.B)f = {f /X }B : [A/X ]B [A /X ]B : B Robin Adams Type Theory for Category Theory 24 July / 28

36 Γ σ : F G : A B Γ τ : G H : A B Γ A : A Γ nat comp (σ, τ, A) : (σ τ) A = σ A τ A : FA HA : B Our rules for categories automatically give us identity natural transformations, and We introduce rules specifying how these behave under application. Γ F : A B Γ A : AΓ nat id (F, A) : (1 F ) A = 1 FA : FA FA : B Robin Adams Type Theory for Category Theory 24 July / 28

37 Special Functors We can define 1 A ΛX : A.X G F ΛX : A.G(FX ) The following equalities then hold: 1 B F F F 1 A F H (G F ) (H G) F Robin Adams Type Theory for Category Theory 24 July / 28

38 And So On... We can go on to introduce: negation, conjunction, disjunction, quantification over sets, quantification over the objects of a category Robin Adams Type Theory for Category Theory 24 July / 28

39 And So On... We can go on to introduce: negation, conjunction, disjunction, quantification over sets, quantification over the objects of a category product sets, disjoint unions, the empty set, natural numbers with elimination over a set Robin Adams Type Theory for Category Theory 24 July / 28

40 And So On... We can go on to introduce: negation, conjunction, disjunction, quantification over sets, quantification over the objects of a category product sets, disjoint unions, the empty set, natural numbers with elimination over a set product categories, sum categories, empty category, unit category Robin Adams Type Theory for Category Theory 24 July / 28

41 And So On... We can go on to introduce: negation, conjunction, disjunction, quantification over sets, quantification over the objects of a category product sets, disjoint unions, the empty set, natural numbers with elimination over a set product categories, sum categories, empty category, unit category set of propositions/truth values, category of sets (behave like universes) Robin Adams Type Theory for Category Theory 24 July / 28

42 And So On... We can go on to introduce: negation, conjunction, disjunction, quantification over sets, quantification over the objects of a category product sets, disjoint unions, the empty set, natural numbers with elimination over a set product categories, sum categories, empty category, unit category set of propositions/truth values, category of sets (behave like universes) sets {x : S φ[x]} and categories ΣX : A.S[X ] Robin Adams Type Theory for Category Theory 24 July / 28

43 And So On... We can go on to introduce: negation, conjunction, disjunction, quantification over sets, quantification over the objects of a category product sets, disjoint unions, the empty set, natural numbers with elimination over a set product categories, sum categories, empty category, unit category set of propositions/truth values, category of sets (behave like universes) sets {x : S φ[x]} and categories ΣX : A.S[X ] comma categories, universals, limits, adjunctions,... Robin Adams Type Theory for Category Theory 24 July / 28

44 Technical Comments We have: an intensional notion of equality between sets, categories and objects; an extensional notion of equality between elements; no notion of equality between proofs. (We are also free to define notions, e.g. equivalence of propositions, isomorphism of objects.) It is thus halfway between good and evil. Robin Adams Type Theory for Category Theory 24 July / 28

45 Categories may not depend on object variables. Sets, objects and categories may not depend on element variables. If an element variable x may occur in an expression E[x], you must have an answer to the question If x = y, what is the relation between E[x] and E[y]? If an object variable X may occur in an expression E[X ], you must have an answer to the question Given an arrow f : X Y, what is the relation between E[X ] and E[Y ]? Robin Adams Type Theory for Category Theory 24 July / 28

46 To Do: Justify system by proving metatheoretic properties giving semantics in terms of category theory formalising loads of examples Extend to 2-categories. Implement this system? Robin Adams Type Theory for Category Theory 24 July / 28

47 To Do: I want to formalise fibred category theory in a non-evil way because the paper would be called Good Fibrations Robin Adams Type Theory for Category Theory 24 July / 28

48 To Do: I want to formalise fibred category theory in a non-evil way because the paper would be called Good Fibrations and Conor hasn t written a paper with that title yet. Robin Adams Type Theory for Category Theory 24 July / 28