Mercurial > hg > Members > kono > Proof > category
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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| date | Fri, 05 Mar 2021 19:08:11 +0900 |
| parents | d81341fad6e1 |
| children | ac4db33b466f |
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--- -- -- A → Sets^A^op : Yoneda Functor -- Contravariant Functor h_a -- Nat(h_a,F) -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> ---- open import Category -- https://github.com/konn/category-agda open import Level open import Category.Sets module yoneda { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) where open import HomReasoning open import cat-utility open import Relation.Binary.Core open import Relation.Binary open import Relation.Binary.PropositionalEquality hiding ( [_] ; sym ) -- Contravariant Functor : op A → Sets ( Obj of Sets^{A^op} ) -- Obj and Hom of Sets^A^op open Functor YObj : Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) YObj = Functor (Category.op A) (Sets {c₂}) YHom : ( f : YObj ) → (g : YObj ) → Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) YHom f g = NTrans (Category.op A) (Sets {c₂}) f g open NTrans Yid : {a : YObj } → YHom a a Yid {a} = record { TMap = λ a → λ x → x ; isNTrans = isNTrans1 {a} } where isNTrans1 : {a : YObj } → IsNTrans (Category.op A) (Sets {c₂}) a a (λ a → λ x → x ) isNTrans1 {a} = record { commute = refl } _+_ : {a b c : YObj} → YHom b c → YHom a b → YHom a c _+_ {a} {b} {c} f g = record { TMap = λ x → Sets [ TMap f x o TMap g x ] ; isNTrans = isNTrans1 } where commute1 : (a b c : YObj ) (f : YHom b c) (g : YHom a b ) (a₁ b₁ : Obj (Category.op A)) (h : Hom (Category.op A) a₁ b₁) → Sets [ Sets [ FMap c h o Sets [ TMap f a₁ o TMap g a₁ ] ] ≈ Sets [ Sets [ TMap f b₁ o TMap g b₁ ] o FMap a h ] ] commute1 a b c f g a₁ b₁ h = let open ≈-Reasoning (Sets {c₂})in begin Sets [ FMap c h o Sets [ TMap f a₁ o TMap g a₁ ] ] ≈⟨ assoc {_} {_} {_} {_} {FMap c h } {TMap f a₁} {TMap g a₁} ⟩ Sets [ Sets [ FMap c h o TMap f a₁ ] o TMap g a₁ ] ≈⟨ car (nat f) ⟩ Sets [ Sets [ TMap f b₁ o FMap b h ] o TMap g a₁ ] ≈↑⟨ assoc {_} {_} {_} {_} { TMap f b₁} {FMap b h } {TMap g a₁}⟩ Sets [ TMap f b₁ o Sets [ FMap b h o TMap g a₁ ] ] ≈⟨ cdr {_} {_} {_} {_} {_} { TMap f b₁} (nat g) ⟩ Sets [ TMap f b₁ o Sets [ TMap g b₁ o FMap a h ] ] ≈↑⟨ assoc {_} {_} {_} {_} {TMap f b₁} {TMap g b₁} { FMap a h} ⟩ Sets [ Sets [ TMap f b₁ o TMap g b₁ ] o FMap a h ] ∎ isNTrans1 : IsNTrans (Category.op A) (Sets {c₂}) a c (λ x → Sets [ TMap f x o TMap g x ]) isNTrans1 = record { commute = λ {a₁ b₁ h} → commute1 a b c f g a₁ b₁ h } _==_ : {a b : YObj} → YHom a b → YHom a b → Set (c₂ ⊔ c₁) _==_ f g = ∀{x : Obj (Category.op A)} → (Sets {c₂}) [ TMap f x ≈ TMap g x ] infix 4 _==_ isSetsAop : IsCategory (YObj) (YHom) _==_ _+_ ( Yid ) isSetsAop = record { isEquivalence = record {refl = refl ; trans = λ {i j k} → trans1 {_} {_} {i} {j} {k} ; sym = λ {i j} → sym1 {_} {_} {i} {j}} ; identityL = refl ; identityR = refl ; o-resp-≈ = λ{a b c f g h i } → o-resp-≈ {a} {b} {c} {f} {g} {h} {i} ; associative = refl } where open ≈-Reasoning (Sets {c₂}) sym1 : {a b : YObj } {i j : YHom a b } → i == j → j == i sym1 {a} {b} {i} {j} eq {x} = sym eq trans1 : {a b : YObj } {i j k : YHom a b} → i == j → j == k → i == k trans1 {a} {b} {i} {j} {k} i=j j=k {x} = trans-hom i=j j=k o-resp-≈ : {A₁ B C : YObj} {f g : YHom A₁ B} {h i : YHom B C} → f == g → h == i → h + f == i + g o-resp-≈ {a} {b} {c} {f} {g} {h} {i} f=g h=i {x} = resp f=g h=i SetsAop : Category (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) (c₂ ⊔ c₁) SetsAop = record { Obj = YObj ; Hom = YHom ; _o_ = _+_ ; _≈_ = _==_ ; Id = Yid ; isCategory = isSetsAop } -- A is Locally small postulate ≈-≡ : { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } {a b : Obj A } { x y : Hom A a b } → (x≈y : A [ x ≈ y ]) → x ≡ y import Axiom.Extensionality.Propositional -- Extensionality a b = {A : Set a} {B : A → Set b} {f g : (x : A) → B x} → (∀ x → f x ≡ g x) → f ≡ g → ( λ x → f x ≡ λ x → g x ) postulate extensionality : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Axiom.Extensionality.Propositional.Extensionality c₂ c₂ ---- -- -- Object mapping in Yoneda Functor -- ---- open import Function y-obj : (a : Obj A) → Functor (Category.op A) (Sets {c₂}) y-obj a = record { FObj = λ b → Hom (Category.op A) a b ; FMap = λ {b c : Obj A } → λ ( f : Hom A c b ) → λ (g : Hom A b a ) → (Category.op A) [ f o g ] ; isFunctor = record { identity = λ {b} → extensionality A ( λ x → lemma-y-obj1 {b} x ) ; distr = λ {a} {b} {c} {f} {g} → extensionality A ( λ x → lemma-y-obj2 a b c f g x ) ; ≈-cong = λ eq → extensionality A ( λ x → lemma-y-obj3 x eq ) } } where lemma-y-obj1 : {b : Obj A } → (x : Hom A b a) → (Category.op A) [ id1 A b o x ] ≡ x lemma-y-obj1 {b} x = let open ≈-Reasoning (Category.op A) in ≈-≡ {_} {_} {_} {A} idL lemma-y-obj2 : (a₁ b c : Obj A) (f : Hom A b a₁) (g : Hom A c b ) → (x : Hom A a₁ a )→ Category.op A [ Category.op A [ g o f ] o x ] ≡ (Sets [ _[_o_] (Category.op A) g o _[_o_] (Category.op A) f ]) x lemma-y-obj2 a₁ b c f g x = let open ≈-Reasoning (Category.op A) in ≈-≡ {_} {_} {_} {A} ( begin Category.op A [ Category.op A [ g o f ] o x ] ≈↑⟨ assoc ⟩ Category.op A [ g o Category.op A [ f o x ] ] ≈⟨⟩ ( λ x → Category.op A [ g o x ] ) ( ( λ x → Category.op A [ f o x ] ) x ) ∎ ) lemma-y-obj3 : {b c : Obj A} {f g : Hom A c b } → (x : Hom A b a ) → A [ f ≈ g ] → Category.op A [ f o x ] ≡ Category.op A [ g o x ] lemma-y-obj3 {_} {_} {f} {g} x eq = let open ≈-Reasoning (Category.op A) in ≈-≡ {_} {_} {_} {A} ( begin Category.op A [ f o x ] ≈⟨ resp refl-hom eq ⟩ Category.op A [ g o x ] ∎ ) ---- -- -- Hom mapping in Yoneda Functor -- ---- y-tmap : ( a b : Obj A ) → (f : Hom A a b ) → (x : Obj (Category.op A)) → FObj (y-obj a) x → FObj (y-obj b ) x y-tmap a b f x = λ ( g : Hom A x a ) → A [ f o g ] -- ( h : Hom A x b ) y-map : {a b : Obj A } → (f : Hom A a b ) → YHom (y-obj a) (y-obj b) y-map {a} {b} f = record { TMap = y-tmap a b f ; isNTrans = isNTrans1 {a} {b} f } where lemma-y-obj4 : {a₁ b₁ : Obj (Category.op A)} {g : Hom (Category.op A) a₁ b₁} → {a b : Obj A } → (f : Hom A a b ) → Sets [ Sets [ FMap (y-obj b) g o y-tmap a b f a₁ ] ≈ Sets [ y-tmap a b f b₁ o FMap (y-obj a) g ] ] lemma-y-obj4 {a₁} {b₁} {g} {a} {b} f = let open ≈-Reasoning A in extensionality A ( λ x → ≈-≡ {_} {_} {_} {A} ( begin A [ A [ f o x ] o g ] ≈↑⟨ assoc ⟩ A [ f o A [ x o g ] ] ∎ ) ) isNTrans1 : {a b : Obj A } → (f : Hom A a b ) → IsNTrans (Category.op A) (Sets {c₂}) (y-obj a) (y-obj b) (y-tmap a b f ) isNTrans1 {a} {b} f = record { commute = λ{a₁ b₁ g } → lemma-y-obj4 {a₁} {b₁} {g} {a} {b} f } ----- -- -- Yoneda Functor itself -- ----- YonedaFunctor : Functor A SetsAop YonedaFunctor = record { FObj = λ a → y-obj a ; FMap = λ f → y-map f ; isFunctor = record { identity = identity ; distr = distr1 ; ≈-cong = ≈-cong } } where ≈-cong : {a b : Obj A} {f g : Hom A a b} → A [ f ≈ g ] → SetsAop [ y-map f ≈ y-map g ] ≈-cong {a} {b} {f} {g} eq = let open ≈-Reasoning A in -- (λ x g₁ → A [ f o g₁ ] ) ≡ (λ x g₁ → A [ g o g₁ ] ) extensionality A ( λ h → ≈-≡ {_} {_} {_} {A} ( begin A [ f o h ] ≈⟨ resp refl-hom eq ⟩ A [ g o h ] ∎ ) ) identity : {a : Obj A} → SetsAop [ y-map (id1 A a) ≈ id1 SetsAop (y-obj a ) ] identity {a} = let open ≈-Reasoning A in -- (λ x g → A [ id1 A a o g ] ) ≡ (λ a₁ x → x) extensionality A ( λ g → ≈-≡ {_} {_} {_} {A} ( begin A [ id1 A a o g ] ≈⟨ idL ⟩ g ∎ ) ) distr1 : {a b c : Obj A} {f : Hom A a b} {g : Hom A b c} → SetsAop [ y-map (A [ g o f ]) ≈ SetsAop [ y-map g o y-map f ] ] distr1 {a} {b} {c} {f} {g} = let open ≈-Reasoning A in -- (λ x g₁ → (A [ (A [ g o f] o g₁ ]))) ≡ (λ x x₁ → A [ g o A [ f o x₁ ] ] ) extensionality A ( λ h → ≈-≡ {_} {_} {_} {A} ( begin A [ A [ g o f ] o h ] ≈↑⟨ assoc ⟩ A [ g o A [ f o h ] ] ∎ ) ) ------ -- -- F : A → Sets ∈ Obj SetsAop -- -- F(a) → Nat(h_a,F) -- x ∈ F(a) , (g : Hom A b a) → ( FMap F g ) x ------ F2Natmap : {a : Obj A} → {F : Obj SetsAop } → {x : FObj F a} → (b : Obj (Category.op A)) → Hom Sets (FObj (y-obj a) b) (FObj F b) F2Natmap {a} {F} {x} b = λ ( g : Hom A b a ) → ( FMap F g ) x F2Nat : {a : Obj A} → {F : Obj SetsAop } → FObj F a → Hom SetsAop (y-obj a) F F2Nat {a} {F} x = record { TMap = F2Natmap {a} {F} {x} ; isNTrans = isNTrans1 } where commute1 : {a₁ b : Obj (Category.op A)} {f : Hom (Category.op A) a₁ b} (g : Hom A a₁ a) → (Sets [ FMap F f o FMap F g ]) x ≡ FMap F (A [ g o f ] ) x commute1 g = let open ≈-Reasoning (Sets) in cong ( λ f → f x ) ( sym ( distr F ) ) commute : {a₁ b : Obj (Category.op A)} {f : Hom (Category.op A) a₁ b} → Sets [ Sets [ FMap F f o F2Natmap {a} {F} {x} a₁ ] ≈ Sets [ F2Natmap {a} {F} {x} b o FMap (y-obj a) f ] ] commute {a₁} {b} {f} = let open ≈-Reasoning (Sets) in begin Sets [ FMap F f o F2Natmap {a} {F} {x} a₁ ] ≈⟨⟩ Sets [ FMap F f o (λ ( g : Hom A a₁ a ) → ( FMap F g ) x) ] ≈⟨ extensionality A ( λ (g : Hom A a₁ a) → commute1 {a₁} {b} {f} g ) ⟩ Sets [ (λ ( g : Hom A b a ) → ( FMap F g ) x) o FMap (y-obj a) f ] ≈⟨⟩ Sets [ F2Natmap {a} {F} {x} b o FMap (y-obj a) f ] ∎ isNTrans1 : IsNTrans (Category.op A) (Sets {c₂}) (y-obj a) F (F2Natmap {a} {F}) isNTrans1 = record { commute = λ {a₁ b f} → commute {a₁} {b} {f} } -- F(a) <- Nat(h_a,F) Nat2F : {a : Obj A} → {F : Obj SetsAop } → Hom SetsAop (y-obj a) F → FObj F a Nat2F {a} {F} ha = ( TMap ha a ) (id1 A a) ---- -- -- Prove Bijection (as routine exercise ...) -- ---- F2Nat→Nat2F : {a : Obj A } → {F : Obj SetsAop } → (fa : FObj F a) → Nat2F {a} {F} (F2Nat {a} {F} fa) ≡ fa F2Nat→Nat2F {a} {F} fa = let open ≈-Reasoning (Sets) in cong ( λ f → f fa ) ( -- FMap F (Category.Category.Id A) fa ≡ fa begin ( FMap F (id1 A _ )) ≈⟨ IsFunctor.identity (isFunctor F) ⟩ id1 Sets (FObj F a) ∎ ) -- open import Relation.Binary.PropositionalEquality ≡-cong = Relation.Binary.PropositionalEquality.cong -- F : op A → Sets -- ha : NTrans (op A) Sets (y-obj {a}) F -- FMap F g o TMap ha a ≈ TMap ha b o FMap (y-obj {a}) g Nat2F→F2Nat : {a : Obj A } → {F : Obj SetsAop } → (ha : Hom SetsAop (y-obj a) F) → SetsAop [ F2Nat {a} {F} (Nat2F {a} {F} ha) ≈ ha ] Nat2F→F2Nat {a} {F} ha {b} = let open ≡-Reasoning in begin TMap (F2Nat {a} {F} (Nat2F {a} {F} ha)) b ≡⟨⟩ (λ g → FMap F g (TMap ha a (Category.Category.Id A))) ≡⟨ extensionality A (λ g → ( begin FMap F g (TMap ha a (Category.Category.Id A)) ≡⟨ ≡-cong (λ f → f (Category.Category.Id A)) (IsNTrans.commute (isNTrans ha)) ⟩ TMap ha b (FMap (y-obj a) g (Category.Category.Id A)) ≡⟨⟩ TMap ha b ( (A Category.o Category.Category.Id A) g ) ≡⟨ ≡-cong ( TMap ha b ) ( ≈-≡ {_} {_} {_} {A} (IsCategory.identityL ( Category.isCategory A ))) ⟩ TMap ha b g ∎ )) ⟩ TMap ha b ∎ -- Yoneda's Lemma -- Yoneda Functor is full and faithfull -- that is FMapp Yoneda is injective and surjective -- λ b g → (A Category.o f₁) g YondaLemma1 : {a a' : Obj A } {f : FObj (FObj YonedaFunctor a) a' } → SetsAop [ F2Nat {a'} {FObj YonedaFunctor a} f ≈ FMap YonedaFunctor f ] YondaLemma1 {a} {a'} {f} = refl domF : (y : Obj SetsAop) → {x : Obj (Category.op A)} → FObj y x → Obj A domF y {x} z = x YondaLemma2 : {a a' b : Obj A } (f : Hom A a a' ) → NTrans (Category.op A) Sets (FObj YonedaFunctor a) (FObj YonedaFunctor a') YondaLemma2 f = FMap YonedaFunctor f YondaLemma3 : {a a' b : Obj A } (f : Hom A a a' ) → Hom A b a → Hom A b a' YondaLemma3 {a} {a'} {b} f = TMap (FMap YonedaFunctor f) b -- YondaLemma4 : {a a' b : Obj A } → (f : Hom A a a' ) → Hom ? (id1 A b) f -- YondaLemma4 {a} {a'} {b} f = Hom A b a → Hom A b a' _^ : {a a' b : Obj A } → (f : Hom A a a' ) → Hom A b a → Hom A b a' _^ {a} {a'} {b} f g = (FMap (FObj YonedaFunctor a') g) f f-unique : {a a' b : Obj A } (f : Hom A a a' ) → f ^ ≡ TMap (FMap YonedaFunctor f) b f-unique {a} {a'} {b} f = extensionality A (λ g → begin (f ^ ) g ≡⟨⟩ (FMap (FObj YonedaFunctor a') g) f ≡⟨⟩ A [ f o g ] ≡⟨⟩ TMap (FMap YonedaFunctor f) b g ∎ ) where open ≡-Reasoning -- {!!} ≡⟨ Relation.Binary.PropositionalEquality.sym (cong (λ k → k (A [ f o z ])) (IsNTrans.commute (isNTrans (invY f)))) ⟩ -- FObj (FObj YonedaFunctor a) c -- c : Obj (Category.op A) -- g : Category.Category.Hom A c a' invY : {a a' b : Obj A } (f : Hom A a a' ) → Hom SetsAop (FObj YonedaFunctor a') (FObj YonedaFunctor a) invY {a} {a'} {b} f = record { TMap = λ c g → A [ {!!} o g ] ; isNTrans = record { commute = {!!} } } inv1 : {a a' b : Obj A } (f : Hom A a a' ) → SetsAop [ SetsAop [ invY {a} {a'} {b} f o FMap YonedaFunctor f ] ≈ id1 SetsAop _ ] inv1 {a} {a'} {b} f = extensionality A (λ z → begin TMap (invY f) (domF (FObj YonedaFunctor a) z) (A [ f o z ] ) ≡⟨ {!!} ⟩ TMap (invY f) (domF (FObj YonedaFunctor a) z) ((FMap (FObj YonedaFunctor a') z) f) ≡⟨ {!!} ⟩ FMap (FObj YonedaFunctor a) z (TMap (invY f) _ f ) ≡⟨ {!!} ⟩ z ∎ ) where open ≡-Reasoning inv2 : {a a' b : Obj A } (f : Hom A a a' ) → SetsAop [ SetsAop [ FMap YonedaFunctor f o invY {a} {a'} {b} f ] ≈ id1 SetsAop _ ] inv2 {a} {a'} {b} f = {!!} -- full (injective ) Yoneda-injective : {a b : Obj A } → {x y : Obj SetsAop} → (g h : Hom SetsAop y (FObj YonedaFunctor a)) (f : Hom A a b ) → SetsAop [ SetsAop [ FMap YonedaFunctor f o g ] ≈ SetsAop [ FMap YonedaFunctor f o h ] ] → SetsAop [ g ≈ h ] Yoneda-injective {a} {b} {x} {y} g h f yg=yh = extensionality A (λ z → begin TMap g _ z ≡⟨ {!!} ⟩ TMap h _ z ∎ ) where open ≡-Reasoning -- faithful (surjective) Yoneda-surjective : {a b : Obj A } → {x y : Obj SetsAop} → (g h : Hom SetsAop (FObj YonedaFunctor b) y) (f : Hom A a b ) → SetsAop [ SetsAop [ g o FMap YonedaFunctor f ] ≈ SetsAop [ h o FMap YonedaFunctor f ] ] → SetsAop [ g ≈ h ] Yoneda-surjective {a} {b} {x} {y} g h f gy=hy = begin g ≈⟨ {!!} ⟩ h ∎ where open ≈-Reasoning SetsAop open import Relation.Binary.HeterogeneousEquality as HE using (_≅_) Yoneda-full-embed : {a b : Obj A } → FObj YonedaFunctor a ≡ FObj YonedaFunctor b → a ≡ b Yoneda-full-embed {a} {b} eq = ylem2 where ylem1 : Hom A a a ≡ Hom A a b ylem1 = cong (λ k → FObj k a) eq ylem3 : (f : Hom A a a ) → Hom A a b ylem3 f = subst ( λ k → k ) ylem1 f postulate ylem0 : Hom A a a ≡ Hom A a b → a ≡ b -- I still don't know how to do it ylem2 : Category.cod A (id1 A a) ≡ b -- ylem2 = begin -- Category.cod A (id1 A a) ≡⟨ HE.cong (λ k → Category.cod A k) (ylem4 (id1 A a)) ⟩ -- Category.cod A (ylem3 (id1 A a)) ≡⟨⟩ -- b ∎ where open ≡-Reasoning -- F2Nat is bijection so FMap YonedaFunctor also ( using functional extensionality ) -- Full embedding of Yoneda Functor requires injective on Object, -- -- But we cannot prove like this -- FObj YonedaFunctor a ≡ FObj YonedaFunctor b → a ≡ b a1 : { c₁ : Level} {A B : Set c₁ } {a : A } { b : B } → a ≅ b → A ≡ B a1 HE.refl = refl open import Relation.Nullary open import Data.Empty --YondaLemma2 : {c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → -- (a b x : Obj A ) → (FObj (FObj (YonedaFunctor A) a) x) ≡ (FObj (FObj (YonedaFunctor A) b ) x) → a ≡ b --YondaLemma2 A a bx eq = {!!} -- N.B = ≡-cong gives you ! a ≡ b, so we cannot cong inv to prove a ≡ b --record Category c₁ c₂ ℓ : Set (suc (c₁ ⊔ c₂ ⊔ ℓ)) where -- infixr 9 _o_ -- infix 4 _≈_ -- field -- Obj : Set c₁ -- Hom : Obj → Obj → Set c₂ --YondaLemma31 : {c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → -- (a b x : Obj A ) → Hom A a x ≡ Hom A b x → a ≡ b --YondaLemma31 A a b x eq = {!!} -- -- Instead we prove only -- inv ( FObj YonedaFunctor a ) ≡ a inv : {a x : Obj A} ( f : FObj (FObj YonedaFunctor a) x) → Obj A inv {a} f = Category.cod A f YonedaLemma21 : {a x : Obj A} ( f : ( FObj (FObj YonedaFunctor a) x) ) → inv f ≡ a YonedaLemma21 {a} {x} f = refl -- YondaLemma3 : {c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → -- (a b x : Obj A ) → Hom A a x ≅ Hom A b x → a ≡ b -- YondaLemma3 A a b x eq = {!!} -- ≡-cong (inv A) ? -- a2 : ( a b : Set ) (f : a → a ) (g : b → a ) -> f ≅ g → a ≡ b -- a2 a b f g eq = {!!} -- YonedaInjective : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} -- → FObj (FObj (YonedaFunctor A ) a ) a ≅ FObj (FObj (YonedaFunctor A ) a ) b -- → a ≡ b -- YonedaInjective A {a} {b} eq = {!!}