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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 02 May 2020 03:03:51 +0900 |
| parents | 232cea484067 |
| children |
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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 where -- { 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) YObj {_} {c₂} A = Functor (Category.op A) (Sets {c₂}) YHom : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (f : YObj A ) → (g : YObj A ) → Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) YHom {_} {c₂} A f g = NTrans (Category.op A) (Sets {c₂}) f g open NTrans Yid : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : YObj A } → YHom A a a Yid {_} {c₂} A {a} = record { TMap = λ a → λ x → x ; isNTrans = isNTrans1 {a} } where isNTrans1 : {a : YObj A } → IsNTrans (Category.op A) (Sets {c₂}) a a (λ a → λ x → x ) isNTrans1 {a} = record { commute = refl } _+_ : { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } {a b c : YObj A} → YHom A b c → YHom A a b → YHom A a c _+_ {_} {c₂} {_} {A} {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 A ) (f : YHom A b c) (g : YHom A 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 } _==_ : { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } {a b : YObj A} → YHom A a b → YHom A a b → Set (c₂ ⊔ c₁) _==_ {_} { c₂} {_} {A} f g = ∀{x : Obj (Category.op A)} → (Sets {c₂}) [ TMap f x ≈ TMap g x ] infix 4 _==_ isSetsAop : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → IsCategory (YObj A) (YHom A) _==_ _+_ ( Yid A ) isSetsAop {_} {c₂} {_} A = 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 A } {i j : YHom A a b } → i == j → j == i sym1 {a} {b} {i} {j} eq {x} = sym eq trans1 : {a b : YObj A } {i j k : YHom A 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 A} {f g : YHom A A₁ B} {h i : YHom A 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Category (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁)) (c₂ ⊔ c₁) SetsAop {_} {c₂} {_} A = record { Obj = YObj A ; Hom = YHom A ; _o_ = _+_ ; _≈_ = _==_ ; Id = Yid A ; isCategory = isSetsAop A } -- 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 Relation.Binary.PropositionalEquality -- 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 ) → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂ ---- -- -- Object mapping in Yoneda Functor -- ---- open import Function y-obj : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a : Obj A) → Functor (Category.op A) (Sets {c₂}) y-obj {_} {c₂} {_} A 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( a b : Obj A ) → (f : Hom A a b ) → (x : Obj (Category.op A)) → FObj (y-obj A a) x → FObj (y-obj A b ) x y-tmap {_} {c₂} {_} A a b f x = λ ( g : Hom A x a ) → A [ f o g ] -- ( h : Hom A x b ) y-map : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A } → (f : Hom A a b ) → YHom A (y-obj A a) (y-obj A b) y-map {_} {c₂} {_} A {a} {b} f = record { TMap = y-tmap A 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 A b) g o y-tmap A a b f a₁ ] ≈ Sets [ y-tmap A a b f b₁ o FMap (y-obj A 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 a) (y-obj A b) (y-tmap A 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Functor A (SetsAop A) YonedaFunctor A = record { FObj = λ a → y-obj A a ; FMap = λ f → y-map A 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 A [ y-map A f ≈ y-map A 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 A [ y-map A (id1 A a) ≈ id1 (SetsAop A) (y-obj A 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 A [ y-map A (A [ g o f ]) ≈ SetsAop A [ y-map A g o y-map A 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A} → {F : Obj ( SetsAop A) } → {x : FObj F a} → (b : Obj (Category.op A)) → Hom Sets (FObj (y-obj A a) b) (FObj F b) F2Natmap A {a} {F} {x} b = λ ( g : Hom A b a ) → ( FMap F g ) x F2Nat : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A} → {F : Obj (SetsAop A )} → FObj F a → Hom (SetsAop A) (y-obj A a) F F2Nat {_} {c₂} A {a} {F} x = record { TMap = F2Natmap A {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 {a} {F} {x} a₁ ] ≈ Sets [ F2Natmap A {a} {F} {x} b o FMap (y-obj A a) f ] ] commute {a₁} {b} {f} = let open ≈-Reasoning (Sets) in begin Sets [ FMap F f o F2Natmap A {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 a) f ] ≈⟨⟩ Sets [ F2Natmap A {a} {F} {x} b o FMap (y-obj A a) f ] ∎ isNTrans1 : IsNTrans (Category.op A) (Sets {c₂}) (y-obj A a) F (F2Natmap A {a} {F}) isNTrans1 = record { commute = λ {a₁ b f} → commute {a₁} {b} {f} } -- F(a) <- Nat(h_a,F) Nat2F : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A} → {F : Obj (SetsAop A) } → Hom (SetsAop A) (y-obj A a) F → FObj F a Nat2F A {a} {F} ha = ( TMap ha a ) (id1 A a) ---- -- -- Prove Bijection (as routine exercise ...) -- ---- F2Nat→Nat2F : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A } → {F : Obj (SetsAop A)} → (fa : FObj F a) → Nat2F A {a} {F} (F2Nat A {a} {F} fa) ≡ fa F2Nat→Nat2F A {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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A } → {F : Obj (SetsAop A)} → (ha : Hom (SetsAop A) (y-obj A a) F) → SetsAop A [ F2Nat A {a} {F} (Nat2F A {a} {F} ha) ≈ ha ] Nat2F→F2Nat A {a} {F} ha {b} = let open ≡-Reasoning in begin TMap (F2Nat A {a} {F} (Nat2F A {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 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a a' : Obj A } {f : FObj (FObj (YonedaFunctor A) a) a' } → SetsAop A [ F2Nat A {a'} {FObj (YonedaFunctor A) a} f ≈ FMap (YonedaFunctor A) f ] YondaLemma1 A {a} {a'} {f} = refl -- 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 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 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a x : Obj A} ( f : FObj (FObj (YonedaFunctor A) a) x) → Obj A inv A {a} f = Category.cod A f YonedaLemma21 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a x : Obj A} ( f : ( FObj (FObj (YonedaFunctor A ) a) x) ) → inv A f ≡ a YonedaLemma21 A {a} {x} f = refl open import Relation.Binary.HeterogeneousEquality a1 : { c₁ : Level} {A B : Set c₁ } {a : A } { b : B } → a ≅ b → A ≡ B a1 refl = 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 = {!!}