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annotate yoneda.agda @ 352:f589e71875ea

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bad approach
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 24 Dec 2014 22:16:20 +0900
parents d6a6dd305da2
children cf9c0f12cec5
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1 ---
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2 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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3 -- A → Sets^A^op : Yoneda Functor
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4 -- Contravariant Functor h_a
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5 -- Nat(h_a,F)
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6 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 ----
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8
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9 open import Category -- https://github.com/konn/category-agda
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10 open import Level
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11 open import Category.Sets
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12 module yoneda where
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13 -- { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where
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14
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15 open import HomReasoning
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16 open import cat-utility
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17 open import Relation.Binary.Core
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18 open import Relation.Binary
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19
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20
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21 -- Contravariant Functor : op A → Sets ( Obj of Sets^{A^op} )
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22 -- Obj and Hom of Sets^A^op
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23
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24 open Functor
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25
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26 YObj : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁))
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27 YObj {_} {c₂} A = Functor (Category.op A) (Sets {c₂})
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28 YHom : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (f : YObj A ) → (g : YObj A ) → Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁))
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29 YHom {_} {c₂} A f g = NTrans (Category.op A) (Sets {c₂}) f g
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30
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31 open NTrans
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32 Yid : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : YObj A } → YHom A a a
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33 Yid {_} {c₂} A {a} = record { TMap = λ a → λ x → x ; isNTrans = isNTrans1 {a} } where
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34 isNTrans1 : {a : YObj A } → IsNTrans (Category.op A) (Sets {c₂}) a a (λ a → λ x → x )
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35 isNTrans1 {a} = record { commute = refl }
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36
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37 _+_ : { 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
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38 _+_ {_} {c₂} {_} {A} {a} {b} {c} f g = record { TMap = λ x → Sets [ TMap f x o TMap g x ] ; isNTrans = isNTrans1 } where
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39 commute1 : (a b c : YObj A ) (f : YHom A b c) (g : YHom A a b )
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40 (a₁ b₁ : Obj (Category.op A)) (h : Hom (Category.op A) a₁ b₁) →
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41 Sets [ Sets [ FMap c h o Sets [ TMap f a₁ o TMap g a₁ ] ] ≈
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42 Sets [ Sets [ TMap f b₁ o TMap g b₁ ] o FMap a h ] ]
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43 commute1 a b c f g a₁ b₁ h = let open ≈-Reasoning (Sets {c₂})in begin
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44 Sets [ FMap c h o Sets [ TMap f a₁ o TMap g a₁ ] ]
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45 ≈⟨ assoc {_} {_} {_} {_} {FMap c h } {TMap f a₁} {TMap g a₁} ⟩
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46 Sets [ Sets [ FMap c h o TMap f a₁ ] o TMap g a₁ ]
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47 ≈⟨ car (nat f) ⟩
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48 Sets [ Sets [ TMap f b₁ o FMap b h ] o TMap g a₁ ]
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49 ≈↑⟨ assoc {_} {_} {_} {_} { TMap f b₁} {FMap b h } {TMap g a₁}⟩
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50 Sets [ TMap f b₁ o Sets [ FMap b h o TMap g a₁ ] ]
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51 ≈⟨ cdr {_} {_} {_} {_} {_} { TMap f b₁} (nat g) ⟩
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52 Sets [ TMap f b₁ o Sets [ TMap g b₁ o FMap a h ] ]
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53 ≈↑⟨ assoc {_} {_} {_} {_} {TMap f b₁} {TMap g b₁} { FMap a h} ⟩
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54 Sets [ Sets [ TMap f b₁ o TMap g b₁ ] o FMap a h ]
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55 ∎
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56 isNTrans1 : IsNTrans (Category.op A) (Sets {c₂}) a c (λ x → Sets [ TMap f x o TMap g x ])
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57 isNTrans1 = record { commute = λ {a₁ b₁ h} → commute1 a b c f g a₁ b₁ h }
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58
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59 _==_ : { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } {a b : YObj A} → YHom A a b → YHom A a b → Set (c₂ ⊔ c₁)
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60 _==_ {_} { c₂} {_} {A} f g = ∀{x : Obj (Category.op A)} → (Sets {c₂}) [ TMap f x ≈ TMap g x ]
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61
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62 infix 4 _==_
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63
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64 isSetsAop : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → IsCategory (YObj A) (YHom A) _==_ _+_ ( Yid A )
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65 isSetsAop {_} {c₂} {_} A =
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66 record { isEquivalence = record {refl = refl ; trans = λ {i j k} → trans1 {_} {_} {i} {j} {k} ; sym = λ {i j} → sym1 {_} {_} {i} {j}}
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67 ; identityL = refl
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68 ; identityR = refl
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69 ; o-resp-≈ = λ{a b c f g h i } → o-resp-≈ {a} {b} {c} {f} {g} {h} {i}
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70 ; associative = refl
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71 } where
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72 sym1 : {a b : YObj A } {i j : YHom A a b } → i == j → j == i
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73 sym1 {a} {b} {i} {j} eq {x} = let open ≈-Reasoning (Sets {c₂}) in begin
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74 TMap j x
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75 ≈⟨ sym eq ⟩
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76 TMap i x
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77 ∎
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78 trans1 : {a b : YObj A } {i j k : YHom A a b} → i == j → j == k → i == k
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79 trans1 {a} {b} {i} {j} {k} i=j j=k {x} = let open ≈-Reasoning (Sets {c₂}) in begin
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80 TMap i x
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81 ≈⟨ i=j ⟩
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82 TMap j x
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83 ≈⟨ j=k ⟩
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84 TMap k x
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85 ∎
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86 o-resp-≈ : {A₁ B C : YObj A} {f g : YHom A A₁ B} {h i : YHom A B C} →
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87 f == g → h == i → h + f == i + g
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88 o-resp-≈ {a} {b} {c} {f} {g} {h} {i} f=g h=i {x} = let open ≈-Reasoning (Sets {c₂}) in begin
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89 (Sets {c₂}) [ TMap h x o TMap f x ]
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90 ≈⟨ resp f=g h=i ⟩
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91 (Sets {c₂}) [ TMap i x o TMap g x ]
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92 ∎
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93
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94 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)
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95 SetsAop {_} {c₂} {_} A =
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96 record { Obj = YObj A
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97 ; Hom = YHom A
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98 ; _o_ = _+_
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99 ; _≈_ = _==_
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100 ; Id = Yid A
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101 ; isCategory = isSetsAop A
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102 }
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103
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104 -- A is Locally small
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105 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
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106
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107 import Relation.Binary.PropositionalEquality
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108 -- 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 )
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109 postulate extensionality : { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } → Relation.Binary.PropositionalEquality.Extensionality c₂ c₂
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110
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111
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112 ----
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113 --
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114 -- Object mapping in Yoneda Functor
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115 --
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
116 ----
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
117
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
118 open import Function
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
119
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
120 y-obj : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a : Obj A) → Functor (Category.op A) (Sets {c₂})
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
121 y-obj {_} {c₂} {_} A a = record {
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
122 FObj = λ b → Hom (Category.op A) a b ;
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
123 FMap = λ {b c : Obj A } → λ ( f : Hom A c b ) → λ (g : Hom A b a ) → (Category.op A) [ f o g ] ;
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
124 isFunctor = record {
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 299
diff changeset
125 identity = λ {b} → extensionality {_} {_} {_} {A} ( λ x → lemma-y-obj1 {b} x ) ;
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
126 distr = λ {a} {b} {c} {f} {g} → extensionality {_} {_} {_} {A} ( λ x → lemma-y-obj2 a b c f g x ) ;
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
127 ≈-cong = λ eq → extensionality {_} {_} {_} {A} ( λ x → lemma-y-obj3 x eq )
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
128 }
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
129 } where
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
130 lemma-y-obj1 : {b : Obj A } → (x : Hom A b a) → (Category.op A) [ id1 A b o x ] ≡ x
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
131 lemma-y-obj1 {b} x = let open ≈-Reasoning (Category.op A) in ≈-≡ {_} {_} {_} {A} idL
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
132 lemma-y-obj2 : (a₁ b c : Obj A) (f : Hom A b a₁) (g : Hom A c b ) → (x : Hom A a₁ a )→
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
133 Category.op A [ Category.op A [ g o f ] o x ] ≡ (Sets [ _[_o_] (Category.op A) g o _[_o_] (Category.op A) f ]) x
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
134 lemma-y-obj2 a₁ b c f g x = let open ≈-Reasoning (Category.op A) in ≈-≡ {_} {_} {_} {A} ( begin
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
135 Category.op A [ Category.op A [ g o f ] o x ]
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
136 ≈↑⟨ assoc ⟩
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
137 Category.op A [ g o Category.op A [ f o x ] ]
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
138 ≈⟨⟩
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
139 ( λ x → Category.op A [ g o x ] ) ( ( λ x → Category.op A [ f o x ] ) x )
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
140 ∎ )
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
141 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 ]
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
142 lemma-y-obj3 {_} {_} {f} {g} x eq = let open ≈-Reasoning (Category.op A) in ≈-≡ {_} {_} {_} {A} ( begin
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
143 Category.op A [ f o x ]
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
144 ≈⟨ resp refl-hom eq ⟩
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
145 Category.op A [ g o x ]
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
146 ∎ )
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
147
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
148
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
149 ----
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
150 --
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
151 -- Hom mapping in Yoneda Functor
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
152 --
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
153 ----
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
154
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
155 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)) →
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
156 FObj (y-obj A a) x → FObj (y-obj A b ) x
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
157 y-tmap {_} {c₂} {_} A a b f x = λ ( g : Hom A x a ) → A [ f o g ] -- ( h : Hom A x b )
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
158
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
159 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)
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
160 y-map {_} {c₂} {_} A {a} {b} f = record { TMap = y-tmap A a b f ; isNTrans = isNTrans1 {a} {b} f } where
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
161 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 ) →
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
162 Sets [ Sets [ FMap (y-obj A b) g o y-tmap A a b f a₁ ] ≈
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
163 Sets [ y-tmap A a b f b₁ o FMap (y-obj A a) g ] ]
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
164 lemma-y-obj4 {a₁} {b₁} {g} {a} {b} f = let open ≈-Reasoning A in extensionality {_} {_} {_} {A} ( λ x → ≈-≡ {_} {_} {_} {A} ( begin
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
165 A [ A [ f o x ] o g ]
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
166 ≈↑⟨ assoc ⟩
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
167 A [ f o A [ x o g ] ]
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
168 ∎ ) )
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
169 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 )
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
170 isNTrans1 {a} {b} f = record { commute = λ{a₁ b₁ g } → lemma-y-obj4 {a₁} {b₁} {g} {a} {b} f }
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
171
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
172 -----
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
173 --
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
174 -- Yoneda Functor itself
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
175 --
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
176 -----
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
177
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
178 YonedaFunctor : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → Functor A (SetsAop A)
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
179 YonedaFunctor {_} {c₂} {_} A = record {
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
180 FObj = λ a → y-obj A a
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
181 ; FMap = λ f → y-map A f
186
b2e01aa0924d y-nat (FMap of Yoneda Functor )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
182 ; isFunctor = record {
187
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
183 identity = identity
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
184 ; distr = distr1
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
185 ; ≈-cong = ≈-cong
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
186
186
b2e01aa0924d y-nat (FMap of Yoneda Functor )
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 185
diff changeset
187 }
187
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
188 } where
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
189 ≈-cong : {a b : Obj A} {f g : Hom A a b} → A [ f ≈ g ] → SetsAop A [ y-map A f ≈ y-map A g ]
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
190 ≈-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 ] )
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
191 extensionality {_} {_} {_} {A} ( λ h → ≈-≡ {_} {_} {_} {A} ( begin
188
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
192 A [ f o h ]
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
193 ≈⟨ resp refl-hom eq ⟩
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
194 A [ g o h ]
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
195 ∎
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
196 ) )
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
197 identity : {a : Obj A} → SetsAop A [ y-map A (id1 A a) ≈ id1 (SetsAop A) (y-obj A a ) ]
188
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
198 identity {a} = let open ≈-Reasoning (A) in -- (λ x g → A [ id1 A a o g ] ) ≡ (λ a₁ x → x)
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
199 extensionality {_} {_} {_} {A} ( λ g → ≈-≡ {_} {_} {_} {A} ( begin
188
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
200 A [ id1 A a o g ]
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
201 ≈⟨ idL ⟩
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
202 g
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
203 ∎
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
204 ) )
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
205 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 ] ]
191
45191dc3f78a nat continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
206 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 ] ] )
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
207 extensionality {_} {_} {_} {A} ( λ h → ≈-≡ {_} {_} {_} {A} ( begin
188
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
208 A [ A [ g o f ] o h ]
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
209 ≈↑⟨ assoc ⟩
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
210 A [ g o A [ f o h ] ]
f4c9d7cbcbb9 Yoneda Functor Constructed
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 187
diff changeset
211 ∎
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
212 ) )
184
d2d749318bee oeration
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 183
diff changeset
213
185
173d078ee443 Yoneda join
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 184
diff changeset
214
190
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
215 ------
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
216 --
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
217 -- F : A → Sets ∈ Obj SetsAop
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
218 --
300
d6a6dd305da2 arrow and lambda fix
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 299
diff changeset
219 -- F(a) → Nat(h_a,F)
191
45191dc3f78a nat continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
220 -- x ∈ F(a) , (g : Hom A b a) → ( FMap F g ) x
190
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
221 ------
187
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 186
diff changeset
222
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
223 F2Natmap : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A} → {F : Obj ( SetsAop A) }
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
224 → {x : FObj F a} → (b : Obj (Category.op A)) → Hom Sets (FObj (y-obj A a) b) (FObj F b)
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
225 F2Natmap A {a} {F} {x} b = λ ( g : Hom A b a ) → ( FMap F g ) x
190
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
226
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
227 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
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
228 F2Nat {_} {c₂} A {a} {F} x = record { TMap = F2Natmap A {a} {F} {x} ; isNTrans = isNTrans1 } where
192
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
229 commute1 : {a₁ b : Obj (Category.op A)} {f : Hom (Category.op A) a₁ b} (g : Hom A a₁ a) →
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
230 (Sets [ FMap F f o FMap F g ]) x ≡ FMap F (A [ g o f ] ) x
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
231 commute1 g = let open ≈-Reasoning (Sets) in
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
232 cong ( λ f → f x ) ( sym ( distr F ) )
191
45191dc3f78a nat continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
233 commute : {a₁ b : Obj (Category.op A)} {f : Hom (Category.op A) a₁ b} →
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
234 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 ] ]
192
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
235 commute {a₁} {b} {f} = let open ≈-Reasoning (Sets) in
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
236 begin
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
237 Sets [ FMap F f o F2Natmap A {a} {F} {x} a₁ ]
192
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
238 ≈⟨⟩
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
239 Sets [ FMap F f o (λ ( g : Hom A a₁ a ) → ( FMap F g ) x) ]
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
240 ≈⟨ extensionality {_} {_} {_} {A} ( λ (g : Hom A a₁ a) → commute1 {a₁} {b} {f} g ) ⟩
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
241 Sets [ (λ ( g : Hom A b a ) → ( FMap F g ) x) o FMap (y-obj A a) f ]
192
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
242 ≈⟨⟩
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
243 Sets [ F2Natmap A {a} {F} {x} b o FMap (y-obj A a) f ]
192
d7e4b7b441be F(a) → Nat(h_a,F) done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 191
diff changeset
244 ∎
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
245 isNTrans1 : IsNTrans (Category.op A) (Sets {c₂}) (y-obj A a) F (F2Natmap A {a} {F})
191
45191dc3f78a nat continue...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 190
diff changeset
246 isNTrans1 = record { commute = λ {a₁ b f} → commute {a₁} {b} {f} }
190
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
247
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
248
199
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
249 -- F(a) <- Nat(h_a,F)
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
250 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
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
251 Nat2F A {a} {F} ha = ( TMap ha a ) (id1 A a)
190
b82d7b054f28 one to one nat
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 189
diff changeset
252
199
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
253 ----
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
254 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
255 -- Prove Bijection (as routine exercise ...)
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
256 --
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
257 ----
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
258
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
259 F2Nat→Nat2F : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a : Obj A } → {F : Obj (SetsAop A)} → (fa : FObj F a)
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
260 → Nat2F A {a} {F} (F2Nat A {a} {F} fa) ≡ fa
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
261 F2Nat→Nat2F A {a} {F} fa = let open ≈-Reasoning (Sets) in cong ( λ f → f fa ) (
199
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
262 -- FMap F (Category.Category.Id A) fa ≡ fa
194
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
263 begin
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
264 ( FMap F (id1 A _ ))
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
265 ≈⟨ IsFunctor.identity (isFunctor F) ⟩
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
266 id1 Sets (FObj F a)
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
267 ∎ )
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
268
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
269 open import Relation.Binary.PropositionalEquality
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
270
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
271 ≡-cong = Relation.Binary.PropositionalEquality.cong
193
4e6f080f0107 isomorphic problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 192
diff changeset
272
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
273 -- F : op A → Sets
197
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
274 -- ha : NTrans (op A) Sets (y-obj {a}) F
ec50ff189f62 clean up
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 196
diff changeset
275 -- FMap F g o TMap ha a ≈ TMap ha b o FMap (y-obj {a}) g
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
276
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
277 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)
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
278 → SetsAop A [ F2Nat A {a} {F} (Nat2F A {a} {F} ha) ≈ ha ]
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
279 Nat2F→F2Nat A {a} {F} ha {b} = let open ≡-Reasoning in
194
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
280 begin
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
281 TMap (F2Nat A {a} {F} (Nat2F A {a} {F} ha)) b
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
282 ≡⟨⟩
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
283 (λ g → FMap F g (TMap ha a (Category.Category.Id A)))
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
284 ≡⟨ extensionality {_} {_} {_} {A} (λ g → (
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
285 begin
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
286 FMap F g (TMap ha a (Category.Category.Id A))
203
1c16d18a8d67 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
287 ≡⟨ ≡-cong (λ f → f (Category.Category.Id A)) (IsNTrans.commute (isNTrans ha)) ⟩
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
288 TMap ha b (FMap (y-obj A a) g (Category.Category.Id A))
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
289 ≡⟨⟩
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
290 TMap ha b ( (A Category.o Category.Category.Id A) g )
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
291 ≡⟨ ≡-cong ( TMap ha b ) ( ≈-≡ {_} {_} {_} {A} (IsCategory.identityL ( Category.isCategory A ))) ⟩
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
292 TMap ha b g
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
293 ∎
202
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
294 )) ⟩
58ee98bbb333 remove an extensionality
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 199
diff changeset
295 TMap ha b
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
296 ∎
194
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
297
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
298 -- Yoneda's Lemma
199
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 198
diff changeset
299 -- Yoneda Functor is full and faithfull
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
300 -- that is FMapp Yoneda is injective and surjective
194
3271d2d04b17 F2Nat→Nat2F done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 193
diff changeset
301
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
302 -- λ b g → (A Category.o f₁) g
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
303 YondaLemma1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a a' : Obj A } {f : FObj (FObj (YonedaFunctor A) a) a' }
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
304 → SetsAop A [ F2Nat A {a'} {FObj (YonedaFunctor A) a} f ≈ FMap (YonedaFunctor A) f ]
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
305 YondaLemma1 A {a} {a'} {f} = refl
195
428d46dfd5aa Nat2F→F2Nat done
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 194
diff changeset
306
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
307 -- F2Nat is bijection so FMap YonedaFunctor also ( using functional extensionality )
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
308
204
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
309 -- Full embedding of Yoneda Functor requires injective on Object,
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
310 --
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
311 -- But we cannot prove like this
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
312 -- FObj YonedaFunctor a ≡ FObj YonedaFunctor b → a ≡ b
204
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
313 -- YondaLemma2 : {a b x : Obj A } → (FObj (FObj YonedaFunctor a) x) ≡ (FObj (FObj YonedaFunctor b ) x) →
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
314 -- a ≡ b
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
315 -- YondaLemma2 {a} {b} eq = {!!}
253
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 204
diff changeset
316 -- N.B = ≡-cong gives you ! a ≡ b, so we cannot cong inv to prove a ≡ b
204
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
317 --
253
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 204
diff changeset
318 -- Instead we prove only
204
b2874c5086ea embedding done?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 203
diff changeset
319 -- inv ( FObj YonedaFunctor a ) ≡ a
196
c040369bd6d4 give up injective on Object?
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 195
diff changeset
320
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
321 inv : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a x : Obj A} ( f : FObj (FObj (YonedaFunctor A) a) x) → Obj A
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
322 inv A {a} f = Category.cod A f
203
1c16d18a8d67 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
323
299
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
324 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
8c72f5284bc8 remove module parameter from yoneda functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 253
diff changeset
325 YonedaLemma21 A {a} {x} f = refl
203
1c16d18a8d67 on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 202
diff changeset
326