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1 open import Level
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2 open import Category
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3 module CCChom where
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4
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5 open import HomReasoning
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6 open import cat-utility
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7 open import Data.Product renaming (_×_ to _∧_)
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8 open import Category.Constructions.Product
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9 open import Relation.Binary.PropositionalEquality
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10
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11 open Functor
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12
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13 -- ccc-1 : Hom A a 1 ≅ {*}
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14 -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b )
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15 -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c
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16
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17 data One : Set where
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18 OneObj : One -- () in Haskell ( or any one object set )
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19
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20 OneCat : Category Level.zero Level.zero Level.zero
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21 OneCat = record {
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22 Obj = One ;
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23 Hom = λ a b → One ;
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24 _o_ = λ{a} {b} {c} x y → OneObj ;
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25 _≈_ = λ x y → x ≡ y ;
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26 Id = λ{a} → OneObj ;
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27 isCategory = record {
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28 isEquivalence = record {refl = refl ; trans = trans ; sym = sym } ;
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29 identityL = λ{a b f} → lemma {a} {b} {f} ;
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30 identityR = λ{a b f} → lemma {a} {b} {f} ;
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31 o-resp-≈ = λ{a b c f g h i} _ _ → refl ;
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32 associative = λ{a b c d f g h } → refl
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33 }
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34 } where
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35 lemma : {a b : One } → { f : One } → OneObj ≡ f
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36 lemma {a} {b} {f} with f
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37 ... | OneObj = refl
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38
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39 record IsoS {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') (a b : Obj A) ( a' b' : Obj B )
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40 : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' ) where
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41 field
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42 ≅→ : Hom A a b → Hom B a' b'
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43 ≅← : Hom B a' b' → Hom A a b
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44 iso→ : {f : Hom B a' b' } → B [ ≅→ ( ≅← f) ≈ f ]
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45 iso← : {f : Hom A a b } → A [ ≅← ( ≅→ f) ≈ f ]
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46 cong→ : {f g : Hom A a b } → A [ f ≈ g ] → B [ ≅→ f ≈ ≅→ g ]
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47 cong← : {f g : Hom B a' b'} → B [ f ≈ g ] → A [ ≅← f ≈ ≅← g ]
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48
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49
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50 record IsCCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (１ : Obj A)
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51 ( _*_ : Obj A → Obj A → Obj A ) ( _^_ : Obj A → Obj A → Obj A ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
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52 field
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53 ccc-1 : {a : Obj A} {b c : Obj OneCat} → -- Hom A a １ ≅ {*}
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54 IsoS A OneCat a １ b c
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55 ccc-2 : {a b c : Obj A} → -- Hom A c ( a * b ) ≅ ( Hom A c a ) * ( Hom A c b )
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56 IsoS A (A × A) c (a * b) (c , c ) (a , b )
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57 ccc-3 : {a b c : Obj A} → -- Hom A a ( c ^ b ) ≅ Hom A ( a * b ) c
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58 IsoS A A a (c ^ b) (a * b) c
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59 nat-2 : {a b c : Obj A} → {f : Hom A (b * c) (b * c) } → {g : Hom A a (b * c) }
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60 → (A × A) [ (A × A) [ IsoS.≅→ ccc-2 f o (g , g) ] ≈ IsoS.≅→ ccc-2 ( A [ f o g ] ) ]
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61
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62 open import CCC
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63
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64
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65 record CCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
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66 field
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67 one : Obj A
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68 _*_ : Obj A → Obj A → Obj A
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69 _^_ : Obj A → Obj A → Obj A
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70 isCCChom : IsCCChom A one _*_ _^_
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71
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72 open import HomReasoning
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73
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74 CCC→hom : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( c : CCC A ) → CCChom A
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75 CCC→hom A c = record {
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76 one = CCC.１ c
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77 ; _*_ = CCC._∧_ c
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78 ; _^_ = CCC._<=_ c
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79 ; isCCChom = record {
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80 ccc-1 = λ {a} {b} {c'} → record { ≅→ = c101 ; ≅← = c102 ; iso→ = c103 {a} {b} {c'} ; iso← = c104 ; cong← = c105 ; cong→ = c106 }
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81 ; ccc-2 = record { ≅→ = c201 ; ≅← = c202 ; iso→ = c203 ; iso← = c204 ; cong← = c205; cong→ = c206 }
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82 ; ccc-3 = record { ≅→ = c301 ; ≅← = c302 ; iso→ = c303 ; iso← = c304 ; cong← = c305 ; cong→ = c306 }
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83 ; nat-2 = nat-2
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84 }
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85 } where
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86 c101 : {a : Obj A} → Hom A a (CCC.１ c) → Hom OneCat OneObj OneObj
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87 c101 _ = OneObj
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88 c102 : {a : Obj A} → Hom OneCat OneObj OneObj → Hom A a (CCC.１ c)
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89 c102 {a} OneObj = CCC.○ c a
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90 c103 : {a : Obj A } {b c : Obj OneCat} {f : Hom OneCat b b } → _[_≈_] OneCat {b} {c} ( c101 {a} (c102 {a} f) ) f
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91 c103 {a} {OneObj} {OneObj} {OneObj} = refl
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92 c104 : {a : Obj A} → {f : Hom A a (CCC.１ c)} → A [ (c102 ( c101 f )) ≈ f ]
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93 c104 {a} {f} = let open ≈-Reasoning A in HomReasoning.≈-Reasoning.sym A (IsCCC.e2 (CCC.isCCC c) f )
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94 c201 : { c₁ a b : Obj A} → Hom A c₁ ((c CCC.∧ a) b) → Hom (A × A) (c₁ , c₁) (a , b)
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95 c201 f = ( A [ CCC.π c o f ] , A [ CCC.π' c o f ] )
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96 c202 : { c₁ a b : Obj A} → Hom (A × A) (c₁ , c₁) (a , b) → Hom A c₁ ((c CCC.∧ a) b)
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97 c202 (f , g ) = CCC.<_,_> c f g
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98 c203 : { c₁ a b : Obj A} → {f : Hom (A × A) (c₁ , c₁) (a , b)} → (A × A) [ (c201 ( c202 f )) ≈ f ]
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99 c203 = ( IsCCC.e3a (CCC.isCCC c) , IsCCC.e3b (CCC.isCCC c))
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100 c204 : { c₁ a b : Obj A} → {f : Hom A c₁ ((c CCC.∧ a) b)} → A [ (c202 ( c201 f )) ≈ f ]
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101 c204 = IsCCC.e3c (CCC.isCCC c)
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102 c301 : { d a b : Obj A} → Hom A a ((c CCC.<= d) b) → Hom A ((c CCC.∧ a) b) d -- a -> d <= b -> (a ∧ b ) -> d
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103 c301 {d} {a} {b} f = A [ CCC.ε c o CCC.<_,_> c ( A [ f o CCC.π c ] ) ( CCC.π' c ) ]
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104 c302 : { d a b : Obj A} → Hom A ((c CCC.∧ a) b) d → Hom A a ((c CCC.<= d) b)
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105 c302 f = CCC._* c f
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106 c303 : { c₁ a b : Obj A} → {f : Hom A ((c CCC.∧ a) b) c₁} → A [ (c301 ( c302 f )) ≈ f ]
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107 c303 = IsCCC.e4a (CCC.isCCC c)
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108 c304 : { c₁ a b : Obj A} → {f : Hom A a ((c CCC.<= c₁) b)} → A [ (c302 ( c301 f )) ≈ f ]
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109 c304 = IsCCC.e4b (CCC.isCCC c)
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110 c105 : {a : Obj A } {f g : Hom OneCat OneObj OneObj} → _[_≈_] OneCat {OneObj} {OneObj} f g → A [ c102 {a} f ≈ c102 {a} g ]
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111 c105 refl = let open ≈-Reasoning A in refl-hom
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112 c106 : { a : Obj A } {f g : Hom A a (CCC.１ c)} → A [ f ≈ g ] → _[_≈_] OneCat {OneObj} {OneObj} OneObj OneObj
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113 c106 _ = refl
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114 c205 : { a b c₁ : Obj A } {f g : Hom (A × A) (c₁ , c₁) (a , b)} → (A × A) [ f ≈ g ] → A [ c202 f ≈ c202 g ]
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115 c205 f=g = IsCCC.π-cong (CCC.isCCC c ) (proj₁ f=g ) (proj₂ f=g )
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116 c206 : { a b c₁ : Obj A } {f g : Hom A c₁ ((c CCC.∧ a) b)} → A [ f ≈ g ] → (A × A) [ c201 f ≈ c201 g ]
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117 c206 {a} {b} {c₁} {f} {g} f=g = ( begin
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118 CCC.π c o f
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119 ≈⟨ cdr f=g ⟩
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120 CCC.π c o g
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121 ∎ ) , ( begin
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122 CCC.π' c o f
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123 ≈⟨ cdr f=g ⟩
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124 CCC.π' c o g
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125 ∎ ) where open ≈-Reasoning A
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126 c305 : { a b c₁ : Obj A } {f g : Hom A ((c CCC.∧ a) b) c₁} → A [ f ≈ g ] → A [ (c CCC.*) f ≈ (c CCC.*) g ]
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127 c305 f=g = IsCCC.*-cong (CCC.isCCC c ) f=g
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128 c306 : { a b c₁ : Obj A } {f g : Hom A a ((c CCC.<= c₁) b)} → A [ f ≈ g ] → A [ c301 f ≈ c301 g ]
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129 c306 {a} {b} {c₁} {f} {g} f=g = begin
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130 CCC.ε c o CCC.<_,_> c ( f o CCC.π c ) ( CCC.π' c )
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131 ≈⟨ cdr ( IsCCC.π-cong (CCC.isCCC c ) (car f=g ) refl-hom) ⟩
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132 CCC.ε c o CCC.<_,_> c ( g o CCC.π c ) ( CCC.π' c )
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133 ∎ where open ≈-Reasoning A
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134 nat-2 : {a b : Obj A} {c = c₁ : Obj A} {f : Hom A ((c CCC.∧ b) c₁) ((c CCC.∧ b) c₁)}
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135 {g : Hom A a ((c CCC.∧ b) c₁)} → (A × A) [ (A × A) [ c201 f o g , g ] ≈ c201 (A [ f o g ]) ]
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136 nat-2 {a} {b} {c₁} {f} {g} = ( begin
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137 ( CCC.π c o f) o g
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138 ≈↑⟨ assoc ⟩
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139 ( CCC.π c ) o (f o g)
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140 ∎ ) , (sym-hom assoc) where open ≈-Reasoning A
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141
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142
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143
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144 open CCChom
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145 open IsCCChom
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146 open IsoS
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147
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148 hom→CCC : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( h : CCChom A ) → CCC A
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149 hom→CCC A h = record {
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150 １ = １
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151 ; ○ = ○
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152 ; _∧_ = _/\_
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153 ; <_,_> = <,>
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154 ; π = π
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155 ; π' = π'
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156 ; _<=_ = _<=_
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157 ; _* = _*
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158 ; ε = ε
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159 ; isCCC = isCCC
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160 } where
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161 １ : Obj A
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162 １ = one h
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163 ○ : (a : Obj A ) → Hom A a １
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164 ○ a = ≅← ( ccc-1 (isCCChom h ) {_} {OneObj} {OneObj} ) OneObj
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165 _/\_ : Obj A → Obj A → Obj A
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166 _/\_ a b = _*_ h a b
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167 <,> : {a b c : Obj A } → Hom A c a → Hom A c b → Hom A c ( a /\ b)
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168 <,> f g = ≅← ( ccc-2 (isCCChom h ) ) ( f , g )
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169 π : {a b : Obj A } → Hom A (a /\ b) a
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170 π {a} {b} = proj₁ ( ≅→ ( ccc-2 (isCCChom h ) ) (id1 A (_*_ h a b) ))
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171 π' : {a b : Obj A } → Hom A (a /\ b) b
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172 π' {a} {b} = proj₂ ( ≅→ ( ccc-2 (isCCChom h ) ) (id1 A (_*_ h a b) ))
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173 _<=_ : (a b : Obj A ) → Obj A
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174 _<=_ = _^_ h
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175 _* : {a b c : Obj A } → Hom A (a /\ b) c → Hom A a (c <= b)
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176 _* = ≅← ( ccc-3 (isCCChom h ) )
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177 ε : {a b : Obj A } → Hom A ((a <= b ) /\ b) a
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178 ε {a} {b} = ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b ))
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179 isCCC : CCC.IsCCC A １ ○ _/\_ <,> π π' _<=_ _* ε
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180 isCCC = record {
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181 e2 = e2
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182 ; e3a = e3a
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183 ; e3b = e3b
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184 ; e3c = e3c
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185 ; π-cong = π-cong
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186 ; e4a = e4a
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187 ; e4b = e4b
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188 ; *-cong = *-cong
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189 } where
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190 e20 : ∀ ( f : Hom OneCat OneObj OneObj ) → _[_≈_] OneCat {OneObj} {OneObj} f OneObj
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191 e20 OneObj = refl
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192 e2 : {a : Obj A} → ∀ ( f : Hom A a １ ) → A [ f ≈ ○ a ]
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193 e2 {a} f = begin
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194 f
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195 ≈↑⟨ iso← ( ccc-1 (isCCChom h )) ⟩
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196 ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj}) ( ≅→ ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) f )
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197 ≈⟨ ≡-cong {Level.zero} {Level.zero} {Level.zero} {OneCat} {OneObj} {OneObj} (
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198 λ y → ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) y ) (e20 ( ≅→ ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) f) ) ⟩
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199 ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) OneObj
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200 ≈⟨⟩
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201 ○ a
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202 ∎ where open ≈-Reasoning A
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203 --
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204 -- g id
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205 -- a -------------> b * c ------> b * c
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206 --
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207 -- a -------------> b * c ------> b
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208 -- a -------------> b * c ------> c
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209 --
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210 cong-proj₁ : {a b c d : Obj A} → { f g : Hom (A × A) ( a , b ) ( c , d ) } → (A × A) [ f ≈ g ] → A [ proj₁ f ≈ proj₁ g ]
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211 cong-proj₁ eq = proj₁ eq
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212 cong-proj₂ : {a b c d : Obj A} → { f g : Hom (A × A) ( a , b ) ( c , d ) } → (A × A) [ f ≈ g ] → A [ proj₂ f ≈ proj₂ g ]
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213 cong-proj₂ eq = proj₂ eq
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214 e3a : {a b c : Obj A} → { f : Hom A c a }{ g : Hom A c b } → A [ A [ π o <,> f g ] ≈ f ]
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215 e3a {a} {b} {c} {f} {g} = begin
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216 π o <,> f g
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217 ≈⟨⟩
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218 proj₁ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) o (≅← (ccc-2 (isCCChom h)) (f , g))
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219 ≈⟨ cong-proj₁ (nat-2 (isCCChom h)) ⟩
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220 proj₁ (≅→ (ccc-2 (isCCChom h)) (( id1 A ( _*_ h a b )) o ( ≅← (ccc-2 (isCCChom h)) (f , g) ) ))
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221 ≈⟨ cong-proj₁ ( cong→ (ccc-2 (isCCChom h)) idL ) ⟩
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222 proj₁ (≅→ (ccc-2 (isCCChom h)) ( ≅← (ccc-2 (isCCChom h)) (f , g) ))
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223 ≈⟨ cong-proj₁ ( iso→ (ccc-2 (isCCChom h))) ⟩
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224 proj₁ ( f , g )
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225 ≈⟨⟩
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226 f
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227 ∎ where open ≈-Reasoning A
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228 e3b : {a b c : Obj A} → { f : Hom A c a }{ g : Hom A c b } → A [ A [ π' o <,> f g ] ≈ g ]
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229 e3b {a} {b} {c} {f} {g} = begin
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230 π' o <,> f g
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231 ≈⟨⟩
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232 proj₂ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) o (≅← (ccc-2 (isCCChom h)) (f , g))
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233 ≈⟨ cong-proj₂ (nat-2 (isCCChom h)) ⟩
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234 proj₂ (≅→ (ccc-2 (isCCChom h)) (( id1 A ( _*_ h a b )) o ( ≅← (ccc-2 (isCCChom h)) (f , g) ) ))
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235 ≈⟨ cong-proj₂ ( cong→ (ccc-2 (isCCChom h)) idL ) ⟩
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236 proj₂ (≅→ (ccc-2 (isCCChom h)) ( ≅← (ccc-2 (isCCChom h)) (f , g) ))
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237 ≈⟨ cong-proj₂ ( iso→ (ccc-2 (isCCChom h))) ⟩
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238 proj₂ ( f , g )
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239 ≈⟨⟩
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240 g
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241 ∎ where open ≈-Reasoning A
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242 e3c : {a b c : Obj A} → { h : Hom A c (a /\ b) } → A [ <,> ( A [ π o h ] ) ( A [ π' o h ] ) ≈ h ]
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243 e3c {a} {b} {c} {f} = begin
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244 <,> ( π o f ) ( π' o f )
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245 ≈⟨⟩
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246 ≅← (ccc-2 (isCCChom h)) ( ( proj₁ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) ))) o f
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247 , ( proj₂ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b)))) o f )
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248 ≈⟨⟩
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249 ≅← (ccc-2 (isCCChom h)) (_[_o_] (A × A) (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) (f , f ) )
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250 ≈⟨ cong← (ccc-2 (isCCChom h)) (nat-2 (isCCChom h)) ⟩
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251 ≅← (ccc-2 (isCCChom h)) (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) o f ))
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252 ≈⟨ cong← (ccc-2 (isCCChom h)) (cong→ (ccc-2 (isCCChom h)) idL ) ⟩
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253 ≅← (ccc-2 (isCCChom h)) (≅→ (ccc-2 (isCCChom h)) f )
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254 ≈⟨ iso← (ccc-2 (isCCChom h)) ⟩
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255 f
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256 ∎ where open ≈-Reasoning A
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785
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257 π-cong : {a b c : Obj A} → { f f' : Hom A c a }{ g g' : Hom A c b } → A [ f ≈ f' ] → A [ g ≈ g' ] → A [ <,> f g ≈ <,> f' g' ]
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786
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258 π-cong {a} {b} {c} {f} {f'} {g} {g'} eq1 eq2 = begin
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259 <,> f g
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260 ≈⟨⟩
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261 ≅← (ccc-2 (isCCChom h)) (f , g)
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787
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262 ≈⟨ cong← (ccc-2 (isCCChom h)) ( eq1 , eq2 ) ⟩
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786
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263 ≅← (ccc-2 (isCCChom h)) (f' , g')
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264 ≈⟨⟩
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265 <,> f' g'
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266 ∎ where open ≈-Reasoning A
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787
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267 e40 : {a c : Obj A} → { f : Hom A (_*_ h a c ) a } → A [ ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) f) ≈ f ]
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268 e40 = iso→ (ccc-3 (isCCChom h))
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269 e41 : {a c : Obj A} → { f : Hom A a (_^_ h c a )} → A [ ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) f) ≈ f ]
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270 e41 = iso← (ccc-3 (isCCChom h))
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786
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271 e4a : {a b c : Obj A} → { k : Hom A (c /\ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ]
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272 e4a {a} {b} {c} {k} = begin
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273 ε o ( <,> ((k *) o π ) π' )
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274 ≈⟨⟩
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787
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275 ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h)) ((( ≅← (ccc-3 (isCCChom h)) k) o π ) , π'))
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786
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276 ≈⟨ {!!} ⟩
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788
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277 ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h))
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278 (_[_o_] (A × A) ( ≅← (ccc-3 (isCCChom h)) k , id1 A b ) ( π , π')))
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279 ≈⟨ {!!} ⟩
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787
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280 ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) k)
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281 ≈⟨ iso→ (ccc-3 (isCCChom h)) ⟩
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786
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282 k
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283 ∎ where open ≈-Reasoning A
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785
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284 e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ]
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787
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285 e4b {a} {b} {c} {k} = begin
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286 ( ε o ( <,> ( k o π ) π' ) ) *
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287 ≈⟨⟩
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288 ≅← (ccc-3 (isCCChom h)) ( ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b )) o (≅← (ccc-2 (isCCChom h)) ( k o π , π')))
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289 ≈⟨ {!!} ⟩
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290 ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) k)
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291 ≈⟨ iso← (ccc-3 (isCCChom h)) ⟩
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292 k
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293 ∎ where open ≈-Reasoning A
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294 *-cong : {a b c : Obj A} {f f' : Hom A (a /\ b) c} → A [ f ≈ f' ] → A [ f * ≈ f' * ]
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295 *-cong eq = cong← ( ccc-3 (isCCChom h )) eq
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785
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296
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787
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297
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298
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