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1 open import Level
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2 open import Category
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3 module CCC where
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4
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6 open import HomReasoning
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7 open import cat-utility
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8 open import Relation.Binary.PropositionalEquality
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11 open import HomReasoning
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12
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13 record IsCCC {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
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14 ( １ : Obj A )
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15 ( ○ : (a : Obj A ) → Hom A a １ )
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16 ( _∧_ : Obj A → Obj A → Obj A )
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17 ( <_,_> : {a b c : Obj A } → Hom A c a → Hom A c b → Hom A c (a ∧ b) )
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18 ( π : {a b : Obj A } → Hom A (a ∧ b) a )
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19 ( π' : {a b : Obj A } → Hom A (a ∧ b) b )
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20 ( _<=_ : (a b : Obj A ) → Obj A )
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21 ( _* : {a b c : Obj A } → Hom A (a ∧ b) c → Hom A a (c <= b) )
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22 ( ε : {a b : Obj A } → Hom A ((a <= b ) ∧ b) a )
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23 : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
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24 field
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25 -- cartesian
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26 e2 : {a : Obj A} → ∀ { f : Hom A a １ } → A [ f ≈ ○ a ]
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27 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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28 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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29 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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30 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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31 -- closed
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32 e4a : {a b c : Obj A} → { h : Hom A (c ∧ b) a } → A [ A [ ε o < A [ (h *) o π ] , π' > ] ≈ h ]
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33 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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34 *-cong : {a b c : Obj A} → { f f' : Hom A (a ∧ b) c } → A [ f ≈ f' ] → A [ f * ≈ f' * ]
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35 open ≈-Reasoning A
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36 e'2 : ○ １ ≈ id1 A １
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37 e'2 = begin
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38 ○ １
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39 ≈↑⟨ e2 ⟩
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40 id1 A １
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41 ∎
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42 e''2 : {a b : Obj A} {f : Hom A a b } → ( ○ b o f ) ≈ ○ a
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43 e''2 {a} {b} {f} = begin
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44 ○ b o f
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45 ≈⟨ e2 ⟩
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46 ○ a
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47 ∎
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48 π-id : {a b : Obj A} → < π , π' > ≈ id1 A (a ∧ b )
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49 π-id {a} {b} = begin
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50 < π , π' >
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51 ≈↑⟨ π-cong idR idR ⟩
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52 < π o id1 A (a ∧ b) , π' o id1 A (a ∧ b) >
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53 ≈⟨ e3c ⟩
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54 id1 A (a ∧ b )
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55 ∎
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56 distr-π : {a b c d : Obj A} {f : Hom A c a }{g : Hom A c b } {h : Hom A d c } → ( < f , g > o h ) ≈ < ( f o h ) , ( g o h ) >
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57 distr-π {a} {b} {c} {d} {f} {g} {h} = begin
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58 < f , g > o h
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59 ≈↑⟨ e3c ⟩
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60 < π o < f , g > o h , π' o < f , g > o h >
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61 ≈⟨ π-cong assoc assoc ⟩
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62 < ( π o < f , g > ) o h , (π' o < f , g > ) o h >
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63 ≈⟨ π-cong (car e3a ) (car e3b) ⟩
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64 < f o h , g o h >
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65 ∎
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66 _×_ : { a b c d : Obj A } ( f : Hom A a c ) (g : Hom A b d ) → Hom A (a ∧ b) ( c ∧ d )
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67 f × g = < ( f o π ) , (g o π' ) >
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68 π-exchg : {a b c : Obj A} {f : Hom A c a }{g : Hom A c b } → < π' , π > o < f , g > ≈ < g , f >
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69 π-exchg {a} {b} {c} {f} {g} = begin
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70 < π' , π > o < f , g >
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71 ≈⟨ distr-π ⟩
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72 < π' o < f , g > , π o < f , g > >
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73 ≈⟨ π-cong e3b e3a ⟩
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74 < g , f >
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75 ∎
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76 π'π : {a b : Obj A} → < π' , π > o < π' , π > ≈ id1 A (a ∧ b)
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77 π'π = trans-hom π-exchg π-id
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78 exchg-π : {a b c d : Obj A} {f : Hom A c a }{g : Hom A d b } → < f o π , g o π' > o < π' , π > ≈ < f o π' , g o π >
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79 exchg-π {a} {b} {c} {d} {f} {g} = begin
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80 < f o π , g o π' > o < π' , π >
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81 ≈⟨ distr-π ⟩
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82 < (f o π) o < π' , π > , (g o π' ) o < π' , π > >
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83 ≈↑⟨ π-cong assoc assoc ⟩
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84 < f o (π o < π' , π > ) , g o (π' o < π' , π >)>
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85 ≈⟨ π-cong (cdr e3a) (cdr e3b) ⟩
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86 < f o π' , g o π >
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87 ∎
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88 π≈ : {a b c : Obj A} {f f' : Hom A c a }{g g' : Hom A c b } → < f , g > ≈ < f' , g' > → f ≈ f'
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89 π≈ {_} {_} {_} {f} {f'} {g} {g'} eq = begin
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90 f ≈↑⟨ e3a ⟩
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91 π o < f , g > ≈⟨ cdr eq ⟩
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92 π o < f' , g' > ≈⟨ e3a ⟩
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93 f'
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94 ∎
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95 π'≈ : {a b c : Obj A} {f f' : Hom A c a }{g g' : Hom A c b } → < f , g > ≈ < f' , g' > → g ≈ g'
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96 π'≈ {_} {_} {_} {f} {f'} {g} {g'} eq = begin
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97 g ≈↑⟨ e3b ⟩
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98 π' o < f , g > ≈⟨ cdr eq ⟩
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99 π' o < f' , g' > ≈⟨ e3b ⟩
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100 g'
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101 ∎
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102 distr-* : {a b c d : Obj A } { h : Hom A (a ∧ b) c } { k : Hom A d a } → ( h * o k ) ≈ ( h o < ( k o π ) , π' > ) *
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103 distr-* {a} {b} {c} {d} {h} {k} = begin
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104 h * o k
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105 ≈↑⟨ e4b ⟩
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106 ( ε o < (h * o k ) o π , π' > ) *
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107 ≈⟨ *-cong ( begin
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108 ε o < (h * o k ) o π , π' >
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109 ≈↑⟨ cdr ( π-cong assoc refl-hom ) ⟩
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110 ε o ( < h * o ( k o π ) , π' > )
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111 ≈↑⟨ cdr ( π-cong (cdr e3a) e3b ) ⟩
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112 ε o ( < h * o ( π o < k o π , π' > ) , π' o < k o π , π' > > )
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113 ≈⟨ cdr ( π-cong assoc refl-hom) ⟩
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114 ε o ( < (h * o π) o < k o π , π' > , π' o < k o π , π' > > )
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115 ≈↑⟨ cdr ( distr-π ) ⟩
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116 ε o ( < h * o π , π' > o < k o π , π' > )
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117 ≈⟨ assoc ⟩
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118 ( ε o < h * o π , π' > ) o < k o π , π' >
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119 ≈⟨ car e4a ⟩
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120 h o < k o π , π' >
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121 ∎ ) ⟩
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122 ( h o < k o π , π' > ) *
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123 ∎
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124 α : {a b c : Obj A } → Hom A (( a ∧ b ) ∧ c ) ( a ∧ ( b ∧ c ) )
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125 α = < ( π o π ) , < ( π' o π ) , π' > >
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126 α' : {a b c : Obj A } → Hom A ( a ∧ ( b ∧ c ) ) (( a ∧ b ) ∧ c )
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127 α' = < < π , ( π o π' ) > , ( π' o π' ) >
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128 β : {a b c d : Obj A } { f : Hom A a b} { g : Hom A a c } { h : Hom A a d } → ( α o < < f , g > , h > ) ≈ < f , < g , h > >
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129 β {a} {b} {c} {d} {f} {g} {h} = begin
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130 α o < < f , g > , h >
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131 ≈⟨⟩
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132 ( < ( π o π ) , < ( π' o π ) , π' > > ) o < < f , g > , h >
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133 ≈⟨ distr-π ⟩
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134 < ( ( π o π ) o < < f , g > , h > ) , ( < ( π' o π ) , π' > o < < f , g > , h > ) >
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135 ≈⟨ π-cong refl-hom distr-π ⟩
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136 < ( ( π o π ) o < < f , g > , h > ) , ( < ( ( π' o π ) o < < f , g > , h > ) , ( π' o < < f , g > , h > ) > ) >
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137 ≈↑⟨ π-cong assoc ( π-cong assoc refl-hom ) ⟩
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138 < ( π o (π o < < f , g > , h >) ) , ( < ( π' o ( π o < < f , g > , h > ) ) , ( π' o < < f , g > , h > ) > ) >
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139 ≈⟨ π-cong (cdr e3a ) ( π-cong (cdr e3a ) e3b ) ⟩
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140 < ( π o < f , g > ) , < ( π' o < f , g > ) , h > >
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141 ≈⟨ π-cong e3a ( π-cong e3b refl-hom ) ⟩
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142 < f , < g , h > >
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143 ∎
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144
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145
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146 record CCC {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
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147 field
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148 １ : Obj A
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149 ○ : (a : Obj A ) → Hom A a １
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150 _∧_ : Obj A → Obj A → Obj A
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151 <_,_> : {a b c : Obj A } → Hom A c a → Hom A c b → Hom A c (a ∧ b)
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152 π : {a b : Obj A } → Hom A (a ∧ b) a
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153 π' : {a b : Obj A } → Hom A (a ∧ b) b
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154 _<=_ : (a b : Obj A ) → Obj A
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155 _* : {a b c : Obj A } → Hom A (a ∧ b) c → Hom A a (c <= b)
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156 ε : {a b : Obj A } → Hom A ((a <= b ) ∧ b) a
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157 isCCC : IsCCC A １ ○ _∧_ <_,_> π π' _<=_ _* ε
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158
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159 open Functor
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160
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161 record CCCFunctor {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ')
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162 (ca : CCC A) (cb : CCC B) (functor : Functor A B)
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163 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ')) where
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164 field
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165 f１ : FObj functor (CCC.１ ca) ≡ CCC.１ cb
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166 f○ : {a : Obj A} → B [ FMap functor (CCC.○ ca a) ≈
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167 subst (λ k → Hom B (FObj functor a) k) (sym f１) (CCC.○ cb (FObj functor a)) ]
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168 f∧ : {a b : Obj A} → FObj functor ( CCC._∧_ ca a b ) ≡ CCC._∧_ cb (FObj functor a ) (FObj functor b)
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169 f<= : {a b : Obj A} → FObj functor ( CCC._<=_ ca a b ) ≡ CCC._<=_ cb (FObj functor a ) (FObj functor b)
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170 f<> : {a b c : Obj A} → (f : Hom A c a ) → (g : Hom A c b )
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171 → B [ FMap functor (CCC.<_,_> ca f g ) ≈
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172 subst (λ k → Hom B (FObj functor c) k ) (sym f∧ ) ( CCC.<_,_> cb (FMap functor f ) ( FMap functor g )) ]
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173 fπ : {a b : Obj A} → B [ FMap functor (CCC.π ca {a} {b}) ≈
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174 subst (λ k → Hom B k (FObj functor a) ) (sym f∧ ) (CCC.π cb {FObj functor a} {FObj functor b}) ]
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175 fπ' : {a b : Obj A} → B [ FMap functor (CCC.π' ca {a} {b}) ≈
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176 subst (λ k → Hom B k (FObj functor b) ) (sym f∧ ) (CCC.π' cb {FObj functor a} {FObj functor b}) ]
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177 f* : {a b c : Obj A} → (f : Hom A (CCC._∧_ ca a b) c ) → B [ FMap functor (CCC._* ca f) ≈
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178 subst (λ k → Hom B (FObj functor a) k) (sym f<=) (CCC._* cb ((subst (λ k → Hom B k (FObj functor c) ) f∧ (FMap functor f) ))) ]
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179 fε : {a b : Obj A} → B [ FMap functor (CCC.ε ca {a} {b} )
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180 ≈ subst (λ k → Hom B k (FObj functor a)) (trans (cong (λ k → CCC._∧_ cb k (FObj functor b)) (sym f<=)) (sym f∧))
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181 (CCC.ε cb {FObj functor a} {FObj functor b}) ]
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182
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183 ----
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184 --
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185 -- Sub Object Classifier as Topos
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186 -- pull back on
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187 --
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188 -- iso ○ b
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189 -- e ⇐====⇒ b -----------→ 1
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190 -- | | |
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191 -- | m | | ⊤
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192 -- | ↓ char m ↓
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193 -- + ------→ a -----------→ Ω
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194 -- ker h h
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195 --
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196 open Equalizer
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197 open import equalizer
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198
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199 record Mono {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {b a : Obj A} (mono : Hom A b a) : Set (c₁ ⊔ c₂ ⊔ ℓ) where
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200 field
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201 isMono : {c : Obj A} ( f g : Hom A c b ) → A [ A [ mono o f ] ≈ A [ mono o g ] ] → A [ f ≈ g ]
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202
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203 open Mono
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204
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205 open import equalizer
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206
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207 record IsTopos {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (c : CCC A)
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208 ( Ω : Obj A )
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209 ( ⊤ : Hom A (CCC.１ c) Ω )
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210 (Ker : {a : Obj A} → ( h : Hom A a Ω ) → Equalizer A h (A [ ⊤ o (CCC.○ c a) ]))
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211 (char : {a b : Obj A} → (m : Hom A b a) → Mono A m → Hom A a Ω) : Set ( suc c₁ ⊔ suc c₂ ⊔ suc ℓ ) where
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212 field
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213 char-uniqueness : {a b : Obj A } {h : Hom A a Ω} (m : Hom A b a) → (mono : Mono A m)
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214 → A [ char (equalizer (Ker h)) (record { isMono = λ f g → monic (Ker h)}) ≈ h ]
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215 ker-iso : {a b : Obj A} → (m : Hom A b a) → (mono : Mono A m) → IsoL A m (equalizer (Ker ( char m mono )))
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216 ker : {a : Obj A} → ( h : Hom A a Ω ) → Hom A ( equalizer-c (Ker h) ) a
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217 ker h = equalizer (Ker h)
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218 mm : {a b : Obj A} → (m : Hom A b a) → (mono : Mono A m ) → A [ A [ equalizer (Ker (char m mono)) o Iso.≅→ (IsoL.iso-L (ker-iso m mono)) ] ≈ m ]
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219 mm m mono = IsoL.L≈iso (ker-iso m mono)
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220 mequ : {a b : Obj A} → (m : Hom A b a) → (mono : Mono A m ) → Equalizer A (char m mono) (A [ ⊤ o (CCC.○ c a) ])
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221 mequ {a} {b} m mono = record { equalizer-c = b ; equalizer = m ; isEqualizer =
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222 subst (λ k → IsEqualizer A k _ _ ) {!!} --
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223 ( equalizer+iso (Ker (char m mono)) (Iso.≅→ (IsoL.iso-L (ker-iso m mono)))
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224 (Iso.≅← (IsoL.iso-L (ker-iso m mono))) (Iso.iso→ (IsoL.iso-L (ker-iso m mono))) (Iso.iso← (IsoL.iso-L (ker-iso m mono))) ) }
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225 char-m=⊤ : {a b : Obj A} → (m : Hom A b a) → (mono : Mono A m) → A [ A [ char m mono o m ] ≈ A [ ⊤ o CCC.○ c b ] ]
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226 char-m=⊤ {a} {b} m mono = begin
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227 char m mono o m ≈⟨ {!!} ⟩
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228 char (ker (char m mono) ) {!!} o m ≈⟨ {!!} ⟩
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229 char m mono o (ker (char m mono) o Iso.≅→ (IsoL.iso-L (ker-iso m mono))) ≈⟨ {!!} ⟩
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230 (char m mono o ker (char m mono) ) o Iso.≅→ (IsoL.iso-L (ker-iso m mono)) ≈⟨ {!!} ⟩
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231 {!!} ≈⟨ {!!} ⟩
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232 (⊤ o CCC.○ c a) o m ≈↑⟨ assoc ⟩
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233 ⊤ o (CCC.○ c a o m ) ≈⟨ cdr (IsCCC.e2 (CCC.isCCC c)) ⟩
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234 ⊤ o CCC.○ c b ∎ where open ≈-Reasoning A
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235
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236 record Topos {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (c : CCC A) : Set ( suc c₁ ⊔ suc c₂ ⊔ suc ℓ ) where
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237 field
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238 Ω : Obj A
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239 ⊤ : Hom A (CCC.１ c) Ω
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240 Ker : {a : Obj A} → ( h : Hom A a Ω ) → Equalizer A h (A [ ⊤ o (CCC.○ c a) ])
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952
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241 char : {a b : Obj A} → (m : Hom A b a ) → Mono A m → Hom A a Ω
|
|
975
|
242 isTopos : IsTopos A c Ω ⊤ Ker char
|
|
973
|
243 Monik : {a : Obj A} (h : Hom A a Ω) → Mono A (equalizer (Ker h))
|
|
|
244 Monik h = record { isMono = λ f g → monic (Ker h ) }
|
|
783
|
245
|
|
963
|
246 record NatD {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( １ : Obj A) : Set ( suc c₁ ⊔ suc c₂ ⊔ suc ℓ ) where
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|
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247 field
|
|
|
248 Nat : Obj A
|
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249 nzero : Hom A １ Nat
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|
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250 nsuc : Hom A Nat Nat
|
|
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251
|
|
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252 open NatD
|
|
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253
|
|
986
|
254 record IsToposNat {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( １ : Obj A) (iNat : NatD A １ )
|
|
|
255 ( initialNat : (nat : NatD A １ ) → Hom A (Nat iNat) (Nat nat) )
|
|
963
|
256 : Set ( suc c₁ ⊔ suc c₂ ⊔ suc ℓ ) where
|
|
|
257 field
|
|
986
|
258 izero : (nat : NatD A １ ) → A [ A [ initialNat nat o nzero iNat ] ≈ nzero nat ]
|
|
|
259 isuc : (nat : NatD A １ ) → A [ A [ initialNat nat o nsuc iNat ] ≈ A [ nsuc nat o initialNat nat ] ]
|
|
963
|
260
|
|
|
261 record ToposNat {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( １ : Obj A) : Set ( suc c₁ ⊔ suc c₂ ⊔ suc ℓ ) where
|
|
|
262 field
|
|
986
|
263 iNat : NatD A １
|
|
|
264 initialNat : (nat : NatD A １ ) → Hom A (Nat iNat) (Nat nat)
|
|
|
265 nat-unique : (nat : NatD A １ ) → {g : Hom A (Nat iNat) (Nat nat) }
|
|
|
266 → A [ A [ g o nzero iNat ] ≈ nzero nat ]
|
|
|
267 → A [ A [ g o nsuc iNat ] ≈ A [ nsuc nat o g ] ]
|
|
|
268 → A [ g ≈ initialNat nat ]
|
|
|
269 isToposN : IsToposNat A １ iNat initialNat
|
|
783
|
270
|
