Mercurial > hg > Members > kono > Proof > category
annotate src/CCChom.agda @ 970:72b6b4577911
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 26 Feb 2021 12:19:58 +0900 |
| parents | ac53803b3b2a |
| children | e50368495cf1 |
| rev | line source |
|---|---|
| 779 | 1 open import Level |
| 2 open import Category | |
| 784 | 3 module CCChom where |
| 779 | 4 |
| 5 open import HomReasoning | |
| 6 open import cat-utility | |
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7 open import Data.Product renaming (_×_ to _/\_ ) hiding ( <_,_> ) |
| 784 | 8 open import Category.Constructions.Product |
| 790 | 9 open import Relation.Binary.PropositionalEquality hiding ( [_] ) |
| 779 | 10 |
| 11 open Functor | |
| 12 | |
| 13 -- ccc-1 : Hom A a 1 ≅ {*} | |
| 14 -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b ) | |
| 15 -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c | |
| 16 | |
| 785 | 17 data One : Set where |
| 779 | 18 OneObj : One -- () in Haskell ( or any one object set ) |
| 19 | |
| 780 | 20 OneCat : Category Level.zero Level.zero Level.zero |
| 21 OneCat = record { | |
| 22 Obj = One ; | |
| 23 Hom = λ a b → One ; | |
| 24 _o_ = λ{a} {b} {c} x y → OneObj ; | |
| 25 _≈_ = λ x y → x ≡ y ; | |
| 26 Id = λ{a} → OneObj ; | |
| 27 isCategory = record { | |
| 28 isEquivalence = record {refl = refl ; trans = trans ; sym = sym } ; | |
| 29 identityL = λ{a b f} → lemma {a} {b} {f} ; | |
| 30 identityR = λ{a b f} → lemma {a} {b} {f} ; | |
| 31 o-resp-≈ = λ{a b c f g h i} _ _ → refl ; | |
| 32 associative = λ{a b c d f g h } → refl | |
| 33 } | |
| 34 } where | |
| 785 | 35 lemma : {a b : One } → { f : One } → OneObj ≡ f |
| 780 | 36 lemma {a} {b} {f} with f |
| 37 ... | OneObj = refl | |
| 38 | |
| 783 | 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 ) |
| 40 : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' ) where | |
| 41 field | |
| 42 ≅→ : Hom A a b → Hom B a' b' | |
| 43 ≅← : Hom B a' b' → Hom A a b | |
| 44 iso→ : {f : Hom B a' b' } → B [ ≅→ ( ≅← f) ≈ f ] | |
| 45 iso← : {f : Hom A a b } → A [ ≅← ( ≅→ f) ≈ f ] | |
| 787 | 46 cong→ : {f g : Hom A a b } → A [ f ≈ g ] → B [ ≅→ f ≈ ≅→ g ] |
| 47 cong← : {f g : Hom B a' b'} → B [ f ≈ g ] → A [ ≅← f ≈ ≅← g ] | |
| 779 | 48 |
| 783 | 49 |
| 784 | 50 record IsCCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (１ : Obj A) |
| 780 | 51 ( _*_ : Obj A → Obj A → Obj A ) ( _^_ : Obj A → Obj A → Obj A ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where |
| 779 | 52 field |
| 785 | 53 ccc-1 : {a : Obj A} {b c : Obj OneCat} → -- Hom A a １ ≅ {*} |
| 54 IsoS A OneCat a １ b c | |
| 780 | 55 ccc-2 : {a b c : Obj A} → -- Hom A c ( a * b ) ≅ ( Hom A c a ) * ( Hom A c b ) |
| 784 | 56 IsoS A (A × A) c (a * b) (c , c ) (a , b ) |
| 780 | 57 ccc-3 : {a b c : Obj A} → -- Hom A a ( c ^ b ) ≅ Hom A ( a * b ) c |
| 58 IsoS A A a (c ^ b) (a * b) c | |
| 788 | 59 nat-2 : {a b c : Obj A} → {f : Hom A (b * c) (b * c) } → {g : Hom A a (b * c) } |
| 60 → (A × A) [ (A × A) [ IsoS.≅→ ccc-2 f o (g , g) ] ≈ IsoS.≅→ ccc-2 ( A [ f o g ] ) ] | |
| 789 | 61 nat-3 : {a b c : Obj A} → { k : Hom A c (a ^ b ) } → A [ A [ IsoS.≅→ (ccc-3) (id1 A (a ^ b)) o |
| 62 (IsoS.≅← (ccc-2 ) (A [ k o (proj₁ ( IsoS.≅→ ccc-2 (id1 A (c * b)))) ] , | |
| 63 (proj₂ ( IsoS.≅→ ccc-2 (id1 A (c * b) ))))) ] ≈ IsoS.≅→ (ccc-3 ) k ] | |
| 788 | 64 |
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65 -------- |
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66 -- CCC satisfies hom natural iso |
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67 -- |
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68 -- ccc-1 : Hom A a 1 ≅ {*} |
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69 -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b ) |
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70 -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c |
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71 -- |
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72 -------- |
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73 |
| 788 | 74 open import CCC |
| 75 | |
| 779 | 76 |
| 784 | 77 record CCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where |
| 781 | 78 field |
| 79 one : Obj A | |
| 80 _*_ : Obj A → Obj A → Obj A | |
| 81 _^_ : Obj A → Obj A → Obj A | |
| 784 | 82 isCCChom : IsCCChom A one _*_ _^_ |
| 783 | 83 |
| 84 open import HomReasoning | |
| 85 | |
| 784 | 86 CCC→hom : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( c : CCC A ) → CCChom A |
| 87 CCC→hom A c = record { | |
| 88 one = CCC.１ c | |
| 89 ; _*_ = CCC._∧_ c | |
| 90 ; _^_ = CCC._<=_ c | |
| 91 ; isCCChom = record { | |
| 787 | 92 ccc-1 = λ {a} {b} {c'} → record { ≅→ = c101 ; ≅← = c102 ; iso→ = c103 {a} {b} {c'} ; iso← = c104 ; cong← = c105 ; cong→ = c106 } |
| 93 ; ccc-2 = record { ≅→ = c201 ; ≅← = c202 ; iso→ = c203 ; iso← = c204 ; cong← = c205; cong→ = c206 } | |
| 94 ; ccc-3 = record { ≅→ = c301 ; ≅← = c302 ; iso→ = c303 ; iso← = c304 ; cong← = c305 ; cong→ = c306 } | |
| 789 | 95 ; nat-2 = nat-2 ; nat-3 = nat-3 |
| 784 | 96 } |
| 794 | 97 } module CCC→HOM where |
| 785 | 98 c101 : {a : Obj A} → Hom A a (CCC.１ c) → Hom OneCat OneObj OneObj |
| 99 c101 _ = OneObj | |
| 100 c102 : {a : Obj A} → Hom OneCat OneObj OneObj → Hom A a (CCC.１ c) | |
| 101 c102 {a} OneObj = CCC.○ c a | |
| 102 c103 : {a : Obj A } {b c : Obj OneCat} {f : Hom OneCat b b } → _[_≈_] OneCat {b} {c} ( c101 {a} (c102 {a} f) ) f | |
| 103 c103 {a} {OneObj} {OneObj} {OneObj} = refl | |
| 104 c104 : {a : Obj A} → {f : Hom A a (CCC.１ c)} → A [ (c102 ( c101 f )) ≈ f ] | |
| 793 | 105 c104 {a} {f} = let open ≈-Reasoning A in HomReasoning.≈-Reasoning.sym A (IsCCC.e2 (CCC.isCCC c) ) |
| 785 | 106 c201 : { c₁ a b : Obj A} → Hom A c₁ ((c CCC.∧ a) b) → Hom (A × A) (c₁ , c₁) (a , b) |
| 107 c201 f = ( A [ CCC.π c o f ] , A [ CCC.π' c o f ] ) | |
| 108 c202 : { c₁ a b : Obj A} → Hom (A × A) (c₁ , c₁) (a , b) → Hom A c₁ ((c CCC.∧ a) b) | |
| 109 c202 (f , g ) = CCC.<_,_> c f g | |
| 110 c203 : { c₁ a b : Obj A} → {f : Hom (A × A) (c₁ , c₁) (a , b)} → (A × A) [ (c201 ( c202 f )) ≈ f ] | |
| 111 c203 = ( IsCCC.e3a (CCC.isCCC c) , IsCCC.e3b (CCC.isCCC c)) | |
| 112 c204 : { c₁ a b : Obj A} → {f : Hom A c₁ ((c CCC.∧ a) b)} → A [ (c202 ( c201 f )) ≈ f ] | |
| 113 c204 = IsCCC.e3c (CCC.isCCC c) | |
| 114 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 | |
| 115 c301 {d} {a} {b} f = A [ CCC.ε c o CCC.<_,_> c ( A [ f o CCC.π c ] ) ( CCC.π' c ) ] | |
| 116 c302 : { d a b : Obj A} → Hom A ((c CCC.∧ a) b) d → Hom A a ((c CCC.<= d) b) | |
| 117 c302 f = CCC._* c f | |
| 118 c303 : { c₁ a b : Obj A} → {f : Hom A ((c CCC.∧ a) b) c₁} → A [ (c301 ( c302 f )) ≈ f ] | |
| 119 c303 = IsCCC.e4a (CCC.isCCC c) | |
| 120 c304 : { c₁ a b : Obj A} → {f : Hom A a ((c CCC.<= c₁) b)} → A [ (c302 ( c301 f )) ≈ f ] | |
| 121 c304 = IsCCC.e4b (CCC.isCCC c) | |
| 787 | 122 c105 : {a : Obj A } {f g : Hom OneCat OneObj OneObj} → _[_≈_] OneCat {OneObj} {OneObj} f g → A [ c102 {a} f ≈ c102 {a} g ] |
| 123 c105 refl = let open ≈-Reasoning A in refl-hom | |
| 124 c106 : { a : Obj A } {f g : Hom A a (CCC.１ c)} → A [ f ≈ g ] → _[_≈_] OneCat {OneObj} {OneObj} OneObj OneObj | |
| 125 c106 _ = refl | |
| 126 c205 : { a b c₁ : Obj A } {f g : Hom (A × A) (c₁ , c₁) (a , b)} → (A × A) [ f ≈ g ] → A [ c202 f ≈ c202 g ] | |
| 127 c205 f=g = IsCCC.π-cong (CCC.isCCC c ) (proj₁ f=g ) (proj₂ f=g ) | |
| 128 c206 : { a b c₁ : Obj A } {f g : Hom A c₁ ((c CCC.∧ a) b)} → A [ f ≈ g ] → (A × A) [ c201 f ≈ c201 g ] | |
| 129 c206 {a} {b} {c₁} {f} {g} f=g = ( begin | |
| 130 CCC.π c o f | |
| 131 ≈⟨ cdr f=g ⟩ | |
| 132 CCC.π c o g | |
| 133 ∎ ) , ( begin | |
| 134 CCC.π' c o f | |
| 135 ≈⟨ cdr f=g ⟩ | |
| 136 CCC.π' c o g | |
| 137 ∎ ) where open ≈-Reasoning A | |
| 138 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 ] | |
| 139 c305 f=g = IsCCC.*-cong (CCC.isCCC c ) f=g | |
| 140 c306 : { a b c₁ : Obj A } {f g : Hom A a ((c CCC.<= c₁) b)} → A [ f ≈ g ] → A [ c301 f ≈ c301 g ] | |
| 141 c306 {a} {b} {c₁} {f} {g} f=g = begin | |
| 142 CCC.ε c o CCC.<_,_> c ( f o CCC.π c ) ( CCC.π' c ) | |
| 143 ≈⟨ cdr ( IsCCC.π-cong (CCC.isCCC c ) (car f=g ) refl-hom) ⟩ | |
| 144 CCC.ε c o CCC.<_,_> c ( g o CCC.π c ) ( CCC.π' c ) | |
| 145 ∎ where open ≈-Reasoning A | |
| 788 | 146 nat-2 : {a b : Obj A} {c = c₁ : Obj A} {f : Hom A ((c CCC.∧ b) c₁) ((c CCC.∧ b) c₁)} |
| 147 {g : Hom A a ((c CCC.∧ b) c₁)} → (A × A) [ (A × A) [ c201 f o g , g ] ≈ c201 (A [ f o g ]) ] | |
| 148 nat-2 {a} {b} {c₁} {f} {g} = ( begin | |
| 149 ( CCC.π c o f) o g | |
| 150 ≈↑⟨ assoc ⟩ | |
| 151 ( CCC.π c ) o (f o g) | |
| 152 ∎ ) , (sym-hom assoc) where open ≈-Reasoning A | |
| 789 | 153 nat-3 : {a b : Obj A} {c = c₁ : Obj A} {k : Hom A c₁ ((c CCC.<= a) b)} → |
| 154 A [ A [ c301 (id1 A ((c CCC.<= a) b)) o c202 (A [ k o proj₁ (c201 (id1 A ((c CCC.∧ c₁) b))) ] , proj₂ (c201 (id1 A ((c CCC.∧ c₁) b)))) ] | |
| 155 ≈ c301 k ] | |
| 156 nat-3 {a} {b} { c₁} {k} = begin | |
| 157 c301 (id1 A ((c CCC.<= a) b)) o c202 ( k o proj₁ (c201 (id1 A ((c CCC.∧ c₁) b))) , proj₂ (c201 (id1 A ((c CCC.∧ c₁) b)))) | |
| 158 ≈⟨⟩ | |
| 159 ( CCC.ε c o CCC.<_,_> c ((id1 A (CCC._<=_ c a b )) o CCC.π c) (CCC.π' c)) | |
| 160 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b)))) | |
| 161 ≈↑⟨ assoc ⟩ | |
| 162 (CCC.ε c) o (( CCC.<_,_> c ((id1 A (CCC._<=_ c a b )) o CCC.π c) (CCC.π' c)) | |
| 163 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))) | |
| 164 ≈⟨ cdr (car (IsCCC.π-cong (CCC.isCCC c ) idL refl-hom ) ) ⟩ | |
| 165 (CCC.ε c) o ( CCC.<_,_> c (CCC.π c) (CCC.π' c) | |
| 166 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))) | |
| 167 ≈⟨ cdr (car (IsCCC.π-id (CCC.isCCC c))) ⟩ | |
| 168 (CCC.ε c) o ( id1 A (CCC._∧_ c ((c CCC.<= a) b) b ) | |
| 169 o (CCC.<_,_> c (k o ( CCC.π c o (id1 A (CCC._∧_ c c₁ b )))) ( CCC.π' c o (id1 A (CCC._∧_ c c₁ b))))) | |
| 170 ≈⟨ cdr ( cdr ( IsCCC.π-cong (CCC.isCCC c) (cdr idR) idR )) ⟩ | |
| 171 (CCC.ε c) o ( id1 A (CCC._∧_ c ((c CCC.<= a) b) b ) o (CCC.<_,_> c (k o ( CCC.π c )) ( CCC.π' c ))) | |
| 172 ≈⟨ cdr idL ⟩ | |
| 173 (CCC.ε c) o (CCC.<_,_> c ( k o (CCC.π c) ) (CCC.π' c)) | |
| 174 ≈⟨⟩ | |
| 175 c301 k | |
| 176 ∎ where open ≈-Reasoning A | |
| 788 | 177 |
| 794 | 178 U_b : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → ( ccc : CCC A ) → (b : Obj A) → Functor A A |
| 179 FObj (U_b A ccc b) = λ a → (CCC._<=_ ccc a b ) | |
| 180 FMap (U_b A ccc b) = λ f → CCC._* ccc ( A [ f o CCC.ε ccc ] ) | |
| 181 isFunctor (U_b A ccc b) = isF where | |
| 948 | 182 open CCC.CCC ccc |
| 183 isc = isCCC | |
| 184 open IsCCC isCCC | |
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185 |
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186 isF : IsFunctor A A ( λ a → (a <= b)) ( λ f → CCC._* ccc ( A [ f o ε ] ) ) |
| 794 | 187 IsFunctor.≈-cong isF f≈g = IsCCC.*-cong (CCC.isCCC ccc) ( car f≈g ) where open ≈-Reasoning A |
| 188 -- e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o < A [ k o π ] , π' > ] ) * ≈ k ] | |
| 189 IsFunctor.identity isF {a} = begin | |
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190 (id1 A a o ε ) * |
| 794 | 191 ≈⟨ IsCCC.*-cong (CCC.isCCC ccc) ( begin |
| 192 id1 A a o CCC.ε ccc | |
| 193 ≈⟨ idL ⟩ | |
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194 ε |
| 794 | 195 ≈↑⟨ idR ⟩ |
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196 ε o id1 A ( ( a <= b ) ∧ b ) |
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197 ≈↑⟨ cdr ( IsCCC.π-id isc) ⟩ |
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198 ε o ( < π , π' > ) |
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199 ≈↑⟨ cdr ( π-cong idL refl-hom) ⟩ |
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200 ε o ( < id1 A (a <= b) o π , π' > ) |
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201 ∎ ) ⟩ |
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202 ( ε o ( < id1 A ( a <= b) o π , π' > ) ) * |
| 794 | 203 ≈⟨ IsCCC.e4b (CCC.isCCC ccc) ⟩ |
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204 id1 A (a <= b) |
| 794 | 205 ∎ where open ≈-Reasoning A |
| 206 IsFunctor.distr isF {x} {y} {z} {f} {g} = begin | |
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207 ( ( g o f ) o ε ) * |
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208 ≈↑⟨ *-cong assoc ⟩ |
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209 ( g o (f o ε ) ) * |
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210 ≈↑⟨ *-cong ( cdr (IsCCC.e4a isc) ) ⟩ |
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211 ( g o ( ε o ( < ( ( f o ε ) * ) o π , π' > ) )) * |
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212 ≈⟨ *-cong assoc ⟩ |
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213 ( ( g o ε ) o ( < ( ( f o ε ) * ) o π , π' > ) ) * |
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214 ≈↑⟨ IsCCC.distr-* isc ⟩ |
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215 ( g o ε ) * o ( f o ε ) * |
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216 ∎ where open ≈-Reasoning A |
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217 |
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218 F_b : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → ( ccc : CCC A ) → (b : Obj A) → Functor A A |
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219 FObj (F_b A ccc b) = λ a → ( CCC._∧_ ccc a b ) |
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220 FMap (F_b A ccc b) = λ f → ( CCC.<_,_> ccc (A [ f o CCC.π ccc ] ) ( CCC.π' ccc) ) |
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221 isFunctor (F_b A ccc b) = isF where |
| 948 | 222 open CCC.CCC ccc |
| 223 isc = isCCC | |
| 224 open IsCCC isCCC | |
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225 |
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226 isF : IsFunctor A A ( λ a → (a ∧ b)) ( λ f → < ( A [ f o π ] ) , π' > ) |
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227 IsFunctor.≈-cong isF f≈g = π-cong ( car f≈g ) refl-hom where open ≈-Reasoning A |
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228 IsFunctor.identity isF {a} = trans-hom (π-cong idL refl-hom ) (IsCCC.π-id isc) where open ≈-Reasoning A |
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229 IsFunctor.distr isF {x} {y} {z} {f} {g} = begin |
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230 < ( g o f ) o π , π' > |
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231 ≈↑⟨ π-cong assoc refl-hom ⟩ |
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232 < g o ( f o π ) , π' > |
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233 ≈↑⟨ π-cong (cdr (IsCCC.e3a isc )) refl-hom ⟩ |
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234 < g o ( π o < ( f o π ) , π' > ) , π' > |
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235 ≈⟨ π-cong assoc ( sym-hom (IsCCC.e3b isc )) ⟩ |
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236 < ( g o π ) o < ( f o π ) , π' > , π' o < ( f o π ) , π' > > |
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237 ≈↑⟨ IsCCC.distr-π isc ⟩ |
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238 < ( g o π ) , π' > o < ( f o π ) , π' > |
| 794 | 239 ∎ where open ≈-Reasoning A |
| 240 | |
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241 ------- |
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242 --- U_b and F_b is an adjunction Functor |
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243 ------- |
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244 |
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245 CCCtoAdj : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( ccc : CCC A ) → (b : Obj A ) → coUniversalMapping A A (F_b A ccc b) |
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246 CCCtoAdj A ccc b = record { |
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247 U = λ a → a <= b |
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248 ; ε = ε' |
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249 ; _*' = solution |
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250 ; iscoUniversalMapping = record { |
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251 couniversalMapping = couniversalMapping |
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252 ; couniquness = couniquness |
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253 } |
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254 } where |
| 948 | 255 open CCC.CCC ccc |
| 256 isc = isCCC | |
| 257 open IsCCC isCCC | |
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258 ε' : (a : Obj A) → Hom A (FObj (F_b A ccc b) (a <= b)) a |
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259 ε' a = ε |
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260 solution : { b' : Obj A} {a : Obj A} → Hom A (FObj (F_b A ccc b) a) b' → Hom A a (b' <= b) |
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261 solution f = f * |
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262 couniversalMapping : {b = b₁ : Obj A} {a : Obj A} |
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263 {f : Hom A (FObj (F_b A ccc b) a) b₁} → |
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264 A [ A [ ε' b₁ o FMap (F_b A ccc b) (solution f) ] ≈ f ] |
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265 couniversalMapping {c} {a} {f} = IsCCC.e4a isc |
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266 couniquness : {b = b₁ : Obj A} {a : Obj A} |
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267 {f : Hom A (FObj (F_b A ccc b) a) b₁} {g : Hom A a (b₁ <= b)} → |
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268 A [ A [ ε' b₁ o FMap (F_b A ccc b) g ] ≈ f ] → A [ solution f ≈ g ] |
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269 couniquness {c} {a} {f} {g} eq = begin |
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270 f * |
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271 ≈↑⟨ *-cong eq ⟩ |
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272 ( ε o FMap (F_b A ccc b) g ) * |
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273 ≈⟨⟩ |
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274 ( ε o < ( g o π ) , π' > ) * |
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275 ≈⟨ IsCCC.e4b isc ⟩ |
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276 g |
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277 ∎ where open ≈-Reasoning A |
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278 |
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279 |
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280 ---------- |
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281 --- Hom A １ ( c ^ b ) ≅ Hom A b c |
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282 ---------- |
| 794 | 283 |
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284 c^b=b<=c : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( ccc : CCC A ) → {a b c : Obj A} → |
| 793 | 285 IsoS A A (CCC.１ ccc ) (CCC._<=_ ccc c b) b c |
| 794 | 286 c^b=b<=c A ccc {a} {b} {c} = record { |
| 287 ≅→ = λ f → A [ CCC.ε ccc o CCC.<_,_> ccc ( A [ f o CCC.○ ccc b ] ) ( id1 A b ) ] --- g ’ (g : 1 → b ^ a) of | |
| 288 ; ≅← = λ f → CCC._* ccc ( A [ f o CCC.π' ccc ] ) --- ┌ f ┐ name of (f : b ^a → 1 ) | |
| 793 | 289 ; iso→ = iso→ |
| 290 ; iso← = iso← | |
| 291 ; cong→ = cong* | |
| 292 ; cong← = cong2 | |
| 293 } where | |
| 294 cc = IsCCChom.ccc-3 ( CCChom.isCCChom (CCC→hom A ccc ) ) | |
| 295 -- e4a : {a b c : Obj A} → { k : Hom A (c /\ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ] | |
| 296 iso→ : {f : Hom A b c} → A [ | |
| 297 (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o (ccc CCC.*) ((A Category.o f) (CCC.π' ccc))) (CCC.○ ccc b) > (Category.Category.Id A)) ≈ f ] | |
| 298 iso→ {f} = begin | |
| 299 CCC.ε ccc o (CCC.<_,_> ccc (CCC._* ccc ( f o (CCC.π' ccc)) o (CCC.○ ccc b)) (id1 A b ) ) | |
| 300 ≈↑⟨ cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) refl-hom (IsCCC.e3b (CCC.isCCC ccc) ) ) ⟩ | |
| 301 CCC.ε ccc o ( CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o CCC.○ ccc b) ((CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) | |
| 302 ≈↑⟨ cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) (cdr (IsCCC.e3a (CCC.isCCC ccc))) refl-hom ) ⟩ | |
| 303 CCC.ε ccc o ( CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o ( CCC.π ccc o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) ((CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) | |
| 304 ≈⟨ cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) assoc refl-hom ) ⟩ | |
| 305 CCC.ε ccc o ( CCC.<_,_> ccc ((CCC._* ccc (f o CCC.π' ccc) o CCC.π ccc ) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ((CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) ) | |
| 794 | 306 ≈↑⟨ cdr ( IsCCC.distr-π ( CCC.isCCC ccc ) ) ⟩ |
| 793 | 307 CCC.ε ccc o ( CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o CCC.π ccc ) (CCC.π' ccc) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) |
| 308 ≈⟨ assoc ⟩ | |
| 309 ( CCC.ε ccc o CCC.<_,_> ccc (CCC._* ccc (f o CCC.π' ccc) o CCC.π ccc ) (CCC.π' ccc) ) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) | |
| 310 ≈⟨ car ( IsCCC.e4a (CCC.isCCC ccc) ) ⟩ | |
| 311 ( f o CCC.π' ccc ) o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) | |
| 312 ≈↑⟨ assoc ⟩ | |
| 313 f o ( CCC.π' ccc o CCC.<_,_> ccc (CCC.○ ccc b) (id1 A b) ) | |
| 314 ≈⟨ cdr (IsCCC.e3b (CCC.isCCC ccc)) ⟩ | |
| 315 f o id1 A b | |
| 316 ≈⟨ idR ⟩ | |
| 317 f | |
| 318 ∎ where open ≈-Reasoning A | |
| 319 lemma : {f : Hom A (CCC.１ ccc) ((ccc CCC.<= c) b)} → A [ A [ A [ f o (CCC.○ ccc b) ] o (CCC.π' ccc) ] ≈ A [ f o (CCC.π ccc) ] ] | |
| 320 lemma {f} = begin | |
| 321 ( f o (CCC.○ ccc b) ) o (CCC.π' ccc) | |
| 322 ≈↑⟨ assoc ⟩ | |
| 323 f o ( (CCC.○ ccc b) o (CCC.π' ccc) ) | |
| 324 ≈⟨ cdr ( IsCCC.e2 (CCC.isCCC ccc) ) ⟩ | |
| 325 f o (CCC.○ ccc ( CCC._∧_ ccc (CCC.１ ccc) b ) ) | |
| 326 ≈↑⟨ cdr ( IsCCC.e2 (CCC.isCCC ccc) ) ⟩ | |
| 327 f o ( (CCC.○ ccc) (CCC.１ ccc) o (CCC.π ccc) ) | |
| 328 ≈↑⟨ cdr ( car ( IsCCC.e2 (CCC.isCCC ccc) )) ⟩ | |
| 329 f o ( id1 A (CCC.１ ccc) o (CCC.π ccc) ) | |
| 330 ≈⟨ cdr (idL) ⟩ | |
| 331 f o (CCC.π ccc) | |
| 332 ∎ where open ≈-Reasoning A | |
| 333 -- e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ] | |
| 334 iso← : {f : Hom A (CCC.１ ccc) ((ccc CCC.<= c) b)} → A [ (ccc CCC.*) ((A Category.o (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o f) (CCC.○ ccc b) > | |
| 335 (Category.Category.Id A))) (CCC.π' ccc)) ≈ f ] | |
| 336 iso← {f} = begin | |
| 337 CCC._* ccc (( CCC.ε ccc o ( CCC.<_,_> ccc ( f o (CCC.○ ccc b) ) (id1 A b ))) o (CCC.π' ccc)) | |
| 338 ≈↑⟨ IsCCC.*-cong ( CCC.isCCC ccc ) assoc ⟩ | |
| 339 CCC._* ccc ( CCC.ε ccc o (( CCC.<_,_> ccc ( f o (CCC.○ ccc b) ) (id1 A b )) o (CCC.π' ccc))) | |
| 794 | 340 ≈⟨ IsCCC.*-cong ( CCC.isCCC ccc ) (cdr ( IsCCC.distr-π ( CCC.isCCC ccc ) ) ) ⟩ |
| 793 | 341 CCC._* ccc ( CCC.ε ccc o CCC.<_,_> ccc ( (f o (CCC.○ ccc b)) o CCC.π' ccc ) (id1 A b o CCC.π' ccc) ) |
| 342 ≈⟨ IsCCC.*-cong ( CCC.isCCC ccc ) (cdr ( IsCCC.π-cong ( CCC.isCCC ccc ) lemma idL )) ⟩ | |
| 343 CCC._* ccc ( CCC.ε ccc o CCC.<_,_> ccc ( f o CCC.π ccc ) (CCC.π' ccc) ) | |
| 344 ≈⟨ IsCCC.e4b (CCC.isCCC ccc) ⟩ | |
| 345 f | |
| 346 ∎ where open ≈-Reasoning A | |
| 347 cong* : {f g : Hom A (CCC.１ ccc) ((ccc CCC.<= c) b)} → | |
| 348 A [ f ≈ g ] → A [ (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o f) (CCC.○ ccc b) > (Category.Category.Id A)) | |
| 349 ≈ (A Category.o CCC.ε ccc) (CCC.< ccc , (A Category.o g) (CCC.○ ccc b) > (Category.Category.Id A)) ] | |
| 350 cong* {f} {g} f≈g = begin | |
| 351 CCC.ε ccc o ( CCC.<_,_> ccc ( f o ( CCC.○ ccc b )) (id1 A b )) | |
| 352 ≈⟨ cdr (IsCCC.π-cong ( CCC.isCCC ccc ) (car f≈g) refl-hom ) ⟩ | |
| 353 CCC.ε ccc o ( CCC.<_,_> ccc ( g o ( CCC.○ ccc b )) (id1 A b )) | |
| 354 ∎ where open ≈-Reasoning A | |
| 355 cong2 : {f g : Hom A b c} → A [ f ≈ g ] → | |
| 356 A [ (ccc CCC.*) ((A Category.o f) (CCC.π' ccc)) ≈ (ccc CCC.*) ((A Category.o g) (CCC.π' ccc)) ] | |
| 357 cong2 {f} {g} f≈g = begin | |
| 358 CCC._* ccc ( f o (CCC.π' ccc) ) | |
| 359 ≈⟨ IsCCC.*-cong ( CCC.isCCC ccc ) (car f≈g ) ⟩ | |
| 360 CCC._* ccc ( g o (CCC.π' ccc) ) | |
| 361 ∎ where open ≈-Reasoning A | |
| 362 | |
| 363 | |
| 785 | 364 open CCChom |
| 365 open IsCCChom | |
| 366 open IsoS | |
| 367 | |
| 368 hom→CCC : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( h : CCChom A ) → CCC A | |
| 369 hom→CCC A h = record { | |
| 370 １ = １ | |
| 371 ; ○ = ○ | |
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372 ; _∧_ = _∧_ |
| 785 | 373 ; <_,_> = <,> |
| 374 ; π = π | |
| 375 ; π' = π' | |
| 376 ; _<=_ = _<=_ | |
| 377 ; _* = _* | |
| 378 ; ε = ε | |
| 379 ; isCCC = isCCC | |
| 380 } where | |
| 381 １ : Obj A | |
| 382 １ = one h | |
| 383 ○ : (a : Obj A ) → Hom A a １ | |
| 384 ○ a = ≅← ( ccc-1 (isCCChom h ) {_} {OneObj} {OneObj} ) OneObj | |
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385 _∧_ : Obj A → Obj A → Obj A |
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386 _∧_ a b = _*_ h a b |
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387 <,> : {a b c : Obj A } → Hom A c a → Hom A c b → Hom A c ( a ∧ b) |
| 785 | 388 <,> f g = ≅← ( ccc-2 (isCCChom h ) ) ( f , g ) |
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389 π : {a b : Obj A } → Hom A (a ∧ b) a |
| 785 | 390 π {a} {b} = proj₁ ( ≅→ ( ccc-2 (isCCChom h ) ) (id1 A (_*_ h a b) )) |
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391 π' : {a b : Obj A } → Hom A (a ∧ b) b |
| 785 | 392 π' {a} {b} = proj₂ ( ≅→ ( ccc-2 (isCCChom h ) ) (id1 A (_*_ h a b) )) |
| 393 _<=_ : (a b : Obj A ) → Obj A | |
| 394 _<=_ = _^_ h | |
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395 _* : {a b c : Obj A } → Hom A (a ∧ b) c → Hom A a (c <= b) |
| 785 | 396 _* = ≅← ( ccc-3 (isCCChom h ) ) |
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397 ε : {a b : Obj A } → Hom A ((a <= b ) ∧ b) a |
| 785 | 398 ε {a} {b} = ≅→ ( ccc-3 (isCCChom h ) {_^_ h a b} {b} ) (id1 A ( _^_ h a b )) |
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399 isCCC : CCC.IsCCC A １ ○ _∧_ <,> π π' _<=_ _* ε |
| 785 | 400 isCCC = record { |
| 401 e2 = e2 | |
| 402 ; e3a = e3a | |
| 403 ; e3b = e3b | |
| 404 ; e3c = e3c | |
| 405 ; π-cong = π-cong | |
| 406 ; e4a = e4a | |
| 407 ; e4b = e4b | |
| 787 | 408 ; *-cong = *-cong |
| 785 | 409 } where |
| 786 | 410 e20 : ∀ ( f : Hom OneCat OneObj OneObj ) → _[_≈_] OneCat {OneObj} {OneObj} f OneObj |
| 411 e20 OneObj = refl | |
| 793 | 412 e2 : {a : Obj A} → ∀ { f : Hom A a １ } → A [ f ≈ ○ a ] |
| 413 e2 {a} {f} = begin | |
| 786 | 414 f |
| 415 ≈↑⟨ iso← ( ccc-1 (isCCChom h )) ⟩ | |
| 416 ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj}) ( ≅→ ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) f ) | |
| 417 ≈⟨ ≡-cong {Level.zero} {Level.zero} {Level.zero} {OneCat} {OneObj} {OneObj} ( | |
| 418 λ y → ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) y ) (e20 ( ≅→ ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) f) ) ⟩ | |
| 419 ≅← ( ccc-1 (isCCChom h ) {a} {OneObj} {OneObj} ) OneObj | |
| 420 ≈⟨⟩ | |
| 421 ○ a | |
| 422 ∎ where open ≈-Reasoning A | |
| 787 | 423 -- |
| 424 -- g id | |
| 425 -- a -------------> b * c ------> b * c | |
| 426 -- | |
| 427 -- a -------------> b * c ------> b | |
| 428 -- a -------------> b * c ------> c | |
| 429 -- | |
| 430 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 ] | |
| 431 cong-proj₁ eq = proj₁ eq | |
| 788 | 432 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 ] |
| 433 cong-proj₂ eq = proj₂ eq | |
| 785 | 434 e3a : {a b c : Obj A} → { f : Hom A c a }{ g : Hom A c b } → A [ A [ π o <,> f g ] ≈ f ] |
| 786 | 435 e3a {a} {b} {c} {f} {g} = begin |
| 436 π o <,> f g | |
| 437 ≈⟨⟩ | |
| 438 proj₁ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) o (≅← (ccc-2 (isCCChom h)) (f , g)) | |
| 788 | 439 ≈⟨ cong-proj₁ (nat-2 (isCCChom h)) ⟩ |
| 787 | 440 proj₁ (≅→ (ccc-2 (isCCChom h)) (( id1 A ( _*_ h a b )) o ( ≅← (ccc-2 (isCCChom h)) (f , g) ) )) |
| 441 ≈⟨ cong-proj₁ ( cong→ (ccc-2 (isCCChom h)) idL ) ⟩ | |
| 442 proj₁ (≅→ (ccc-2 (isCCChom h)) ( ≅← (ccc-2 (isCCChom h)) (f , g) )) | |
| 443 ≈⟨ cong-proj₁ ( iso→ (ccc-2 (isCCChom h))) ⟩ | |
| 786 | 444 proj₁ ( f , g ) |
| 445 ≈⟨⟩ | |
| 446 f | |
| 447 ∎ where open ≈-Reasoning A | |
| 785 | 448 e3b : {a b c : Obj A} → { f : Hom A c a }{ g : Hom A c b } → A [ A [ π' o <,> f g ] ≈ g ] |
| 788 | 449 e3b {a} {b} {c} {f} {g} = begin |
| 450 π' o <,> f g | |
| 451 ≈⟨⟩ | |
| 452 proj₂ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) o (≅← (ccc-2 (isCCChom h)) (f , g)) | |
| 453 ≈⟨ cong-proj₂ (nat-2 (isCCChom h)) ⟩ | |
| 454 proj₂ (≅→ (ccc-2 (isCCChom h)) (( id1 A ( _*_ h a b )) o ( ≅← (ccc-2 (isCCChom h)) (f , g) ) )) | |
| 455 ≈⟨ cong-proj₂ ( cong→ (ccc-2 (isCCChom h)) idL ) ⟩ | |
| 456 proj₂ (≅→ (ccc-2 (isCCChom h)) ( ≅← (ccc-2 (isCCChom h)) (f , g) )) | |
| 457 ≈⟨ cong-proj₂ ( iso→ (ccc-2 (isCCChom h))) ⟩ | |
| 458 proj₂ ( f , g ) | |
| 459 ≈⟨⟩ | |
| 460 g | |
| 461 ∎ where open ≈-Reasoning A | |
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462 e3c : {a b c : Obj A} → { h : Hom A c (a ∧ b) } → A [ <,> ( A [ π o h ] ) ( A [ π' o h ] ) ≈ h ] |
| 788 | 463 e3c {a} {b} {c} {f} = begin |
| 464 <,> ( π o f ) ( π' o f ) | |
| 465 ≈⟨⟩ | |
| 466 ≅← (ccc-2 (isCCChom h)) ( ( proj₁ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) ))) o f | |
| 467 , ( proj₂ (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b)))) o f ) | |
| 468 ≈⟨⟩ | |
| 469 ≅← (ccc-2 (isCCChom h)) (_[_o_] (A × A) (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) )) (f , f ) ) | |
| 470 ≈⟨ cong← (ccc-2 (isCCChom h)) (nat-2 (isCCChom h)) ⟩ | |
| 471 ≅← (ccc-2 (isCCChom h)) (≅→ (ccc-2 (isCCChom h)) (id1 A (_*_ h a b) o f )) | |
| 472 ≈⟨ cong← (ccc-2 (isCCChom h)) (cong→ (ccc-2 (isCCChom h)) idL ) ⟩ | |
| 473 ≅← (ccc-2 (isCCChom h)) (≅→ (ccc-2 (isCCChom h)) f ) | |
| 474 ≈⟨ iso← (ccc-2 (isCCChom h)) ⟩ | |
| 475 f | |
| 476 ∎ where open ≈-Reasoning A | |
| 785 | 477 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' ] |
| 786 | 478 π-cong {a} {b} {c} {f} {f'} {g} {g'} eq1 eq2 = begin |
| 479 <,> f g | |
| 480 ≈⟨⟩ | |
| 481 ≅← (ccc-2 (isCCChom h)) (f , g) | |
| 787 | 482 ≈⟨ cong← (ccc-2 (isCCChom h)) ( eq1 , eq2 ) ⟩ |
| 786 | 483 ≅← (ccc-2 (isCCChom h)) (f' , g') |
| 484 ≈⟨⟩ | |
| 485 <,> f' g' | |
| 486 ∎ where open ≈-Reasoning A | |
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487 e4a : {a b c : Obj A} → { k : Hom A (c ∧ b) a } → A [ A [ ε o ( <,> ( A [ (k *) o π ] ) π') ] ≈ k ] |
| 786 | 488 e4a {a} {b} {c} {k} = begin |
| 489 ε o ( <,> ((k *) o π ) π' ) | |
| 490 ≈⟨⟩ | |
| 787 | 491 ≅→ (ccc-3 (isCCChom h)) (id1 A (_^_ h a b)) o (≅← (ccc-2 (isCCChom h)) ((( ≅← (ccc-3 (isCCChom h)) k) o π ) , π')) |
| 789 | 492 ≈⟨ nat-3 (isCCChom h) ⟩ |
| 787 | 493 ≅→ (ccc-3 (isCCChom h)) (≅← (ccc-3 (isCCChom h)) k) |
| 494 ≈⟨ iso→ (ccc-3 (isCCChom h)) ⟩ | |
| 786 | 495 k |
| 496 ∎ where open ≈-Reasoning A | |
| 785 | 497 e4b : {a b c : Obj A} → { k : Hom A c (a <= b ) } → A [ ( A [ ε o ( <,> ( A [ k o π ] ) π' ) ] ) * ≈ k ] |
| 787 | 498 e4b {a} {b} {c} {k} = begin |
| 499 ( ε o ( <,> ( k o π ) π' ) ) * | |
| 500 ≈⟨⟩ | |
| 501 ≅← (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'))) | |
| 789 | 502 ≈⟨ cong← (ccc-3 (isCCChom h)) (nat-3 (isCCChom h)) ⟩ |
| 787 | 503 ≅← (ccc-3 (isCCChom h)) (≅→ (ccc-3 (isCCChom h)) k) |
| 504 ≈⟨ iso← (ccc-3 (isCCChom h)) ⟩ | |
| 505 k | |
| 506 ∎ where open ≈-Reasoning A | |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
794
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|
507 *-cong : {a b c : Obj A} {f f' : Hom A (a ∧ b) c} → A [ f ≈ f' ] → A [ f * ≈ f' * ] |
| 787 | 508 *-cong eq = cong← ( ccc-3 (isCCChom h )) eq |
| 785 | 509 |