Mercurial > hg > Members > kono > Proof > category
annotate nat.agda @ 75:8e665152c306
Comparison Functor
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 26 Jul 2013 19:52:19 +0900 |
| parents | 49e6eb3ef9c0 |
| children | 6c6c3dd8ef12 |
| rev | line source |
|---|---|
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parents:
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1 |
| 0 | 2 |
| 3 -- Monad | |
| 4 -- Category A | |
| 5 -- A = Category | |
| 22 | 6 -- Functor T : A → A |
| 0 | 7 --T(a) = t(a) |
| 8 --T(f) = tf(f) | |
| 9 | |
| 2 | 10 open import Category -- https://github.com/konn/category-agda |
| 0 | 11 open import Level |
| 56 | 12 --open import Category.HomReasoning |
| 13 open import HomReasoning | |
| 14 open import cat-utility | |
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15 open import Category.Cat |
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16 |
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17 module nat { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } |
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18 { T : Functor A A } |
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19 { η : NTrans A A identityFunctor T } |
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20 { μ : NTrans A A (T ○ T) T } |
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21 { M : Monad A T η μ } |
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22 { K : Kleisli A T η μ M } where |
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23 |
| 0 | 24 |
| 1 | 25 --T(g f) = T(g) T(f) |
| 26 | |
| 31 | 27 open Functor |
| 22 | 28 Lemma1 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} (T : Functor A A) → {a b c : Obj A} {g : Hom A b c} { f : Hom A a b } |
| 29 → A [ ( FMap T (A [ g o f ] )) ≈ (A [ FMap T g o FMap T f ]) ] | |
| 30 Lemma1 = \t → IsFunctor.distr ( isFunctor t ) | |
| 0 | 31 |
| 32 | |
| 7 | 33 open NTrans |
| 1 | 34 Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A} |
| 22 | 35 → (μ : NTrans A A F G) → {a b : Obj A} { f : Hom A a b } |
| 30 | 36 → A [ A [ FMap G f o TMap μ a ] ≈ A [ TMap μ b o FMap F f ] ] |
| 22 | 37 Lemma2 = \n → IsNTrans.naturality ( isNTrans n ) |
| 0 | 38 |
| 22 | 39 -- η : 1_A → T |
| 40 -- μ : TT → T | |
| 0 | 41 -- μ(a)η(T(a)) = a |
| 42 -- μ(a)T(η(a)) = a | |
| 43 -- μ(a)(μ(T(a))) = μ(a)T(μ(a)) | |
| 44 | |
| 45 | |
| 2 | 46 open Monad |
| 47 Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 48 { T : Functor A A } | |
| 7 | 49 { η : NTrans A A identityFunctor T } |
| 50 { μ : NTrans A A (T ○ T) T } | |
| 22 | 51 { a : Obj A } → |
| 2 | 52 ( M : Monad A T η μ ) |
| 30 | 53 → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈ A [ TMap μ a o FMap T (TMap μ a) ] ] |
| 22 | 54 Lemma3 = \m → IsMonad.assoc ( isMonad m ) |
| 2 | 55 |
| 56 | |
| 57 Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b} | |
| 22 | 58 → A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ] |
| 59 Lemma4 = \a → IsCategory.identityL ( Category.isCategory a ) | |
| 0 | 60 |
| 3 | 61 Lemma5 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} |
| 62 { T : Functor A A } | |
| 7 | 63 { η : NTrans A A identityFunctor T } |
| 64 { μ : NTrans A A (T ○ T) T } | |
| 22 | 65 { a : Obj A } → |
| 3 | 66 ( M : Monad A T η μ ) |
| 30 | 67 → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 22 | 68 Lemma5 = \m → IsMonad.unity1 ( isMonad m ) |
| 3 | 69 |
| 70 Lemma6 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 71 { T : Functor A A } | |
| 7 | 72 { η : NTrans A A identityFunctor T } |
| 73 { μ : NTrans A A (T ○ T) T } | |
| 22 | 74 { a : Obj A } → |
| 3 | 75 ( M : Monad A T η μ ) |
| 30 | 76 → A [ A [ TMap μ a o (FMap T (TMap η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 22 | 77 Lemma6 = \m → IsMonad.unity2 ( isMonad m ) |
| 3 | 78 |
| 79 -- T = M x A | |
| 0 | 80 -- nat of η |
| 81 -- g ○ f = μ(c) T(g) f | |
| 82 -- η(b) ○ f = f | |
| 83 -- f ○ η(a) = f | |
| 22 | 84 -- h ○ (g ○ f) = (h ○ g) ○ f |
| 0 | 85 |
| 7 | 86 |
| 22 | 87 lemma12 : {c₁ c₂ ℓ : Level} (L : Category c₁ c₂ ℓ) { a b c : Obj L } → |
| 88 ( x : Hom L c a ) → ( y : Hom L b c ) → L [ L [ x o y ] ≈ L [ x o y ] ] | |
| 18 | 89 lemma12 L x y = |
| 90 let open ≈-Reasoning ( L ) in | |
| 91 begin L [ x o y ] ∎ | |
| 11 | 92 |
| 4 | 93 open Kleisli |
| 22 | 94 -- η(b) ○ f = f |
| 73 | 95 Lemma7 : { a : Obj A } { b : Obj A } ( f : Hom A a ( FObj T b) ) |
| 96 → A [ join K (TMap η b) f ≈ f ] | |
| 97 Lemma7 {_} {b} f = | |
| 98 let open ≈-Reasoning (A) in | |
| 21 | 99 begin |
| 73 | 100 join K (TMap η b) f |
| 21 | 101 ≈⟨ refl-hom ⟩ |
| 73 | 102 A [ TMap μ b o A [ FMap T ((TMap η b)) o f ] ] |
| 103 ≈⟨ IsCategory.associative (Category.isCategory A) ⟩ | |
| 104 A [ A [ TMap μ b o FMap T ((TMap η b)) ] o f ] | |
| 105 ≈⟨ car ( IsMonad.unity2 ( isMonad ( monad K )) ) ⟩ | |
| 106 A [ id (FObj T b) o f ] | |
| 107 ≈⟨ IsCategory.identityL (Category.isCategory A) ⟩ | |
| 21 | 108 f |
| 109 ∎ | |
| 7 | 110 |
| 22 | 111 -- f ○ η(a) = f |
| 73 | 112 Lemma8 : { a : Obj A } { b : Obj A } |
| 22 | 113 ( f : Hom A a ( FObj T b) ) |
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parents:
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114 → A [ join K f (TMap η a) ≈ f ] |
| 73 | 115 Lemma8 {a} {b} f = |
| 22 | 116 begin |
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72
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parents:
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117 join K f (TMap η a) |
| 22 | 118 ≈⟨ refl-hom ⟩ |
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parents:
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119 A [ TMap μ b o A [ FMap T f o (TMap η a) ] ] |
| 66 | 120 ≈⟨ cdr ( nat η ) ⟩ |
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parents:
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121 A [ TMap μ b o A [ (TMap η ( FObj T b)) o f ] ] |
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parents:
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122 ≈⟨ IsCategory.associative (Category.isCategory A) ⟩ |
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parents:
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123 A [ A [ TMap μ b o (TMap η ( FObj T b)) ] o f ] |
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parents:
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124 ≈⟨ car ( IsMonad.unity1 ( isMonad ( monad K )) ) ⟩ |
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parents:
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125 A [ id (FObj T b) o f ] |
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126 ≈⟨ IsCategory.identityL (Category.isCategory A) ⟩ |
| 22 | 127 f |
| 128 ∎ where | |
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129 open ≈-Reasoning (A) |
| 5 | 130 |
| 22 | 131 -- h ○ (g ○ f) = (h ○ g) ○ f |
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132 Lemma9 : { a b c d : Obj A } |
| 73 | 133 ( h : Hom A c ( FObj T d) ) |
| 23 | 134 ( g : Hom A b ( FObj T c) ) |
| 73 | 135 ( f : Hom A a ( FObj T b) ) |
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136 → A [ join K h (join K g f) ≈ join K ( join K h g) f ] |
| 73 | 137 Lemma9 {a} {b} {c} {d} h g f = |
| 24 | 138 begin |
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139 join K h (join K g f) |
| 30 | 140 ≈⟨⟩ |
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141 join K h ( ( TMap μ c o ( FMap T g o f ) ) ) |
| 28 | 142 ≈⟨ refl-hom ⟩ |
| 30 | 143 ( TMap μ d o ( FMap T h o ( TMap μ c o ( FMap T g o f ) ) ) ) |
| 28 | 144 ≈⟨ cdr ( cdr ( assoc )) ⟩ |
| 30 | 145 ( TMap μ d o ( FMap T h o ( ( TMap μ c o FMap T g ) o f ) ) ) |
| 28 | 146 ≈⟨ assoc ⟩ --- ( f o ( g o h ) ) = ( ( f o g ) o h ) |
| 30 | 147 ( ( TMap μ d o FMap T h ) o ( (TMap μ c o FMap T g ) o f ) ) |
| 25 | 148 ≈⟨ assoc ⟩ |
| 30 | 149 ( ( ( TMap μ d o FMap T h ) o (TMap μ c o FMap T g ) ) o f ) |
| 28 | 150 ≈⟨ car (sym assoc) ⟩ |
| 30 | 151 ( ( TMap μ d o ( FMap T h o ( TMap μ c o FMap T g ) ) ) o f ) |
| 28 | 152 ≈⟨ car ( cdr (assoc) ) ⟩ |
| 30 | 153 ( ( TMap μ d o ( ( FMap T h o TMap μ c ) o FMap T g ) ) o f ) |
| 28 | 154 ≈⟨ car assoc ⟩ |
| 30 | 155 ( ( ( TMap μ d o ( FMap T h o TMap μ c ) ) o FMap T g ) o f ) |
| 28 | 156 ≈⟨ car (car ( cdr ( begin |
| 30 | 157 ( FMap T h o TMap μ c ) |
| 66 | 158 ≈⟨ nat μ ⟩ |
| 30 | 159 ( TMap μ (FObj T d) o FMap T (FMap T h) ) |
| 25 | 160 ∎ |
| 161 ))) ⟩ | |
| 30 | 162 ( ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) ) o FMap T g ) o f ) |
| 28 | 163 ≈⟨ car (sym assoc) ⟩ |
| 30 | 164 ( ( TMap μ d o ( ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) o FMap T g ) ) o f ) |
| 28 | 165 ≈⟨ car ( cdr (sym assoc) ) ⟩ |
| 30 | 166 ( ( TMap μ d o ( TMap μ ( FObj T d) o ( FMap T ( FMap T h ) o FMap T g ) ) ) o f ) |
| 28 | 167 ≈⟨ car ( cdr (cdr (sym (distr T )))) ⟩ |
| 30 | 168 ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 28 | 169 ≈⟨ car assoc ⟩ |
| 30 | 170 ( ( ( TMap μ d o TMap μ ( FObj T d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 171 ≈⟨ car ( car ( |
| 27 | 172 begin |
| 30 | 173 ( TMap μ d o TMap μ (FObj T d) ) |
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174 ≈⟨ IsMonad.assoc ( isMonad M) ⟩ |
| 30 | 175 ( TMap μ d o FMap T (TMap μ d) ) |
| 27 | 176 ∎ |
| 177 )) ⟩ | |
| 30 | 178 ( ( ( TMap μ d o FMap T ( TMap μ d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 179 ≈⟨ car (sym assoc) ⟩ |
| 30 | 180 ( ( TMap μ d o ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 24 | 181 ≈⟨ sym assoc ⟩ |
| 30 | 182 ( TMap μ d o ( ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) o f ) ) |
| 28 | 183 ≈⟨ cdr ( car ( sym ( distr T ))) ⟩ |
| 30 | 184 ( TMap μ d o ( FMap T ( ( ( TMap μ d ) o ( FMap T h o g ) ) ) o f ) ) |
| 23 | 185 ≈⟨ refl-hom ⟩ |
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186 join K ( ( TMap μ d o ( FMap T h o g ) ) ) f |
| 23 | 187 ≈⟨ refl-hom ⟩ |
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188 join K ( join K h g) f |
| 24 | 189 ∎ where open ≈-Reasoning (A) |
| 3 | 190 |
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191 Lemma10 : {a b c : Obj A} -> (f g : Hom A a (FObj T b) ) → (h i : Hom A b (FObj T c) ) → |
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192 A [ f ≈ g ] → A [ h ≈ i ] → A [ (join K h f) ≈ (join K i g) ] |
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193 Lemma10 {a} {b} {c} f g h i eq-fg eq-hi = let open ≈-Reasoning (A) in |
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194 begin |
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195 join K h f |
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196 ≈⟨⟩ |
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197 TMap μ c o ( FMap T h o f ) |
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198 ≈⟨ cdr ( IsCategory.o-resp-≈ (Category.isCategory A) eq-fg ((IsFunctor.≈-cong (isFunctor T)) eq-hi )) ⟩ |
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199 TMap μ c o ( FMap T i o g ) |
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200 ≈⟨⟩ |
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201 join K i g |
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202 ∎ |
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203 |
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204 |
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205 record KHom (a : Obj A) (b : Obj A) |
| 70 | 206 : Set (suc (c₂ ⊔ c₁)) where |
| 207 field | |
| 208 KMap : Hom A a ( FObj T b ) | |
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209 |
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210 K-id : {a : Obj A} → KHom a a |
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211 K-id {a = a} = record { KMap = TMap η a } |
| 56 | 212 |
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213 open import Relation.Binary.Core |
| 73 | 214 open KHom |
| 56 | 215 |
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216 _⋍_ : { a : Obj A } { b : Obj A } (f g : KHom a b ) -> Set ℓ |
| 73 | 217 _⋍_ {a} {b} f g = A [ KMap f ≈ KMap g ] |
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218 |
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219 _*_ : { a b : Obj A } → { c : Obj A } → |
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220 ( KHom b c) → ( KHom a b) → KHom a c |
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221 _*_ {a} {b} {c} g f = record { KMap = join K {a} {b} {c} (KMap g) (KMap f) } |
| 70 | 222 |
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223 isKleisliCategory : IsCategory ( Obj A ) KHom _⋍_ _*_ K-id |
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224 isKleisliCategory = record { isEquivalence = isEquivalence |
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225 ; identityL = KidL |
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226 ; identityR = KidR |
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227 ; o-resp-≈ = Ko-resp |
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228 ; associative = Kassoc |
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229 } |
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230 where |
| 73 | 231 open ≈-Reasoning (A) |
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232 isEquivalence : { a b : Obj A } -> |
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233 IsEquivalence {_} {_} {KHom a b} _⋍_ |
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234 isEquivalence {C} {D} = |
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235 record { refl = λ {F} → ⋍-refl {F} |
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236 ; sym = λ {F} {G} → ⋍-sym {F} {G} |
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237 ; trans = λ {F} {G} {H} → ⋍-trans {F} {G} {H} |
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238 } where |
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239 ⋍-refl : {F : KHom C D} → F ⋍ F |
| 73 | 240 ⋍-refl = refl-hom |
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241 ⋍-sym : {F G : KHom C D} → F ⋍ G → G ⋍ F |
| 73 | 242 ⋍-sym = sym |
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243 ⋍-trans : {F G H : KHom C D} → F ⋍ G → G ⋍ H → F ⋍ H |
| 73 | 244 ⋍-trans = trans-hom |
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245 KidL : {C D : Obj A} -> {f : KHom C D} → (K-id * f) ⋍ f |
| 73 | 246 KidL {_} {_} {f} = Lemma7 (KMap f) |
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247 KidR : {C D : Obj A} -> {f : KHom C D} → (f * K-id) ⋍ f |
| 73 | 248 KidR {_} {_} {f} = Lemma8 (KMap f) |
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249 Ko-resp : {a b c : Obj A} -> {f g : KHom a b } → {h i : KHom b c } → |
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250 f ⋍ g → h ⋍ i → (h * f) ⋍ (i * g) |
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251 Ko-resp {a} {b} {c} {f} {g} {h} {i} eq-fg eq-hi = Lemma10 {a} {b} {c} (KMap f) (KMap g) (KMap h) (KMap i) eq-fg eq-hi |
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252 Kassoc : {a b c d : Obj A} -> {f : KHom c d } → {g : KHom b c } → {h : KHom a b } → |
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253 (f * (g * h)) ⋍ ((f * g) * h) |
| 73 | 254 Kassoc {_} {_} {_} {_} {f} {g} {h} = Lemma9 (KMap f) (KMap g) (KMap h) |
| 3 | 255 |
| 75 | 256 KleisliCategory : Category c₁ (suc (c₂ ⊔ c₁)) ℓ |
| 257 KleisliCategory = | |
| 258 record { Obj = Obj A | |
| 259 ; Hom = KHom | |
| 260 ; _o_ = _*_ | |
| 261 ; _≈_ = _⋍_ | |
| 262 ; Id = K-id | |
| 263 ; isCategory = isKleisliCategory | |
| 264 } | |
| 56 | 265 |
| 75 | 266 U_T : Functor KleisliCategory A |
| 267 U_T = record { | |
| 268 FObj = FObj T | |
| 269 ; FMap = ufmap | |
| 270 ; isFunctor = record | |
| 271 { ≈-cong = ≈-cong | |
| 272 ; identity = identity | |
| 273 ; distr = distr | |
| 274 } | |
| 275 } where | |
| 276 ufmap : {a b : Obj A} (f : KHom a b ) -> Hom A (FObj T a) (FObj T b) | |
| 277 ufmap {a} {b} f = A [ TMap μ (b) o FMap T (KMap f) ] | |
| 278 identity : {a : Obj A} → A [ ufmap (K-id {a}) ≈ id1 A (FObj T a) ] | |
| 279 identity {a} = let open ≈-Reasoning (A) in | |
| 280 begin | |
| 281 TMap μ (a) o FMap T (TMap η a) | |
| 282 ≈⟨ IsMonad.unity2 (isMonad M) ⟩ | |
| 283 id1 A (FObj T a) | |
| 284 ∎ | |
| 285 ≈-cong : {a b : Obj A} {f g : KHom a b} → A [ KMap f ≈ KMap g ] → A [ ufmap f ≈ ufmap g ] | |
| 286 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning (A) in | |
| 287 begin | |
| 288 TMap μ (b) o FMap T (KMap f) | |
| 289 ≈⟨ cdr (fcong T f≈g) ⟩ | |
| 290 TMap μ (b) o FMap T (KMap g) | |
| 291 ∎ | |
| 292 distr : {a b c : Obj A} {f : KHom a b} {g : KHom b c} → A [ ufmap (g * f) ≈ (A [ ufmap g o ufmap f ] )] | |
| 293 distr {_} {_} {c} {f} {g} = let open ≈-Reasoning (A) in | |
| 294 begin | |
| 295 ufmap (g * f) | |
| 296 -- ≈⟨⟩ | |
| 297 -- TMap μ (FObj T c) o FMap T ( (TMap μ (c) o FMap T (KMap g)) o (KMap f)) | |
| 298 ≈⟨ {!!} ⟩ | |
| 299 ufmap g o ufmap f | |
| 300 ∎ | |
| 301 | |
| 302 | |
| 303 ffmap : {a b : Obj A} (f : Hom A a b) -> KHom a b | |
| 304 ffmap f = record { KMap = A [ TMap η (Category.cod A f) o f ] } | |
| 305 | |
| 306 F_T : Functor A KleisliCategory | |
| 307 F_T = record { | |
| 308 FObj = \a -> a | |
| 309 ; FMap = ffmap | |
| 310 ; isFunctor = record | |
| 311 { ≈-cong = ≈-cong | |
| 312 ; identity = identity | |
| 313 ; distr = distr | |
| 314 } | |
| 315 } where | |
| 316 identity : {a : Obj A} → A [ A [ TMap η a o id1 A a ] ≈ TMap η a ] | |
| 317 identity {a} = IsCategory.identityR ( Category.isCategory A) | |
| 318 lemma1 : {a b : Obj A} {f g : Hom A a b} → A [ f ≈ g ] → A [ TMap η b ≈ TMap η b ] | |
| 319 lemma1 f≈g = IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory A )) | |
| 320 ≈-cong : {a b : Obj A} {f g : Hom A a b} → A [ f ≈ g ] → A [ A [ TMap η (Category.cod A f) o f ] ≈ A [ TMap η (Category.cod A g) o g ] ] | |
| 321 ≈-cong f≈g = (IsCategory.o-resp-≈ (Category.isCategory A)) f≈g ( lemma1 f≈g ) | |
| 322 distr : {a b c : Obj A} {f : Hom A a b} {g : Hom A b c} → | |
| 323 ( ffmap (A [ g o f ]) ⋍ ( ffmap g * ffmap f ) ) | |
| 324 distr {_} {_} {_} {f} {g} = let open ≈-Reasoning (A) in | |
| 325 begin | |
| 326 KMap (ffmap (A [ g o f ])) | |
| 327 ≈⟨ {!!} ⟩ | |
| 328 KMap ( ffmap g * ffmap f ) | |
| 329 ∎ | |
| 330 | |
| 331 | |
| 332 | |
| 333 | |
| 334 | |
| 335 |