Mercurial > hg > Members > kono > Proof > category
annotate kleisli.agda @ 639:4cf8f982dc5b
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 02 Jul 2017 02:18:57 +0900 |
| parents | 2d32ded94aaf |
| children | a5f2ca67e7c5 |
| rev | line source |
|---|---|
| 82 | 1 -- -- -- -- -- -- -- -- |
| 153 | 2 -- Monad to Kelisli Category |
| 82 | 3 -- defines U_T and F_T as a resolution of Monad |
| 4 -- checks Adjointness | |
| 5 -- | |
| 6 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 7 -- -- -- -- -- -- -- -- | |
| 0 | 8 |
| 9 -- Monad | |
| 10 -- Category A | |
| 11 -- A = Category | |
| 22 | 12 -- Functor T : A → A |
| 0 | 13 --T(a) = t(a) |
| 14 --T(f) = tf(f) | |
| 15 | |
| 2 | 16 open import Category -- https://github.com/konn/category-agda |
| 0 | 17 open import Level |
| 56 | 18 --open import Category.HomReasoning |
| 19 open import HomReasoning | |
| 20 open import cat-utility | |
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21 open import Category.Cat |
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22 |
| 152 | 23 module kleisli { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } |
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24 { T : Functor A A } |
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25 { η : NTrans A A identityFunctor T } |
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26 { μ : NTrans A A (T ○ T) T } |
| 130 | 27 { M : Monad A T η μ } where |
| 0 | 28 |
| 1 | 29 --T(g f) = T(g) T(f) |
| 30 | |
| 31 | 31 open Functor |
| 22 | 32 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 } |
| 33 → A [ ( FMap T (A [ g o f ] )) ≈ (A [ FMap T g o FMap T f ]) ] | |
| 300 | 34 Lemma1 = λ t → IsFunctor.distr ( isFunctor t ) |
| 0 | 35 |
| 36 | |
| 7 | 37 open NTrans |
| 1 | 38 Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A} |
| 22 | 39 → (μ : NTrans A A F G) → {a b : Obj A} { f : Hom A a b } |
| 30 | 40 → A [ A [ FMap G f o TMap μ a ] ≈ A [ TMap μ b o FMap F f ] ] |
| 300 | 41 Lemma2 = λ n → IsNTrans.commute ( isNTrans n ) |
| 0 | 42 |
| 82 | 43 -- Laws of Monad |
| 44 -- | |
| 22 | 45 -- η : 1_A → T |
| 46 -- μ : TT → T | |
| 82 | 47 -- μ(a)η(T(a)) = a -- unity law 1 |
| 48 -- μ(a)T(η(a)) = a -- unity law 2 | |
| 49 -- μ(a)(μ(T(a))) = μ(a)T(μ(a)) -- association law | |
| 0 | 50 |
| 51 | |
| 2 | 52 open Monad |
| 53 Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 54 { T : Functor A A } | |
| 7 | 55 { η : NTrans A A identityFunctor T } |
| 56 { μ : NTrans A A (T ○ T) T } | |
| 22 | 57 { a : Obj A } → |
| 2 | 58 ( M : Monad A T η μ ) |
| 30 | 59 → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈ A [ TMap μ a o FMap T (TMap μ a) ] ] |
| 300 | 60 Lemma3 = λ m → IsMonad.assoc ( isMonad m ) |
| 2 | 61 |
| 62 | |
| 63 Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b} | |
| 22 | 64 → A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ] |
| 300 | 65 Lemma4 = λ a → IsCategory.identityL ( Category.isCategory a ) |
| 0 | 66 |
| 3 | 67 Lemma5 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} |
| 68 { T : Functor A A } | |
| 7 | 69 { η : NTrans A A identityFunctor T } |
| 70 { μ : NTrans A A (T ○ T) T } | |
| 22 | 71 { a : Obj A } → |
| 3 | 72 ( M : Monad A T η μ ) |
| 30 | 73 → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 300 | 74 Lemma5 = λ m → IsMonad.unity1 ( isMonad m ) |
| 3 | 75 |
| 76 Lemma6 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} | |
| 77 { T : Functor A A } | |
| 7 | 78 { η : NTrans A A identityFunctor T } |
| 79 { μ : NTrans A A (T ○ T) T } | |
| 22 | 80 { a : Obj A } → |
| 3 | 81 ( M : Monad A T η μ ) |
| 30 | 82 → A [ A [ TMap μ a o (FMap T (TMap η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] |
| 300 | 83 Lemma6 = λ m → IsMonad.unity2 ( isMonad m ) |
| 3 | 84 |
| 82 | 85 -- Laws of Kleisli Category |
| 86 -- | |
| 0 | 87 -- nat of η |
| 82 | 88 -- g ○ f = μ(c) T(g) f -- join definition |
| 89 -- η(b) ○ f = f -- left id (Lemma7) | |
| 90 -- f ○ η(a) = f -- right id (Lemma8) | |
| 91 -- h ○ (g ○ f) = (h ○ g) ○ f -- assocation law (Lemma9) | |
| 11 | 92 |
| 22 | 93 -- η(b) ○ f = f |
| 73 | 94 Lemma7 : { a : Obj A } { b : Obj A } ( f : Hom A a ( FObj T b) ) |
| 130 | 95 → A [ join M (TMap η b) f ≈ f ] |
| 73 | 96 Lemma7 {_} {b} f = |
| 97 let open ≈-Reasoning (A) in | |
| 21 | 98 begin |
| 130 | 99 join M (TMap η b) f |
| 21 | 100 ≈⟨ refl-hom ⟩ |
| 73 | 101 A [ TMap μ b o A [ FMap T ((TMap η b)) o f ] ] |
| 102 ≈⟨ IsCategory.associative (Category.isCategory A) ⟩ | |
| 103 A [ A [ TMap μ b o FMap T ((TMap η b)) ] o f ] | |
| 130 | 104 ≈⟨ car ( IsMonad.unity2 ( isMonad M) ) ⟩ |
| 73 | 105 A [ id (FObj T b) o f ] |
| 106 ≈⟨ IsCategory.identityL (Category.isCategory A) ⟩ | |
| 21 | 107 f |
| 108 ∎ | |
| 7 | 109 |
| 22 | 110 -- f ○ η(a) = f |
| 73 | 111 Lemma8 : { a : Obj A } { b : Obj A } |
| 22 | 112 ( f : Hom A a ( FObj T b) ) |
| 130 | 113 → A [ join M f (TMap η a) ≈ f ] |
| 73 | 114 Lemma8 {a} {b} f = |
| 22 | 115 begin |
| 130 | 116 join M f (TMap η a) |
| 22 | 117 ≈⟨ refl-hom ⟩ |
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118 A [ TMap μ b o A [ FMap T f o (TMap η a) ] ] |
| 66 | 119 ≈⟨ cdr ( nat η ) ⟩ |
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120 A [ TMap μ b o A [ (TMap η ( FObj T b)) o f ] ] |
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121 ≈⟨ IsCategory.associative (Category.isCategory A) ⟩ |
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122 A [ A [ TMap μ b o (TMap η ( FObj T b)) ] o f ] |
| 130 | 123 ≈⟨ car ( IsMonad.unity1 ( isMonad M) ) ⟩ |
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124 A [ id (FObj T b) o f ] |
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125 ≈⟨ IsCategory.identityL (Category.isCategory A) ⟩ |
| 22 | 126 f |
| 127 ∎ where | |
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128 open ≈-Reasoning (A) |
| 5 | 129 |
| 22 | 130 -- h ○ (g ○ f) = (h ○ g) ○ f |
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131 Lemma9 : { a b c d : Obj A } |
| 73 | 132 ( h : Hom A c ( FObj T d) ) |
| 23 | 133 ( g : Hom A b ( FObj T c) ) |
| 73 | 134 ( f : Hom A a ( FObj T b) ) |
| 130 | 135 → A [ join M h (join M g f) ≈ join M ( join M h g) f ] |
| 73 | 136 Lemma9 {a} {b} {c} {d} h g f = |
| 24 | 137 begin |
| 130 | 138 join M h (join M g f) |
| 30 | 139 ≈⟨⟩ |
| 130 | 140 join M h ( ( TMap μ c o ( FMap T g o f ) ) ) |
| 28 | 141 ≈⟨ refl-hom ⟩ |
| 30 | 142 ( TMap μ d o ( FMap T h o ( TMap μ c o ( FMap T g o f ) ) ) ) |
| 28 | 143 ≈⟨ cdr ( cdr ( assoc )) ⟩ |
| 30 | 144 ( TMap μ d o ( FMap T h o ( ( TMap μ c o FMap T g ) o f ) ) ) |
| 28 | 145 ≈⟨ assoc ⟩ --- ( f o ( g o h ) ) = ( ( f o g ) o h ) |
| 30 | 146 ( ( TMap μ d o FMap T h ) o ( (TMap μ c o FMap T g ) o f ) ) |
| 25 | 147 ≈⟨ assoc ⟩ |
| 30 | 148 ( ( ( TMap μ d o FMap T h ) o (TMap μ c o FMap T g ) ) o f ) |
| 253 | 149 ≈↑⟨ car assoc ⟩ |
| 30 | 150 ( ( TMap μ d o ( FMap T h o ( TMap μ c o FMap T g ) ) ) o f ) |
| 28 | 151 ≈⟨ car ( cdr (assoc) ) ⟩ |
| 30 | 152 ( ( TMap μ d o ( ( FMap T h o TMap μ c ) o FMap T g ) ) o f ) |
| 28 | 153 ≈⟨ car assoc ⟩ |
| 30 | 154 ( ( ( TMap μ d o ( FMap T h o TMap μ c ) ) o FMap T g ) o f ) |
| 28 | 155 ≈⟨ car (car ( cdr ( begin |
| 30 | 156 ( FMap T h o TMap μ c ) |
| 66 | 157 ≈⟨ nat μ ⟩ |
| 30 | 158 ( TMap μ (FObj T d) o FMap T (FMap T h) ) |
| 25 | 159 ∎ |
| 160 ))) ⟩ | |
| 30 | 161 ( ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) ) o FMap T g ) o f ) |
| 253 | 162 ≈↑⟨ car assoc ⟩ |
| 30 | 163 ( ( TMap μ d o ( ( TMap μ ( FObj T d) o FMap T ( FMap T h ) ) o FMap T g ) ) o f ) |
| 253 | 164 ≈↑⟨ car ( cdr assoc ) ⟩ |
| 30 | 165 ( ( TMap μ d o ( TMap μ ( FObj T d) o ( FMap T ( FMap T h ) o FMap T g ) ) ) o f ) |
| 253 | 166 ≈↑⟨ car ( cdr (cdr (distr T ))) ⟩ |
| 30 | 167 ( ( TMap μ d o ( TMap μ ( FObj T d) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 28 | 168 ≈⟨ car assoc ⟩ |
| 30 | 169 ( ( ( TMap μ d o TMap μ ( FObj T d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 28 | 170 ≈⟨ car ( car ( |
| 27 | 171 begin |
| 30 | 172 ( TMap μ d o TMap μ (FObj T d) ) |
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173 ≈⟨ IsMonad.assoc ( isMonad M) ⟩ |
| 30 | 174 ( TMap μ d o FMap T (TMap μ d) ) |
| 27 | 175 ∎ |
| 176 )) ⟩ | |
| 30 | 177 ( ( ( TMap μ d o FMap T ( TMap μ d) ) o FMap T ( ( FMap T h o g ) ) ) o f ) |
| 253 | 178 ≈↑⟨ car assoc ⟩ |
| 30 | 179 ( ( TMap μ d o ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) ) o f ) |
| 253 | 180 ≈↑⟨ assoc ⟩ |
| 30 | 181 ( TMap μ d o ( ( FMap T ( TMap μ d ) o FMap T ( ( FMap T h o g ) ) ) o f ) ) |
| 253 | 182 ≈↑⟨ cdr ( car ( distr T )) ⟩ |
| 30 | 183 ( TMap μ d o ( FMap T ( ( ( TMap μ d ) o ( FMap T h o g ) ) ) o f ) ) |
| 23 | 184 ≈⟨ refl-hom ⟩ |
| 130 | 185 join M ( ( TMap μ d o ( FMap T h o g ) ) ) f |
| 23 | 186 ≈⟨ refl-hom ⟩ |
| 130 | 187 join M ( join M h g) f |
| 24 | 188 ∎ where open ≈-Reasoning (A) |
| 3 | 189 |
| 82 | 190 -- |
| 191 -- o-resp in Kleisli Category ( for Functor definitions ) | |
| 192 -- | |
| 300 | 193 Lemma10 : {a b c : Obj A} → (f g : Hom A a (FObj T b) ) → (h i : Hom A b (FObj T c) ) → |
| 130 | 194 A [ f ≈ g ] → A [ h ≈ i ] → A [ (join M h f) ≈ (join M i g) ] |
| 474 | 195 Lemma10 {a} {b} {c} f g h i f≈g h≈i = let open ≈-Reasoning (A) in |
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196 begin |
| 130 | 197 join M h f |
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198 ≈⟨⟩ |
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199 TMap μ c o ( FMap T h o f ) |
| 474 | 200 ≈⟨ cdr ( car ( fcong T h≈i )) ⟩ |
| 201 TMap μ c o ( FMap T i o f ) | |
| 202 ≈⟨ cdr ( cdr f≈g ) ⟩ | |
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203 TMap μ c o ( FMap T i o g ) |
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204 ≈⟨⟩ |
| 130 | 205 join M i g |
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206 ∎ |
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207 |
| 82 | 208 -- |
| 209 -- Hom in Kleisli Category | |
| 210 -- | |
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211 |
| 87 | 212 |
| 213 record KleisliHom { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } { T : Functor A A } (a : Obj A) (b : Obj A) | |
| 467 | 214 : Set c₂ where |
| 215 field | |
| 216 KMap : Hom A a ( FObj T b ) | |
| 217 | |
| 218 open KleisliHom | |
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219 |
| 467 | 220 -- we need this to make agda check stop |
| 300 | 221 KHom = λ (a b : Obj A) → KleisliHom {c₁} {c₂} {ℓ} {A} {T} a b |
| 87 | 222 |
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223 K-id : {a : Obj A} → KHom a a |
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224 K-id {a = a} = record { KMap = TMap η a } |
| 56 | 225 |
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226 open import Relation.Binary.Core |
| 56 | 227 |
| 300 | 228 _⋍_ : { a : Obj A } { b : Obj A } (f g : KHom a b ) → Set ℓ |
| 73 | 229 _⋍_ {a} {b} f g = A [ KMap f ≈ KMap g ] |
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230 |
| 108 | 231 _*_ : { a b c : Obj A } → ( KHom b c) → ( KHom a b) → KHom a c |
| 130 | 232 _*_ {a} {b} {c} g f = record { KMap = join M {a} {b} {c} (KMap g) (KMap f) } |
| 70 | 233 |
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234 isKleisliCategory : IsCategory ( Obj A ) KHom _⋍_ _*_ K-id |
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235 isKleisliCategory = record { isEquivalence = isEquivalence |
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236 ; identityL = KidL |
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237 ; identityR = KidR |
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238 ; o-resp-≈ = Ko-resp |
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239 ; associative = Kassoc |
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240 } |
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241 where |
| 73 | 242 open ≈-Reasoning (A) |
| 300 | 243 isEquivalence : { a b : Obj A } → |
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244 IsEquivalence {_} {_} {KHom a b} _⋍_ |
| 82 | 245 isEquivalence {C} {D} = -- this is the same function as A's equivalence but has different types |
| 246 record { refl = refl-hom | |
| 247 ; sym = sym | |
| 248 ; trans = trans-hom | |
| 249 } | |
| 300 | 250 KidL : {C D : Obj A} → {f : KHom C D} → (K-id * f) ⋍ f |
| 73 | 251 KidL {_} {_} {f} = Lemma7 (KMap f) |
| 300 | 252 KidR : {C D : Obj A} → {f : KHom C D} → (f * K-id) ⋍ f |
| 73 | 253 KidR {_} {_} {f} = Lemma8 (KMap f) |
| 300 | 254 Ko-resp : {a b c : Obj A} → {f g : KHom a b } → {h i : KHom b c } → |
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255 f ⋍ g → h ⋍ i → (h * f) ⋍ (i * g) |
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256 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 |
| 300 | 257 Kassoc : {a b c d : Obj A} → {f : KHom c d } → {g : KHom b c } → {h : KHom a b } → |
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258 (f * (g * h)) ⋍ ((f * g) * h) |
| 73 | 259 Kassoc {_} {_} {_} {_} {f} {g} {h} = Lemma9 (KMap f) (KMap g) (KMap h) |
| 3 | 260 |
| 78 | 261 KleisliCategory : Category c₁ c₂ ℓ |
| 75 | 262 KleisliCategory = |
| 263 record { Obj = Obj A | |
| 264 ; Hom = KHom | |
| 265 ; _o_ = _*_ | |
| 266 ; _≈_ = _⋍_ | |
| 267 ; Id = K-id | |
| 268 ; isCategory = isKleisliCategory | |
| 269 } | |
| 56 | 270 |
| 82 | 271 -- |
| 272 -- Resolution | |
| 273 -- T = U_T U_F | |
| 274 -- nat-ε | |
| 275 -- nat-η | |
| 276 -- | |
| 277 | |
| 300 | 278 ufmap : {a b : Obj A} (f : KHom a b ) → Hom A (FObj T a) (FObj T b) |
| 80 | 279 ufmap {a} {b} f = A [ TMap μ (b) o FMap T (KMap f) ] |
| 280 | |
| 75 | 281 U_T : Functor KleisliCategory A |
| 282 U_T = record { | |
| 283 FObj = FObj T | |
| 284 ; FMap = ufmap | |
| 285 ; isFunctor = record | |
| 286 { ≈-cong = ≈-cong | |
| 287 ; identity = identity | |
| 76 | 288 ; distr = distr1 |
| 75 | 289 } |
| 290 } where | |
| 291 identity : {a : Obj A} → A [ ufmap (K-id {a}) ≈ id1 A (FObj T a) ] | |
| 292 identity {a} = let open ≈-Reasoning (A) in | |
| 293 begin | |
| 294 TMap μ (a) o FMap T (TMap η a) | |
| 295 ≈⟨ IsMonad.unity2 (isMonad M) ⟩ | |
| 253 | 296 id (FObj T a) |
| 75 | 297 ∎ |
| 298 ≈-cong : {a b : Obj A} {f g : KHom a b} → A [ KMap f ≈ KMap g ] → A [ ufmap f ≈ ufmap g ] | |
| 299 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning (A) in | |
| 300 begin | |
| 301 TMap μ (b) o FMap T (KMap f) | |
| 302 ≈⟨ cdr (fcong T f≈g) ⟩ | |
| 303 TMap μ (b) o FMap T (KMap g) | |
| 304 ∎ | |
| 76 | 305 distr1 : {a b c : Obj A} {f : KHom a b} {g : KHom b c} → A [ ufmap (g * f) ≈ (A [ ufmap g o ufmap f ] )] |
| 306 distr1 {a} {b} {c} {f} {g} = let open ≈-Reasoning (A) in | |
| 75 | 307 begin |
| 308 ufmap (g * f) | |
| 76 | 309 ≈⟨⟩ |
| 310 ufmap {a} {c} ( record { KMap = TMap μ (c) o (FMap T (KMap g) o (KMap f)) } ) | |
| 311 ≈⟨⟩ | |
| 312 TMap μ (c) o FMap T ( TMap μ (c) o (FMap T (KMap g) o (KMap f))) | |
| 313 ≈⟨ cdr ( distr T) ⟩ | |
| 314 TMap μ (c) o (( FMap T ( TMap μ (c)) o FMap T (FMap T (KMap g) o (KMap f)))) | |
| 315 ≈⟨ assoc ⟩ | |
| 316 (TMap μ (c) o ( FMap T ( TMap μ (c)))) o FMap T (FMap T (KMap g) o (KMap f)) | |
| 317 ≈⟨ car (sym (IsMonad.assoc (isMonad M))) ⟩ | |
| 318 (TMap μ (c) o ( TMap μ (FObj T c))) o FMap T (FMap T (KMap g) o (KMap f)) | |
| 319 ≈⟨ sym assoc ⟩ | |
| 320 TMap μ (c) o (( TMap μ (FObj T c)) o FMap T (FMap T (KMap g) o (KMap f))) | |
| 321 ≈⟨ cdr (cdr (distr T)) ⟩ | |
| 322 TMap μ (c) o (( TMap μ (FObj T c)) o (FMap T (FMap T (KMap g)) o FMap T (KMap f))) | |
| 323 ≈⟨ cdr (assoc) ⟩ | |
| 324 TMap μ (c) o ((( TMap μ (FObj T c)) o (FMap T (FMap T (KMap g)))) o FMap T (KMap f)) | |
| 325 ≈⟨ sym (cdr (car (nat μ ))) ⟩ | |
| 326 TMap μ (c) o ((FMap T (KMap g ) o TMap μ (b)) o FMap T (KMap f )) | |
| 327 ≈⟨ cdr (sym assoc) ⟩ | |
| 328 TMap μ (c) o (FMap T (KMap g ) o ( TMap μ (b) o FMap T (KMap f ))) | |
| 329 ≈⟨ assoc ⟩ | |
| 330 ( TMap μ (c) o FMap T (KMap g ) ) o ( TMap μ (b) o FMap T (KMap f ) ) | |
| 331 ≈⟨⟩ | |
| 75 | 332 ufmap g o ufmap f |
| 333 ∎ | |
| 334 | |
| 335 | |
| 300 | 336 ffmap : {a b : Obj A} (f : Hom A a b) → KHom a b |
| 474 | 337 ffmap {a} {b} f = record { KMap = A [ TMap η b o f ] } |
| 75 | 338 |
| 339 F_T : Functor A KleisliCategory | |
| 340 F_T = record { | |
| 300 | 341 FObj = λ a → a |
| 75 | 342 ; FMap = ffmap |
| 343 ; isFunctor = record | |
| 344 { ≈-cong = ≈-cong | |
| 345 ; identity = identity | |
| 76 | 346 ; distr = distr1 |
| 75 | 347 } |
| 348 } where | |
| 349 identity : {a : Obj A} → A [ A [ TMap η a o id1 A a ] ≈ TMap η a ] | |
| 350 identity {a} = IsCategory.identityR ( Category.isCategory A) | |
| 82 | 351 -- Category.cod A f and Category.cod A g are actualy the same b, so we don't need cong-≈, just refl |
| 474 | 352 lemma1 : {b : Obj A} → A [ TMap η b ≈ TMap η b ] |
| 353 lemma1 = IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory A )) | |
| 354 ≈-cong : {a b : Obj A} {f g : Hom A a b} → A [ f ≈ g ] → A [ A [ TMap η b o f ] ≈ A [ TMap η b o g ] ] | |
| 355 ≈-cong f≈g = (IsCategory.o-resp-≈ (Category.isCategory A)) f≈g lemma1 | |
| 76 | 356 distr1 : {a b c : Obj A} {f : Hom A a b} {g : Hom A b c} → |
| 75 | 357 ( ffmap (A [ g o f ]) ⋍ ( ffmap g * ffmap f ) ) |
| 77 | 358 distr1 {a} {b} {c} {f} {g} = let open ≈-Reasoning (A) in |
| 75 | 359 begin |
| 360 KMap (ffmap (A [ g o f ])) | |
| 77 | 361 ≈⟨⟩ |
| 362 TMap η (c) o (g o f) | |
| 363 ≈⟨ assoc ⟩ | |
| 364 (TMap η (c) o g) o f | |
| 365 ≈⟨ car (sym (nat η)) ⟩ | |
| 366 (FMap T g o TMap η (b)) o f | |
| 367 ≈⟨ sym idL ⟩ | |
| 253 | 368 id (FObj T c) o ((FMap T g o TMap η (b)) o f) |
| 77 | 369 ≈⟨ sym (car (IsMonad.unity2 (isMonad M))) ⟩ |
| 370 (TMap μ c o FMap T (TMap η c)) o ((FMap T g o TMap η (b)) o f) | |
| 371 ≈⟨ sym assoc ⟩ | |
| 372 TMap μ c o ( FMap T (TMap η c) o ((FMap T g o TMap η (b)) o f) ) | |
| 373 ≈⟨ cdr (cdr (sym assoc)) ⟩ | |
| 374 TMap μ c o ( FMap T (TMap η c) o (FMap T g o (TMap η (b) o f))) | |
| 375 ≈⟨ cdr assoc ⟩ | |
| 376 TMap μ c o ( ( FMap T (TMap η c) o FMap T g ) o (TMap η (b) o f) ) | |
| 377 ≈⟨ sym (cdr ( car ( distr T ))) ⟩ | |
| 378 TMap μ c o ( ( FMap T (TMap η c o g)) o (TMap η (b) o f)) | |
| 379 ≈⟨ assoc ⟩ | |
| 380 (TMap μ c o ( FMap T (TMap η c o g))) o (TMap η (b) o f) | |
| 381 ≈⟨ assoc ⟩ | |
| 382 ((TMap μ c o (FMap T (TMap η c o g))) o TMap η b) o f | |
| 383 ≈⟨ sym assoc ⟩ | |
| 384 (TMap μ c o (FMap T (TMap η c o g))) o (TMap η b o f) | |
| 385 ≈⟨ sym assoc ⟩ | |
| 386 TMap μ c o ( (FMap T (TMap η c o g)) o (TMap η b o f) ) | |
| 387 ≈⟨⟩ | |
| 130 | 388 join M (TMap η c o g) (TMap η b o f) |
| 77 | 389 ≈⟨⟩ |
| 75 | 390 KMap ( ffmap g * ffmap f ) |
| 391 ∎ | |
| 392 | |
| 82 | 393 -- |
| 394 -- T = (U_T ○ F_T) | |
| 395 -- | |
| 396 | |
| 300 | 397 Lemma11-1 : ∀{a b : Obj A} → (f : Hom A a b ) → A [ FMap T f ≈ FMap (U_T ○ F_T) f ] |
| 79 | 398 Lemma11-1 {a} {b} f = let open ≈-Reasoning (A) in |
| 399 sym ( begin | |
| 400 FMap (U_T ○ F_T) f | |
| 401 ≈⟨⟩ | |
| 402 FMap U_T ( FMap F_T f ) | |
| 403 ≈⟨⟩ | |
| 404 TMap μ (b) o FMap T (KMap ( ffmap f ) ) | |
| 405 ≈⟨⟩ | |
| 406 TMap μ (b) o FMap T (TMap η (b) o f) | |
| 407 ≈⟨ cdr (distr T ) ⟩ | |
| 408 TMap μ (b) o ( FMap T (TMap η (b)) o FMap T f) | |
| 409 ≈⟨ assoc ⟩ | |
| 410 (TMap μ (b) o FMap T (TMap η (b))) o FMap T f | |
| 411 ≈⟨ car (IsMonad.unity2 (isMonad M)) ⟩ | |
| 253 | 412 id (FObj T b) o FMap T f |
| 79 | 413 ≈⟨ idL ⟩ |
| 414 FMap T f | |
| 415 ∎ ) | |
| 416 | |
| 82 | 417 -- construct ≃ |
| 418 | |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
419 Lemma11 : T ≃ (U_T ○ F_T) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
420 Lemma11 f = Category.Cat.refl (Lemma11-1 f) |
| 78 | 421 |
| 82 | 422 -- |
| 423 -- natural transformations | |
| 424 -- | |
| 425 | |
| 78 | 426 nat-η : NTrans A A identityFunctor ( U_T ○ F_T ) |
| 467 | 427 nat-η = record { TMap = λ x → TMap η x ; isNTrans = record { commute = commute } } where |
| 428 commute : {a b : Obj A} {f : Hom A a b} | |
| 79 | 429 → A [ A [ ( FMap ( U_T ○ F_T ) f ) o ( TMap η a ) ] ≈ A [ (TMap η b ) o f ] ] |
| 467 | 430 commute {a} {b} {f} = let open ≈-Reasoning (A) in |
| 79 | 431 begin |
| 432 ( FMap ( U_T ○ F_T ) f ) o ( TMap η a ) | |
| 84 | 433 ≈⟨ sym (resp refl-hom (Lemma11-1 f)) ⟩ |
| 434 FMap T f o TMap η a | |
| 79 | 435 ≈⟨ nat η ⟩ |
| 84 | 436 TMap η b o f |
| 79 | 437 ∎ |
| 77 | 438 |
| 300 | 439 tmap-ε : (a : Obj A) → KHom (FObj T a) a |
| 78 | 440 tmap-ε a = record { KMap = id1 A (FObj T a) } |
| 441 | |
| 442 nat-ε : NTrans KleisliCategory KleisliCategory ( F_T ○ U_T ) identityFunctor | |
| 300 | 443 nat-ε = record { TMap = λ a → tmap-ε a; isNTrans = isNTrans1 } where |
| 130 | 444 commute1 : {a b : Obj A} {f : KHom a b} |
| 78 | 445 → (f * ( tmap-ε a ) ) ⋍ (( tmap-ε b ) * ( FMap (F_T ○ U_T) f ) ) |
| 130 | 446 commute1 {a} {b} {f} = let open ≈-Reasoning (A) in |
| 79 | 447 sym ( begin |
| 448 KMap (( tmap-ε b ) * ( FMap (F_T ○ U_T) f )) | |
| 80 | 449 ≈⟨⟩ |
| 253 | 450 TMap μ b o ( FMap T ( id (FObj T b) ) o ( KMap (FMap (F_T ○ U_T) f ) )) |
| 80 | 451 ≈⟨ cdr ( cdr ( |
| 452 begin | |
| 453 KMap (FMap (F_T ○ U_T) f ) | |
| 454 ≈⟨⟩ | |
| 455 KMap (FMap F_T (FMap U_T f)) | |
| 456 ≈⟨⟩ | |
| 457 TMap η (FObj T b) o (TMap μ (b) o FMap T (KMap f)) | |
| 458 ∎ | |
| 459 )) ⟩ | |
| 253 | 460 TMap μ b o ( FMap T ( id (FObj T b) ) o (TMap η (FObj T b) o (TMap μ (b) o FMap T (KMap f)))) |
| 80 | 461 ≈⟨ cdr (car (IsFunctor.identity (isFunctor T))) ⟩ |
| 253 | 462 TMap μ b o ( id (FObj T (FObj T b) ) o (TMap η (FObj T b) o (TMap μ (b) o FMap T (KMap f)))) |
| 80 | 463 ≈⟨ cdr idL ⟩ |
| 464 TMap μ b o (TMap η (FObj T b) o (TMap μ (b) o FMap T (KMap f))) | |
| 465 ≈⟨ assoc ⟩ | |
| 466 (TMap μ b o (TMap η (FObj T b))) o (TMap μ (b) o FMap T (KMap f)) | |
| 467 ≈⟨ (car (IsMonad.unity1 (isMonad M))) ⟩ | |
| 253 | 468 id (FObj T b) o (TMap μ (b) o FMap T (KMap f)) |
| 80 | 469 ≈⟨ idL ⟩ |
| 470 TMap μ b o FMap T (KMap f) | |
| 471 ≈⟨ cdr (sym idR) ⟩ | |
| 253 | 472 TMap μ b o ( FMap T (KMap f) o id (FObj T a) ) |
| 80 | 473 ≈⟨⟩ |
| 79 | 474 KMap (f * ( tmap-ε a )) |
| 475 ∎ ) | |
| 300 | 476 isNTrans1 : IsNTrans KleisliCategory KleisliCategory ( F_T ○ U_T ) identityFunctor (λ a → tmap-ε a ) |
| 130 | 477 isNTrans1 = record { commute = commute1 } |
| 77 | 478 |
| 82 | 479 -- |
| 480 -- μ = U_T ε U_F | |
| 481 -- | |
| 300 | 482 tmap-μ : (a : Obj A) → Hom A (FObj ( U_T ○ F_T ) (FObj ( U_T ○ F_T ) a)) (FObj ( U_T ○ F_T ) a) |
| 95 | 483 tmap-μ a = FMap U_T ( TMap nat-ε ( FObj F_T a )) |
| 484 | |
| 485 nat-μ : NTrans A A (( U_T ○ F_T ) ○ ( U_T ○ F_T )) ( U_T ○ F_T ) | |
| 486 nat-μ = record { TMap = tmap-μ ; isNTrans = isNTrans1 } where | |
| 130 | 487 commute1 : {a b : Obj A} {f : Hom A a b} |
| 95 | 488 → A [ A [ (FMap (U_T ○ F_T) f) o ( tmap-μ a) ] ≈ A [ ( tmap-μ b ) o FMap (U_T ○ F_T) ( FMap (U_T ○ F_T) f) ] ] |
| 130 | 489 commute1 {a} {b} {f} = let open ≈-Reasoning (A) in |
| 95 | 490 sym ( begin |
| 491 ( tmap-μ b ) o FMap (U_T ○ F_T) ( FMap (U_T ○ F_T) f) | |
| 492 ≈⟨⟩ | |
| 493 ( FMap U_T ( TMap nat-ε ( FObj F_T b )) ) o FMap (U_T ○ F_T) ( FMap (U_T ○ F_T) f) | |
| 494 ≈⟨ sym ( distr U_T) ⟩ | |
| 495 FMap U_T ( KleisliCategory [ TMap nat-ε ( FObj F_T b ) o FMap F_T ( FMap (U_T ○ F_T) f) ] ) | |
| 496 ≈⟨ fcong U_T (sym (nat nat-ε)) ⟩ | |
| 497 FMap U_T ( KleisliCategory [ (FMap F_T f) o TMap nat-ε ( FObj F_T a ) ] ) | |
| 498 ≈⟨ distr U_T ⟩ | |
| 499 (FMap U_T (FMap F_T f)) o FMap U_T ( TMap nat-ε ( FObj F_T a )) | |
| 500 ≈⟨⟩ | |
| 501 (FMap (U_T ○ F_T) f) o ( tmap-μ a) | |
| 502 ∎ ) | |
| 503 isNTrans1 : IsNTrans A A (( U_T ○ F_T ) ○ ( U_T ○ F_T )) ( U_T ○ F_T ) tmap-μ | |
| 130 | 504 isNTrans1 = record { commute = commute1 } |
| 95 | 505 |
| 300 | 506 Lemma12 : {x : Obj A } → A [ TMap nat-μ x ≈ FMap U_T ( TMap nat-ε ( FObj F_T x ) ) ] |
| 80 | 507 Lemma12 {x} = let open ≈-Reasoning (A) in |
| 508 sym ( begin | |
| 509 FMap U_T ( TMap nat-ε ( FObj F_T x ) ) | |
| 510 ≈⟨⟩ | |
| 95 | 511 tmap-μ x |
| 512 ≈⟨⟩ | |
| 513 TMap nat-μ x | |
| 514 ∎ ) | |
| 515 | |
| 300 | 516 Lemma13 : {x : Obj A } → A [ TMap μ x ≈ FMap U_T ( TMap nat-ε ( FObj F_T x ) ) ] |
| 95 | 517 Lemma13 {x} = let open ≈-Reasoning (A) in |
| 518 sym ( begin | |
| 519 FMap U_T ( TMap nat-ε ( FObj F_T x ) ) | |
| 520 ≈⟨⟩ | |
| 253 | 521 TMap μ x o FMap T (id (FObj T x) ) |
| 80 | 522 ≈⟨ cdr (IsFunctor.identity (isFunctor T)) ⟩ |
| 253 | 523 TMap μ x o id (FObj T (FObj T x) ) |
| 80 | 524 ≈⟨ idR ⟩ |
| 525 TMap μ x | |
| 526 ∎ ) | |
| 78 | 527 |
| 84 | 528 Adj_T : Adjunction A KleisliCategory U_T F_T nat-η nat-ε |
| 529 Adj_T = record { | |
| 80 | 530 isAdjunction = record { |
| 531 adjoint1 = adjoint1 ; | |
| 532 adjoint2 = adjoint2 | |
| 533 } | |
| 534 } where | |
| 535 adjoint1 : { b : Obj KleisliCategory } → | |
| 536 A [ A [ ( FMap U_T ( TMap nat-ε b)) o ( TMap nat-η ( FObj U_T b )) ] ≈ id1 A (FObj U_T b) ] | |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
537 adjoint1 {b} = let open ≈-Reasoning (A) in |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
538 begin |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
539 ( FMap U_T ( TMap nat-ε b)) o ( TMap nat-η ( FObj U_T b )) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
540 ≈⟨⟩ |
| 253 | 541 (TMap μ (b) o FMap T (id (FObj T b ))) o ( TMap η ( FObj T b )) |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
542 ≈⟨ car ( cdr (IsFunctor.identity (isFunctor T))) ⟩ |
| 253 | 543 (TMap μ (b) o (id (FObj T (FObj T b )))) o ( TMap η ( FObj T b )) |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
544 ≈⟨ car idR ⟩ |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
545 TMap μ (b) o ( TMap η ( FObj T b )) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
546 ≈⟨ IsMonad.unity1 (isMonad M) ⟩ |
| 253 | 547 id (FObj U_T b) |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
548 ∎ |
| 80 | 549 adjoint2 : {a : Obj A} → |
| 550 KleisliCategory [ KleisliCategory [ ( TMap nat-ε ( FObj F_T a )) o ( FMap F_T ( TMap nat-η a )) ] | |
| 87 | 551 ≈ id1 KleisliCategory (FObj F_T a) ] |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
552 adjoint2 {a} = let open ≈-Reasoning (A) in |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
553 begin |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
554 KMap (( TMap nat-ε ( FObj F_T a )) * ( FMap F_T ( TMap nat-η a )) ) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
555 ≈⟨⟩ |
| 253 | 556 TMap μ a o (FMap T (id (FObj T a) ) o ((TMap η (FObj T a)) o (TMap η a))) |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
557 ≈⟨ cdr ( car ( IsFunctor.identity (isFunctor T))) ⟩ |
| 253 | 558 TMap μ a o ((id (FObj T (FObj T a) )) o ((TMap η (FObj T a)) o (TMap η a))) |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
559 ≈⟨ cdr ( idL ) ⟩ |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
560 TMap μ a o ((TMap η (FObj T a)) o (TMap η a)) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
561 ≈⟨ assoc ⟩ |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
562 (TMap μ a o (TMap η (FObj T a))) o (TMap η a) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
563 ≈⟨ car (IsMonad.unity1 (isMonad M)) ⟩ |
| 253 | 564 id (FObj T a) o (TMap η a) |
|
81
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
565 ≈⟨ idL ⟩ |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
566 TMap η a |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
567 ≈⟨⟩ |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
568 TMap η (FObj F_T a) |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
569 ≈⟨⟩ |
|
0404b2ba7db6
Resolution Adjoint proved.
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
80
diff
changeset
|
570 KMap (id1 KleisliCategory (FObj F_T a)) |
| 82 | 571 ∎ |
| 77 | 572 |
| 87 | 573 open MResolution |
| 84 | 574 |
| 95 | 575 Resolution_T : MResolution A KleisliCategory T U_T F_T {nat-η} {nat-ε} {nat-μ} Adj_T |
| 84 | 576 Resolution_T = record { |
| 87 | 577 T=UF = Lemma11; |
| 578 μ=UεF = Lemma12 | |
| 84 | 579 } |
| 580 | |
|
97
2feec58bb02d
seprate comparison functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
96
diff
changeset
|
581 -- end |