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1 module nat where
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2
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3 -- Monad
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4 -- Category A
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5 -- A = Category
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6 -- Functor T : A -> A
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7 --T(a) = t(a)
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8 --T(f) = tf(f)
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9
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10 open import Category -- https://github.com/konn/category-agda
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11 open import Level
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12 open Functor
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13
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14 --T(g f) = T(g) T(f)
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15
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16 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 }
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17 -> A [ ( FMap T (A [ g o f ] )) ≈ (A [ FMap T g o FMap T f ]) ]
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18 Lemma1 = \t -> IsFunctor.distr ( isFunctor t )
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19
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20 -- F(f)
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21 -- F(a) ----> F(b)
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22 -- | |
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23 -- |t(a) |t(b) G(f)t(a) = t(b)F(f)
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24 -- | |
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25 -- v v
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26 -- G(a) ----> G(b)
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27 -- G(f)
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28
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29 record IsNTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (D : Category c₁ c₂ ℓ) (C : Category c₁′ c₂′ ℓ′)
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30 ( F G : Functor D C )
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31 (Trans : (A : Obj D) → Hom C (FObj F A) (FObj G A))
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32 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where
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33 field
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34 naturality : {a b : Obj D} {f : Hom D a b}
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35 → C [ C [ ( FMap G f ) o ( Trans a ) ] ≈ C [ (Trans b ) o (FMap F f) ] ]
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36 -- how to write uniquness?
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37 -- uniqness : {d : Obj D}
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38 -- → ∃{e : Trans d} -> ∀{a : Trans d} → C [ e ≈ a ]
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39
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40
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41 record NTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (domain : Category c₁ c₂ ℓ) (codomain : Category c₁′ c₂′ ℓ′) (F G : Functor domain codomain )
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42 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where
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43 field
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44 Trans : (A : Obj domain) → Hom codomain (FObj F A) (FObj G A)
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45 isNTrans : IsNTrans domain codomain F G Trans
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46
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47 open NTrans
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48 Lemma2 : {c₁ c₂ l : Level} {A : Category c₁ c₂ l} {F G : Functor A A}
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49 -> (μ : NTrans A A F G) -> {a b : Obj A} { f : Hom A a b }
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50 -> A [ A [ FMap G f o Trans μ a ] ≈ A [ Trans μ b o FMap F f ] ]
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51 Lemma2 = \n -> IsNTrans.naturality ( isNTrans n )
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52
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53 open import Category.Cat
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54
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55 -- η : 1_A -> T
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56 -- μ : TT -> T
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57 -- μ(a)η(T(a)) = a
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58 -- μ(a)T(η(a)) = a
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59 -- μ(a)(μ(T(a))) = μ(a)T(μ(a))
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60
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61 record IsMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ)
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62 ( T : Functor A A )
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63 ( η : NTrans A A identityFunctor T )
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64 ( μ : NTrans A A (T ○ T) T)
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65 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
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66 field
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67 assoc : {a : Obj A} -> A [ A [ Trans μ a o Trans μ ( FObj T a ) ] ≈ A [ Trans μ a o FMap T (Trans μ a) ] ]
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68 unity1 : {a : Obj A} -> A [ A [ Trans μ a o Trans η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
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69 unity2 : {a : Obj A} -> A [ A [ Trans μ a o (FMap T (Trans η a )) ] ≈ Id {_} {_} {_} {A} (FObj T a) ]
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70
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71 record Monad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (T : Functor A A) (η : NTrans A A identityFunctor T) (μ : NTrans A A (T ○ T) T)
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72 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where
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73 field
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74 isMonad : IsMonad A T η μ
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75
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76 open Monad
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77 Lemma3 : {c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ}
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78 { T : Functor A A }
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79 { η : NTrans A A identityFunctor T }
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80 { μ : NTrans A A (T ○ T) T }
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81 { a : Obj A } ->
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82 ( M : Monad A T η μ )
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83 -> A [ A [ Trans μ a o Trans μ ( FObj T a ) ] ≈ A [ Trans μ a o FMap T (Trans μ a) ] ]
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84 Lemma3 = \m -> IsMonad.assoc ( isMonad m )
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85
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86
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87 Lemma4 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b}
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88 -> A [ A [ Id {_} {_} {_} {A} b o f ] ≈ f ]
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89 Lemma4 = \a -> IsCategory.identityL ( Category.isCategory a )
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90
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91 -- nat of η
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92
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93
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94 -- g ○ f = μ(c) T(g) f
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95
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96 -- h ○ (g ○ f) = (h ○ g) ○ f
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97
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98 -- η(b) ○ f = f
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99 -- f ○ η(a) = f
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100
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101
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