Mercurial > hg > Members > kono > Proof > category
annotate monoidal.agda @ 704:b48c2d1c0796
clean up
author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
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date | Wed, 22 Nov 2017 04:42:48 +0900 |
parents | df3f1aae984f |
children | 73a998711118 |
rev | line source |
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696
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Monoidal category and applicative functor
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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1 open import Level |
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2 open import Level |
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3 open import Level |
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4 open import Category |
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5 module monoidal where |
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6 |
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7 open import Data.Product renaming (_×_ to _*_) |
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8 open import Category.Constructions.Product |
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9 open import HomReasoning |
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10 open import cat-utility |
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11 open import Relation.Binary.Core |
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12 open import Relation.Binary |
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13 |
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14 open Functor |
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15 |
698 | 16 record Iso {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) |
17 (x y : Obj C ) | |
18 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where | |
696
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19 field |
698 | 20 ≅→ : Hom C x y |
21 ≅← : Hom C y x | |
22 iso→ : C [ C [ ≅← o ≅→ ] ≈ id1 C x ] | |
23 iso← : C [ C [ ≅→ o ≅← ] ≈ id1 C y ] | |
696
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24 |
698 | 25 record IsStrictMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (I : Obj C) ( BI : Functor ( C × C ) C ) |
696
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26 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
700
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27 infixr 9 _□_ |
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28 _□_ : ( x y : Obj C ) → Obj C |
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29 _□_ x y = FObj BI ( x , y ) |
696
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30 field |
700
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31 mα : {a b c : Obj C} → ( a □ b) □ c ≡ a □ ( b □ c ) |
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32 mλ : (a : Obj C) → I □ a ≡ a |
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33 mρ : (a : Obj C) → a □ I ≡ a |
698 | 34 |
35 record StrictMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) | |
36 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where | |
37 field | |
38 m-i : Obj C | |
39 m-bi : Functor ( C × C ) C | |
40 isMonoidal : IsStrictMonoidal C m-i m-bi | |
41 | |
696
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42 |
698 | 43 -- non strict version includes 5 naturalities |
44 record IsMonoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (I : Obj C) ( BI : Functor ( C × C ) C ) | |
45 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where | |
46 open Iso | |
700
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47 infixr 9 _□_ _■_ |
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48 _□_ : ( x y : Obj C ) → Obj C |
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49 _□_ x y = FObj BI ( x , y ) |
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50 _■_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a □ b ) ( c □ d ) |
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51 _■_ f g = FMap BI ( f , g ) |
698 | 52 field |
700
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53 mα-iso : {a b c : Obj C} → Iso C ( ( a □ b) □ c) ( a □ ( b □ c ) ) |
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54 mλ-iso : {a : Obj C} → Iso C ( I □ a) a |
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55 mρ-iso : {a : Obj C} → Iso C ( a □ I) a |
698 | 56 mα→nat1 : {a a' b c : Obj C} → ( f : Hom C a a' ) → |
700
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57 C [ C [ ( f ■ id1 C ( b □ c )) o ≅→ (mα-iso {a} {b} {c}) ] ≈ |
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58 C [ ≅→ (mα-iso ) o ( (f ■ id1 C b ) ■ id1 C c ) ] ] |
698 | 59 mα→nat2 : {a b b' c : Obj C} → ( f : Hom C b b' ) → |
700
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60 C [ C [ ( id1 C a ■ ( f ■ id1 C c ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈ |
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61 C [ ≅→ (mα-iso ) o ( (id1 C a ■ f ) ■ id1 C c ) ] ] |
698 | 62 mα→nat3 : {a b c c' : Obj C} → ( f : Hom C c c' ) → |
700
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parents:
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63 C [ C [ ( id1 C a ■ ( id1 C b ■ f ) ) o ≅→ (mα-iso {a} {b} {c} ) ] ≈ |
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64 C [ ≅→ (mα-iso ) o ( id1 C ( a □ b ) ■ f ) ] ] |
698 | 65 mλ→nat : {a a' : Obj C} → ( f : Hom C a a' ) → |
66 C [ C [ f o ≅→ (mλ-iso {a} ) ] ≈ | |
700
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67 C [ ≅→ (mλ-iso ) o ( id1 C I ■ f ) ] ] |
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68 mρ→nat : {a a' : Obj C} → ( f : Hom C a a' ) → |
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69 C [ C [ f o ≅→ (mρ-iso {a} ) ] ≈ |
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70 C [ ≅→ (mρ-iso ) o ( f ■ id1 C I ) ] ] |
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71 αABC□1D : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ (b □ c)) □ d) |
701
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Monoidal Functor on going ...
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700
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72 αABC□1D {a} {b} {c} {d} {e} = ( ≅→ mα-iso ■ id1 C d ) |
700
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73 αAB□CD : {a b c d e : Obj C } → Hom C ((a □ (b □ c)) □ d) (a □ ((b □ c ) □ d)) |
701
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Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
700
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74 αAB□CD {a} {b} {c} {d} {e} = ≅→ mα-iso |
700
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75 1A□BCD : {a b c d e : Obj C } → Hom C (a □ ((b □ c ) □ d)) (a □ (b □ ( c □ d) )) |
701
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Monoidal Functor on going ...
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700
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76 1A□BCD {a} {b} {c} {d} {e} = ( id1 C a ■ ≅→ mα-iso ) |
700
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77 αABC□D : {a b c d e : Obj C } → Hom C (a □ (b □ ( c □ d) )) ((a □ b ) □ (c □ d)) |
701
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Monoidal Functor on going ...
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700
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78 αABC□D {a} {b} {c} {d} {e} = ≅← mα-iso |
700
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79 αA□BCD : {a b c d e : Obj C } → Hom C (((a □ b) □ c ) □ d) ((a □ b ) □ (c □ d)) |
701
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Monoidal Functor on going ...
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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700
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80 αA□BCD {a} {b} {c} {d} {e} = ≅→ mα-iso |
700
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81 αAIB : {a b : Obj C } → Hom C (( a □ I ) □ b ) (a □ ( I □ b )) |
701
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700
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82 αAIB {a} {b} = ≅→ mα-iso |
700
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83 1A□λB : {a b : Obj C } → Hom C (a □ ( I □ b )) ( a □ b ) |
701
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84 1A□λB {a} {b} = id1 C a ■ ≅→ mλ-iso |
700
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85 ρA□IB : {a b : Obj C } → Hom C (( a □ I ) □ b ) ( a □ b ) |
701
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86 ρA□IB {a} {b} = ≅→ mρ-iso ■ id1 C b |
700
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87 |
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88 field |
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89 comm-penta : {a b c d e : Obj C} |
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90 → C [ C [ αABC□D {a} {b} {c} {d} {e} o C [ 1A□BCD {a} {b} {c} {d} {e} o C [ αAB□CD {a} {b} {c} {d} {e} o αABC□1D {a} {b} {c} {d} {e} ] ] ] |
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91 ≈ αA□BCD {a} {b} {c} {d} {e} ] |
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92 comm-unit : {a b : Obj C} |
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93 → C [ C [ 1A□λB {a} {b} o αAIB ] ≈ ρA□IB {a} {b} ] |
696
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94 |
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95 record Monoidal {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) |
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96 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
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97 field |
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98 m-i : Obj C |
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99 m-bi : Functor ( C × C ) C |
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100 isMonoidal : IsMonoidal C m-i m-bi |
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101 |
703 | 102 Functor● : {c₁ c₂ ℓ : Level} (C D : Category c₁ c₂ ℓ) ( N : Monoidal D ) |
103 ( MF : Functor C D ) → Functor ( C × C ) D | |
104 Functor● C D N MF = record { | |
105 FObj = λ x → (FObj MF (proj₁ x) ) ● (FObj MF (proj₂ x) ) | |
106 ; FMap = λ {x : Obj ( C × C ) } {y} f → FMap (Monoidal.m-bi N) ( FMap MF (proj₁ f ) , FMap MF (proj₂ f) ) | |
107 ; isFunctor = record { | |
108 ≈-cong = ≈-cong | |
109 ; identity = identity | |
110 ; distr = distr | |
111 } | |
112 } where | |
113 _●_ : (x y : Obj D ) → Obj D | |
114 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y | |
704 | 115 _■_ : {a b c d : Obj D } ( f : Hom D a c ) ( g : Hom D b d ) → Hom D ( a ● b ) ( c ● d ) |
116 _■_ f g = FMap (Monoidal.m-bi N) ( f , g ) | |
117 F : { a b : Obj C } → ( f : Hom C a b ) → Hom D (FObj MF a) (FObj MF b ) | |
118 F f = FMap MF f | |
703 | 119 ≈-cong : {a b : Obj (C × C)} {f g : Hom (C × C) a b} → (C × C) [ f ≈ g ] → |
704 | 120 D [ (F (proj₁ f) ■ F (proj₂ f)) ≈ (F (proj₁ g) ■ F (proj₂ g)) ] |
703 | 121 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning D in begin |
704 | 122 F (proj₁ f) ■ F (proj₂ f) |
703 | 123 ≈⟨ fcong (Monoidal.m-bi N) ( fcong MF ( proj₁ f≈g ) , fcong MF ( proj₂ f≈g )) ⟩ |
704 | 124 F (proj₁ g) ■ F (proj₂ g) |
703 | 125 ∎ |
704 | 126 identity : {a : Obj (C × C)} → D [ (F (proj₁ (id1 (C × C) a)) ■ F (proj₂ (id1 (C × C) a))) |
703 | 127 ≈ id1 D (FObj MF (proj₁ a) ● FObj MF (proj₂ a)) ] |
128 identity {a} = let open ≈-Reasoning D in begin | |
704 | 129 F (proj₁ (id1 (C × C) a)) ■ F (proj₂ (id1 (C × C) a)) |
703 | 130 ≈⟨ fcong (Monoidal.m-bi N) ( IsFunctor.identity (isFunctor MF ) , IsFunctor.identity (isFunctor MF )) ⟩ |
704 | 131 id1 D (FObj MF (proj₁ a)) ■ id1 D (FObj MF (proj₂ a)) |
703 | 132 ≈⟨ IsFunctor.identity (isFunctor (Monoidal.m-bi N)) ⟩ |
133 id1 D (FObj MF (proj₁ a) ● FObj MF (proj₂ a)) | |
134 ∎ | |
135 distr : {a b c : Obj (C × C)} {f : Hom (C × C) a b} {g : Hom (C × C) b c} → | |
704 | 136 D [ (F (proj₁ ((C × C) [ g o f ])) ■ F (proj₂ ((C × C) [ g o f ]))) |
137 ≈ D [ (F (proj₁ g) ■ F (proj₂ g)) o (F (proj₁ f) ■ F (proj₂ f)) ] ] | |
703 | 138 distr {a} {b} {c} {f} {g} = let open ≈-Reasoning D in begin |
704 | 139 (F (proj₁ ((C × C) [ g o f ])) ■ F (proj₂ ((C × C) [ g o f ]))) |
703 | 140 ≈⟨ fcong (Monoidal.m-bi N) ( IsFunctor.distr ( isFunctor MF) , IsFunctor.distr ( isFunctor MF )) ⟩ |
704 | 141 ( F (proj₁ g) o F (proj₁ f) ) ■ ( F (proj₂ g) o F (proj₂ f) ) |
703 | 142 ≈⟨ IsFunctor.distr ( isFunctor (Monoidal.m-bi N)) ⟩ |
704 | 143 (F (proj₁ g) ■ F (proj₂ g)) o (F (proj₁ f) ■ F (proj₂ f)) |
703 | 144 ∎ |
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145 |
703 | 146 Functor⊗ : {c₁ c₂ ℓ : Level} (C D : Category c₁ c₂ ℓ) ( M : Monoidal C ) |
147 ( MF : Functor C D ) → Functor ( C × C ) D | |
148 Functor⊗ C D M MF = record { | |
149 FObj = λ x → FObj MF ( proj₁ x ⊗ proj₂ x ) | |
704 | 150 ; FMap = λ {a} {b} f → F ( FMap (Monoidal.m-bi M) ( proj₁ f , proj₂ f) ) |
703 | 151 ; isFunctor = record { |
152 ≈-cong = ≈-cong | |
153 ; identity = identity | |
154 ; distr = distr | |
155 } | |
156 } where | |
157 _⊗_ : (x y : Obj C ) → Obj C | |
158 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y | |
704 | 159 _□_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a ⊗ b ) ( c ⊗ d ) |
160 _□_ f g = FMap (Monoidal.m-bi M) ( f , g ) | |
161 F : { a b : Obj C } → ( f : Hom C a b ) → Hom D (FObj MF a) (FObj MF b ) | |
162 F f = FMap MF f | |
703 | 163 ≈-cong : {a b : Obj (C × C)} {f g : Hom (C × C) a b} → (C × C) [ f ≈ g ] → |
704 | 164 D [ F ( (proj₁ f □ proj₂ f)) ≈ F ( (proj₁ g □ proj₂ g)) ] |
703 | 165 ≈-cong {a} {b} {f} {g} f≈g = IsFunctor.≈-cong (isFunctor MF ) ( IsFunctor.≈-cong (isFunctor (Monoidal.m-bi M) ) f≈g ) |
704 | 166 identity : {a : Obj (C × C)} → D [ F ( (proj₁ (id1 (C × C) a) □ proj₂ (id1 (C × C) a))) |
703 | 167 ≈ id1 D (FObj MF (proj₁ a ⊗ proj₂ a)) ] |
168 identity {a} = let open ≈-Reasoning D in begin | |
704 | 169 F ( (proj₁ (id1 (C × C) a) □ proj₂ (id1 (C × C) a))) |
703 | 170 ≈⟨⟩ |
704 | 171 F (FMap (Monoidal.m-bi M) (id1 (C × C) a ) ) |
703 | 172 ≈⟨ fcong MF ( IsFunctor.identity (isFunctor (Monoidal.m-bi M) )) ⟩ |
704 | 173 F (id1 C (proj₁ a ⊗ proj₂ a)) |
703 | 174 ≈⟨ IsFunctor.identity (isFunctor MF) ⟩ |
175 id1 D (FObj MF (proj₁ a ⊗ proj₂ a)) | |
176 ∎ | |
177 distr : {a b c : Obj (C × C)} {f : Hom (C × C) a b} {g : Hom (C × C) b c} → D [ | |
704 | 178 F ( (proj₁ ((C × C) [ g o f ]) □ proj₂ ((C × C) [ g o f ]))) |
179 ≈ D [ F ( (proj₁ g □ proj₂ g)) o F ( (proj₁ f □ proj₂ f)) ] ] | |
703 | 180 distr {a} {b} {c} {f} {g} = let open ≈-Reasoning D in begin |
704 | 181 F ( (proj₁ ((C × C) [ g o f ]) □ proj₂ ((C × C) [ g o f ]))) |
703 | 182 ≈⟨⟩ |
704 | 183 F (FMap (Monoidal.m-bi M) ( (C × C) [ g o f ] )) |
703 | 184 ≈⟨ fcong MF ( IsFunctor.distr (isFunctor (Monoidal.m-bi M))) ⟩ |
704 | 185 F (C [ FMap (Monoidal.m-bi M) g o FMap (Monoidal.m-bi M) f ]) |
703 | 186 ≈⟨ IsFunctor.distr ( isFunctor MF ) ⟩ |
704 | 187 F ( proj₁ g □ proj₂ g) o F ( proj₁ f □ proj₂ f) |
703 | 188 ∎ |
189 | |
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190 |
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191 record IsMonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D ) |
698 | 192 ( MF : Functor C D ) |
702
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193 ( ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) ) |
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194 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
698 | 195 _⊗_ : (x y : Obj C ) → Obj C |
700
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196 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y |
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197 _□_ : {a b c d : Obj C } ( f : Hom C a c ) ( g : Hom C b d ) → Hom C ( a ⊗ b ) ( c ⊗ d ) |
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198 _□_ f g = FMap (Monoidal.m-bi M) ( f , g ) |
698 | 199 _●_ : (x y : Obj D ) → Obj D |
700
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200 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y |
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201 _■_ : {a b c d : Obj D } ( f : Hom D a c ) ( g : Hom D b d ) → Hom D ( a ● b ) ( c ● d ) |
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202 _■_ f g = FMap (Monoidal.m-bi N) ( f , g ) |
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203 F● : Functor ( C × C ) D |
703 | 204 F● = Functor● C D N MF |
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205 F⊗ : Functor ( C × C ) D |
703 | 206 F⊗ = Functor⊗ C D M MF |
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207 field |
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208 φab : NTrans ( C × C ) D F● F⊗ |
698 | 209 open Iso |
210 open Monoidal | |
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211 open IsMonoidal hiding ( _■_ ; _□_ ) |
699 | 212 αC : {a b c : Obj C} → Hom C (( a ⊗ b ) ⊗ c ) ( a ⊗ ( b ⊗ c ) ) |
213 αC {a} {b} {c} = ≅→ (mα-iso (isMonoidal M) {a} {b} {c}) | |
214 αD : {a b c : Obj D} → Hom D (( a ● b ) ● c ) ( a ● ( b ● c ) ) | |
215 αD {a} {b} {c} = ≅→ (mα-iso (isMonoidal N) {a} {b} {c}) | |
701
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216 F : Obj C → Obj D |
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217 F x = FObj MF x |
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218 φ : ( x y : Obj C ) → Hom D ( FObj F● (x , y) ) ( FObj F⊗ ( x , y )) |
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219 φ x y = NTrans.TMap φab ( x , y ) |
701
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220 1●φBC : {a b c : Obj C} → Hom D ( F a ● ( F b ● F c ) ) ( F a ● ( F ( b ⊗ c ) )) |
702
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221 1●φBC {a} {b} {c} = id1 D (F a) ■ φ b c |
701
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222 φAB⊗C : {a b c : Obj C} → Hom D ( F a ● ( F ( b ⊗ c ) )) (F ( a ⊗ ( b ⊗ c ))) |
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223 φAB⊗C {a} {b} {c} = φ a (b ⊗ c ) |
701
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224 φAB●1 : {a b c : Obj C} → Hom D ( ( F a ● F b ) ● F c ) ( F ( a ⊗ b ) ● F c ) |
702
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225 φAB●1 {a} {b} {c} = φ a b ■ id1 D (F c) |
701
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226 φA⊗BC : {a b c : Obj C} → Hom D ( F ( a ⊗ b ) ● F c ) (F ( (a ⊗ b ) ⊗ c )) |
702
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227 φA⊗BC {a} {b} {c} = φ ( a ⊗ b ) c |
701
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228 FαC : {a b c : Obj C} → Hom D (F ( (a ⊗ b ) ⊗ c )) (F ( a ⊗ ( b ⊗ c ))) |
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229 FαC {a} {b} {c} = FMap MF ( ≅→ (mα-iso (isMonoidal M) {a} {b} {c}) ) |
702
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230 1●ψ : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ● F ( Monoidal.m-i M ) ) |
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231 1●ψ{a} {b} = id1 D (F a) ■ ψ |
701
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232 φAIC : { a b : Obj C } → Hom D ( F a ● F ( Monoidal.m-i M ) ) (F ( a ⊗ Monoidal.m-i M )) |
702
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233 φAIC {a} {b} = φ a ( Monoidal.m-i M ) |
701
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234 FρC : { a b : Obj C } → Hom D (F ( a ⊗ Monoidal.m-i M ))( F a ) |
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235 FρC {a} {b} = FMap MF ( ≅→ (mρ-iso (isMonoidal M) {a} ) ) |
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236 ρD : { a b : Obj C } → Hom D (F a ● Monoidal.m-i N ) ( F a ) |
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237 ρD {a} {b} = ≅→ (mρ-iso (isMonoidal N) {F a} ) |
702
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238 ψ●1 : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) ( F ( Monoidal.m-i M ) ● F b ) |
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239 ψ●1 {a} {b} = ψ ■ id1 D (F b) |
701
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240 φICB : { a b : Obj C } → Hom D ( F ( Monoidal.m-i M ) ● F b ) ( F ( ( Monoidal.m-i M ) ⊗ b ) ) |
702
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241 φICB {a} {b} = φ ( Monoidal.m-i M ) b |
701
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242 FλD : { a b : Obj C } → Hom D ( F ( ( Monoidal.m-i M ) ⊗ b ) ) (F b ) |
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243 FλD {a} {b} = FMap MF ( ≅→ (mλ-iso (isMonoidal M) {b} ) ) |
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244 λD : { a b : Obj C } → Hom D (Monoidal.m-i N ● F b ) (F b ) |
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245 λD {a} {b} = ≅→ (mλ-iso (isMonoidal N) {F b} ) |
696
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246 field |
700
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247 comm1 : {a b c : Obj C } → D [ D [ φAB⊗C {a} {b} {c} o D [ 1●φBC o αD ] ] ≈ D [ FαC o D [ φA⊗BC o φAB●1 ] ] ] |
702
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248 comm-idr : {a b : Obj C } → D [ D [ FρC {a} {b} o D [ φAIC {a} {b} o 1●ψ{a} {b} ] ] ≈ ρD {a} {b} ] |
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249 comm-idl : {a b : Obj C } → D [ D [ FλD {a} {b} o D [ φICB {a} {b} o ψ●1 {a} {b} ] ] ≈ λD {a} {b} ] |
696
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250 |
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251 record MonoidalFunctor {c₁ c₂ ℓ : Level} {C D : Category c₁ c₂ ℓ} ( M : Monoidal C ) ( N : Monoidal D ) |
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252 : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where |
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253 _⊗_ : (x y : Obj C ) → Obj C |
700
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254 _⊗_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal M) ) x y |
696
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255 _●_ : (x y : Obj D ) → Obj D |
700
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256 _●_ x y = (IsMonoidal._□_ (Monoidal.isMonoidal N) ) x y |
696
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257 field |
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258 MF : Functor C D |
702
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259 ψ : Hom D (Monoidal.m-i N) (FObj MF (Monoidal.m-i M) ) |
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260 isMonodailFunctor : IsMonoidalFunctor M N MF ψ |