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safe fix done
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Sun, 07 Jul 2024 22:28:50 +0900
parents 5620d4a85069
children
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1 {-# OPTIONS --cubical-compatible --safe #-}
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2 module Category.Ring where
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3 open import Category
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4 open import Algebra
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5 open import Algebra.Structures
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6 open import Algebra.Morphism
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7 import Relation.Binary.Reasoning.Setoid as EqR
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8 open import Relation.Binary.Core
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9 open import Relation.Binary
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10 open import Level
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11 open import Data.Product
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12
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13 RingObj : ∀{c ℓ} → Set (suc (c ⊔ ℓ))
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14 RingObj {c} {ℓ} = Ring c ℓ
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15
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16
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17 -- open import Algebra.Morphism.Definitions
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18
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19 record _-Ring⟶_ {c ℓ c' ℓ' : Level } (From : RingObj {c} {ℓ}) (To : RingObj {c'} {ℓ'}) : Set ((c ⊔ ℓ ⊔ c' ⊔ ℓ')) where
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20 module F = Ring From
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21 module T = Ring To
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22 open Definitions (F.Carrier) (T.Carrier) T._≈_
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23 field
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24 ⟦_⟧ : Ring.Carrier From → Ring.Carrier To
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25 ⟦⟧-cong : ⟦_⟧ Preserves F._≈_ ⟶ T._≈_
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26 +-homo : Homomorphic₂ ⟦_⟧ F._+_ T._+_
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27 *-homo : Homomorphic₂ ⟦_⟧ F._*_ T._*_
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28 1-homo : Homomorphic₀ ⟦_⟧ F.1# T.1#
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29
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30 infixr 5 _-Ring⟶_
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31
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32 RingHom : ∀{c ℓ} → Set (suc (c ⊔ ℓ))
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33 RingHom {c} {ℓ} = {R₁ R₂ : Ring c ℓ} → R₁ -Ring⟶ R₂
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34
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35 RingId : ∀{c ℓ} {R : Ring c ℓ} → R -Ring⟶ R
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36 RingId {R = R} = record { ⟦_⟧ = ⟦_⟧ ; ⟦⟧-cong = ⟦⟧-cong ; +-homo = +-homo
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37 ; *-homo = *-homo ; 1-homo = 1-homo
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38 }
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39 where
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40 open Algebra.Ring R
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41 open Definitions Carrier Carrier _≈_
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42 ⟦_⟧ : Carrier → Carrier
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43 ⟦_⟧ = λ x → x
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44 ⟦⟧-cong : ⟦_⟧ Preserves _≈_ ⟶ _≈_
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45 ⟦⟧-cong = λ x → x
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46 +-homo : Homomorphic₂ ⟦_⟧ _+_ _+_
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47 +-homo x y = IsEquivalence.refl isEquivalence
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48 *-homo : Homomorphic₂ ⟦_⟧ _*_ _*_
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49 *-homo x y = IsEquivalence.refl isEquivalence
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50 1-homo : Homomorphic₀ ⟦_⟧ 1# 1#
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51 1-homo = IsEquivalence.refl isEquivalence
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52
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53 [_]_ : ∀{c ℓ c′ ℓ′} {R₁ : Ring c ℓ} {R₂ : Ring c′ ℓ′} → R₁ -Ring⟶ R₂ → Ring.Carrier R₁ → Ring.Carrier R₂
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54 [ f ] x = _-Ring⟶_.⟦_⟧ f x
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55
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56 _∘_ : ∀{c₁ ℓ₁ c₂ ℓ₂ c₃ ℓ₃} {R₁ : Ring c₁ ℓ₁} {R₂ : Ring c₂ ℓ₂} {R₃ : Ring c₃ ℓ₃}
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57 → R₂ -Ring⟶ R₃ → R₁ -Ring⟶ R₂ → R₁ -Ring⟶ R₃
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58 _∘_ {R₁ = R₁} {R₂} {R₃} g f =
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59 record { ⟦_⟧ = ⟦_⟧ ; ⟦⟧-cong = ⟦⟧-cong ; +-homo = +-homo
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60 ; *-homo = *-homo ; 1-homo = 1-homo
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61 }
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62 where
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63 module F = Ring R₁
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64 module M = Ring R₂
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65 module T = Ring R₃
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66 module homo = _-Ring⟶_
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67 open Definitions F.Carrier T.Carrier T._≈_
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68 ⟦_⟧ : Morphism
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69 ⟦ x ⟧ = [ g ] ([ f ] x)
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70 ⟦⟧-cong : ⟦_⟧ Preserves F._≈_ ⟶ T._≈_
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71 ⟦⟧-cong x = homo.⟦⟧-cong g (homo.⟦⟧-cong f x)
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72 +-homo : Homomorphic₂ ⟦_⟧ F._+_ T._+_
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73 +-homo x y =
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74 begin
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75 ⟦ F._+_ x y ⟧ ≈⟨ homo.⟦⟧-cong g (homo.+-homo f x y) ⟩
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76 [ g ] ( M._+_ ([ f ] x) ([ f ] y) ) ≈⟨ homo.+-homo g ([ f ] x) ([ f ] y) ⟩
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77 T._+_ ⟦ x ⟧ ⟦ y ⟧
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78 ∎
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79 where
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80 open IsEquivalence T.isEquivalence
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81 open EqR T.setoid
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82 *-homo : Homomorphic₂ ⟦_⟧ F._*_ T._*_
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83 *-homo x y =
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84 begin
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85 ⟦ F._*_ x y ⟧ ≈⟨ homo.⟦⟧-cong g (homo.*-homo f x y) ⟩
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86 [ g ] ( M._*_ ([ f ] x) ([ f ] y) ) ≈⟨ homo.*-homo g ([ f ] x) ([ f ] y) ⟩
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87 T._*_ ⟦ x ⟧ ⟦ y ⟧
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88 ∎
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89 where
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90 open IsEquivalence T.isEquivalence
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91 open EqR T.setoid
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92 1-homo : Homomorphic₀ ⟦_⟧ F.1# T.1#
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93 1-homo =
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94 begin
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95 ⟦ F.1# ⟧ ≈⟨ homo.⟦⟧-cong g (homo.1-homo f) ⟩
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96 [ g ] M.1# ≈⟨ homo.1-homo g ⟩
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97 T.1#
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98 ∎
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99 where
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100 open IsEquivalence T.isEquivalence
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101 open EqR T.setoid
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102
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103 _≈_ : ∀{c ℓ c′ ℓ′} {R₁ : Ring c ℓ} {R₂ : Ring c′ ℓ′} → Rel (R₁ -Ring⟶ R₂) (ℓ′ ⊔ c)
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104 _≈_ {R₁ = R₁} {R₂} φ ψ = ∀(x : F.Carrier) → T._≈_ ([ φ ] x) ([ ψ ] x)
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105 where
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106 module F = Ring R₁
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107 module T = Ring R₂
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108
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109 ≈-refl : ∀{c ℓ c′ ℓ′} {R₁ : Ring c ℓ} {R₂ : Ring c′ ℓ′} {F : R₁ -Ring⟶ R₂} → F ≈ F
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110 ≈-refl {R₁ = R₁} {F = F} _ = _-Ring⟶_.⟦⟧-cong F (IsEquivalence.refl (Ring.isEquivalence R₁))
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111
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112 ≈-sym : ∀{c ℓ c′ ℓ′} {R₁ : Ring c ℓ} {R₂ : Ring c′ ℓ′} {F G : R₁ -Ring⟶ R₂} → F ≈ G → G ≈ F
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113 ≈-sym {R₁ = R₁} {R₂} {F} {G} F≈G x = IsEquivalence.sym (Ring.isEquivalence R₂)(F≈G x)
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114
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115 ≈-trans : ∀{c ℓ c′ ℓ′} {R₁ : Ring c ℓ} {R₂ : Ring c′ ℓ′} {F G H : R₁ -Ring⟶ R₂}
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116 → F ≈ G → G ≈ H → F ≈ H
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117 ≈-trans {R₁ = R₁} {R₂} {F} {G} F≈G G≈H x = IsEquivalence.trans (Ring.isEquivalence R₂) (F≈G x) (G≈H x)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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118
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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119 ≈-isEquivalence : ∀{c ℓ c′ ℓ′} {R₁ : Ring c ℓ} {R₂ : Ring c′ ℓ′}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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120 → IsEquivalence {A = R₁ -Ring⟶ R₂} _≈_
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121 ≈-isEquivalence {R₁ = R₁} {R₂} = record { refl = λ {F} → ≈-refl {R₁ = R₁} {R₂} {F}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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122 ; sym = λ{F} {G} → ≈-sym {R₁ = R₁} {R₂} {F} {G}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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123 ; trans = λ{F} {G} {H} → ≈-trans {R₁ = R₁} {R₂} {F} {G} {H}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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124 }
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125
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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126
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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127 RingCat : ∀{c ℓ} → Category _ _ _
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128 RingCat {c} {ℓ} =
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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129 record { Obj = Ring c ℓ
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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130 ; Hom = _-Ring⟶_
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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131 ; Id = RingId
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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132 ; _o_ = _∘_
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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133 ; _≈_ = _≈_
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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134 ; isCategory = isCategory
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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135 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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136 where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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137 isCategory : IsCategory (Ring c ℓ) _-Ring⟶_ _≈_ _∘_ RingId
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
138 isCategory =
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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139 record { isEquivalence = ≈-isEquivalence {c} {ℓ} {c} {ℓ}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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140 ; identityL = λ {R₁ R₂ f} → identityL {R₁} {R₂} {f}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
141 ; identityR = λ {R₁ R₂ f} → identityR {R₁} {R₂} {f}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
142 ; o-resp-≈ = λ {A} {B} {C} {f} {g} {h} {i} → o-resp-≈ {A} {B} {C} {f} {g} {h} {i}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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143 ; associative = λ {R₁} {R₂} {R₃} {R₄} {f} {g} {h} → associative {R₁} {R₂} {R₃} {R₄} {f} {g} {h}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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144 }
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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145 where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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146 identityL : {R₁ R₂ : Ring c ℓ} {f : R₁ -Ring⟶ R₂} → (RingId ∘ f) ≈ f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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147 identityL {f = f} = ≈-refl {F = f}
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parents:
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148 identityR : {R₁ R₂ : Ring c ℓ} {f : R₁ -Ring⟶ R₂} → (f ∘ RingId) ≈ f
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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149 identityR {f = f} = ≈-refl {F = f}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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150 o-resp-≈ : {R₁ R₂ R₃ : Ring c ℓ} {f g : R₁ -Ring⟶ R₂} {h i : R₂ -Ring⟶ R₃}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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151 → f ≈ g → h ≈ i → (h ∘ f) ≈ (i ∘ g)
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parents:
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152 o-resp-≈ {R₃ = R₃} {f} {g} {h} {i} f≈g h≈i x =
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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153 begin
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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154 ([ h ∘ f ] x) ≈⟨ h≈i ([ f ] x) ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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155 ([ i ∘ f ] x) ≈⟨ _-Ring⟶_.⟦⟧-cong i (f≈g x) ⟩
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
156 ([ i ∘ g ] x)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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157 ∎
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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158 where
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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159 module T = Ring R₃
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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160 open IsEquivalence T.isEquivalence
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
161 open EqR T.setoid
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
162 associative : {R₁ R₂ R₃ R₄ : Ring c ℓ} {f : R₃ -Ring⟶ R₄} {g : R₂ -Ring⟶ R₃} {h : R₁ -Ring⟶ R₂}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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163 → (f ∘ (g ∘ h)) ≈ ((f ∘ g) ∘ h)
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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164 associative {f = f} {g} {h} = ≈-refl {F = f ∘ (g ∘ h)}
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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165
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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166