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annotate src/HomReasoning.agda @ 953:eb62812b5885
Topos written
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 19 Feb 2021 12:09:48 +0900 |
| parents | ac53803b3b2a |
| children | 270f0ba65b88 |
| rev | line source |
|---|---|
| 56 | 1 module HomReasoning where |
| 31 | 2 |
| 3 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 4 | |
| 5 open import Category -- https://github.com/konn/category-agda | |
| 6 open import Level | |
| 7 open Functor | |
| 8 | |
| 9 -- F(f) | |
| 10 -- F(a) ---→ F(b) | |
| 11 -- | | | |
| 12 -- |t(a) |t(b) G(f)t(a) = t(b)F(f) | |
| 13 -- | | | |
| 14 -- v v | |
| 15 -- G(a) ---→ G(b) | |
| 16 -- G(f) | |
| 17 | |
| 18 record IsNTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (D : Category c₁ c₂ ℓ) (C : Category c₁′ c₂′ ℓ′) | |
| 19 ( F G : Functor D C ) | |
| 20 (TMap : (A : Obj D) → Hom C (FObj F A) (FObj G A)) | |
| 21 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where | |
| 22 field | |
| 130 | 23 commute : {a b : Obj D} {f : Hom D a b} |
| 31 | 24 → C [ C [ ( FMap G f ) o ( TMap a ) ] ≈ C [ (TMap b ) o (FMap F f) ] ] |
| 25 | |
| 130 | 26 record NTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (domain : Category c₁ c₂ ℓ) (codomain : Category c₁′ c₂′ ℓ′) |
| 27 (F G : Functor domain codomain ) | |
| 31 | 28 : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where |
| 29 field | |
| 30 TMap : (A : Obj domain) → Hom codomain (FObj F A) (FObj G A) | |
| 31 isNTrans : IsNTrans domain codomain F G TMap | |
| 32 | |
| 33 | |
| 34 module ≈-Reasoning {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) where | |
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35 open import Relation.Binary |
| 31 | 36 |
| 37 _o_ : {a b c : Obj A } ( x : Hom A a b ) ( y : Hom A c a ) → Hom A c b | |
| 38 x o y = A [ x o y ] | |
| 39 | |
| 40 _≈_ : {a b : Obj A } → Rel (Hom A a b) ℓ | |
| 41 x ≈ y = A [ x ≈ y ] | |
| 42 | |
| 43 infixr 9 _o_ | |
| 44 infix 4 _≈_ | |
| 45 | |
| 46 refl-hom : {a b : Obj A } { x : Hom A a b } → x ≈ x | |
| 47 refl-hom = IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory A )) | |
| 48 | |
| 49 trans-hom : {a b : Obj A } { x y z : Hom A a b } → | |
| 50 x ≈ y → y ≈ z → x ≈ z | |
| 51 trans-hom b c = ( IsEquivalence.trans (IsCategory.isEquivalence ( Category.isCategory A ))) b c | |
| 52 | |
| 53 -- some short cuts | |
| 54 | |
| 55 car : {a b c : Obj A } {x y : Hom A a b } { f : Hom A c a } → | |
| 56 x ≈ y → ( x o f ) ≈ ( y o f ) | |
| 787 | 57 car eq = ( IsCategory.o-resp-≈ ( Category.isCategory A )) ( refl-hom ) eq |
| 58 | |
| 59 car1 : { c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b c : Obj A } {x y : Hom A a b } { f : Hom A c a } → | |
| 60 A [ x ≈ y ] → A [ A [ x o f ] ≈ A [ y o f ] ] | |
| 61 car1 A eq = ( IsCategory.o-resp-≈ ( Category.isCategory A )) ( IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory A )) ) eq | |
| 31 | 62 |
| 63 cdr : {a b c : Obj A } {x y : Hom A a b } { f : Hom A b c } → | |
| 64 x ≈ y → f o x ≈ f o y | |
| 787 | 65 cdr eq = ( IsCategory.o-resp-≈ ( Category.isCategory A )) eq (refl-hom ) |
| 66 | |
| 67 cdr1 : { c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b c : Obj A } {x y : Hom A a b } { f : Hom A b c } → | |
| 68 A [ x ≈ y ] → A [ A [ f o x ] ≈ A [ f o y ] ] | |
| 69 cdr1 A eq = ( IsCategory.o-resp-≈ ( Category.isCategory A )) eq (IsEquivalence.refl (IsCategory.isEquivalence ( Category.isCategory A )) ) | |
| 31 | 70 |
| 71 id : (a : Obj A ) → Hom A a a | |
| 72 id a = (Id {_} {_} {_} {A} a) | |
| 73 | |
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74 idL : {a b : Obj A } { f : Hom A a b } → id b o f ≈ f |
| 31 | 75 idL = IsCategory.identityL (Category.isCategory A) |
| 76 | |
| 77 idR : {a b : Obj A } { f : Hom A a b } → f o id a ≈ f | |
| 78 idR = IsCategory.identityR (Category.isCategory A) | |
| 79 | |
| 80 sym : {a b : Obj A } { f g : Hom A a b } → f ≈ g → g ≈ f | |
| 81 sym = IsEquivalence.sym (IsCategory.isEquivalence (Category.isCategory A)) | |
| 82 | |
| 455 | 83 sym-hom = sym |
| 84 | |
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85 -- working on another cateogry |
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86 idL1 : { c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A b a } → A [ A [ Id {_} {_} {_} {A} a o f ] ≈ f ] |
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87 idL1 A = IsCategory.identityL (Category.isCategory A) |
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88 |
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89 idR1 : { c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b : Obj A } { f : Hom A a b } → A [ A [ f o Id {_} {_} {_} {A} a ] ≈ f ] |
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90 idR1 A = IsCategory.identityR (Category.isCategory A) |
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91 |
| 781 | 92 open import Relation.Binary.PropositionalEquality using ( _≡_ ) |
| 693 | 93 ≈←≡ : {a b : Obj A } { x y : Hom A a b } → (x≈y : x ≡ y ) → x ≈ y |
| 781 | 94 ≈←≡ _≡_.refl = refl-hom |
| 75 | 95 |
| 693 | 96 -- Ho← to prove this? |
| 97 -- ≡←≈ : {a b : Obj A } { x y : Hom A a b } → (x≈y : x ≈ y ) → x ≡ y | |
| 98 -- ≡←≈ x≈y = irr x≈y | |
| 99 | |
| 75 | 100 ≡-cong : { c₁′ c₂′ ℓ′ : Level} {B : Category c₁′ c₂′ ℓ′} {x y : Obj B } { a b : Hom B x y } {x' y' : Obj A } → |
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101 (f : Hom B x y → Hom A x' y' ) → a ≡ b → f a ≈ f b |
| 781 | 102 ≡-cong f _≡_.refl = ≈←≡ _≡_.refl |
| 75 | 103 |
| 67 | 104 -- cong-≈ : { c₁′ c₂′ ℓ′ : Level} {B : Category c₁′ c₂′ ℓ′} {x y : Obj B } { a b : Hom B x y } {x' y' : Obj A } → |
| 105 -- B [ a ≈ b ] → (f : Hom B x y → Hom A x' y' ) → f a ≈ f b | |
| 75 | 106 -- cong-≈ eq f = {!!} |
| 67 | 107 |
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108 assoc : {a b c d : Obj A } {f : Hom A c d} {g : Hom A b c} {h : Hom A a b} |
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109 → f o ( g o h ) ≈ ( f o g ) o h |
| 31 | 110 assoc = IsCategory.associative (Category.isCategory A) |
| 111 | |
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112 -- working on another cateogry |
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113 assoc1 : { c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) {a b c d : Obj A } {f : Hom A c d} {g : Hom A b c} {h : Hom A a b} |
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114 → A [ A [ f o ( A [ g o h ] ) ] ≈ A [ ( A [ f o g ] ) o h ] ] |
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115 assoc1 A = IsCategory.associative (Category.isCategory A) |
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116 |
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117 distr : { c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} |
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118 { c₁′ c₂′ ℓ′ : Level} {D : Category c₁′ c₂′ ℓ′} (T : Functor D A) → {a b c : Obj D} {g : Hom D b c} { f : Hom D a b } |
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119 → A [ FMap T ( D [ g o f ] ) ≈ A [ FMap T g o FMap T f ] ] |
| 75 | 120 distr T = IsFunctor.distr ( isFunctor T ) |
| 121 | |
| 122 resp : {a b c : Obj A} {f g : Hom A a b} {h i : Hom A b c} → f ≈ g → h ≈ i → (h o f) ≈ (i o g) | |
| 123 resp = IsCategory.o-resp-≈ (Category.isCategory A) | |
| 124 | |
| 125 fcong : { c₁ c₂ ℓ : Level} {C : Category c₁ c₂ ℓ} | |
| 126 { c₁′ c₂′ ℓ′ : Level} {D : Category c₁′ c₂′ ℓ′} {a b : Obj C} {f g : Hom C a b} → (T : Functor C D) → C [ f ≈ g ] → D [ FMap T f ≈ FMap T g ] | |
| 127 fcong T = IsFunctor.≈-cong (isFunctor T) | |
| 31 | 128 |
| 129 open NTrans | |
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130 nat : { c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} { c₁′ c₂′ ℓ′ : Level} {D : Category c₁′ c₂′ ℓ′} |
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131 {a b : Obj D} {f : Hom D a b} {F G : Functor D A } |
| 31 | 132 → (η : NTrans D A F G ) |
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133 → A [ A [ FMap G f o TMap η a ] ≈ A [ TMap η b o FMap F f ] ] |
| 130 | 134 nat η = IsNTrans.commute ( isNTrans η ) |
| 31 | 135 |
| 460 | 136 nat1 : { c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} { c₁′ c₂′ ℓ′ : Level} {D : Category c₁′ c₂′ ℓ′} |
| 137 {a b : Obj D} {F G : Functor D A } | |
| 138 → (η : NTrans D A F G ) → (f : Hom D a b) | |
| 139 → A [ A [ FMap G f o TMap η a ] ≈ A [ TMap η b o FMap F f ] ] | |
| 140 nat1 η f = IsNTrans.commute ( isNTrans η ) | |
| 141 | |
| 606 | 142 infix 3 _∎ |
| 31 | 143 infixr 2 _≈⟨_⟩_ _≈⟨⟩_ |
| 168 | 144 infixr 2 _≈↑⟨_⟩_ |
| 31 | 145 infix 1 begin_ |
| 146 | |
| 147 ------ If we have this, for example, as an axiom of a category, we can use ≡-Reasoning directly | |
| 148 -- ≈-to-≡ : {a b : Obj A } { x y : Hom A a b } → A [ x ≈ y ] → x ≡ y | |
| 149 -- ≈-to-≡ refl-hom = refl | |
| 150 | |
| 151 data _IsRelatedTo_ { a b : Obj A } ( x y : Hom A a b ) : | |
| 152 Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where | |
| 153 relTo : (x≈y : x ≈ y ) → x IsRelatedTo y | |
| 154 | |
| 155 begin_ : { a b : Obj A } { x y : Hom A a b } → | |
| 156 x IsRelatedTo y → x ≈ y | |
| 157 begin relTo x≈y = x≈y | |
| 158 | |
| 159 _≈⟨_⟩_ : { a b : Obj A } ( x : Hom A a b ) → { y z : Hom A a b } → | |
| 160 x ≈ y → y IsRelatedTo z → x IsRelatedTo z | |
| 161 _ ≈⟨ x≈y ⟩ relTo y≈z = relTo (trans-hom x≈y y≈z) | |
| 162 | |
| 168 | 163 _≈↑⟨_⟩_ : { a b : Obj A } ( x : Hom A a b ) → { y z : Hom A a b } → |
| 164 y ≈ x → y IsRelatedTo z → x IsRelatedTo z | |
| 165 _ ≈↑⟨ y≈x ⟩ relTo y≈z = relTo (trans-hom ( sym y≈x ) y≈z) | |
| 166 | |
| 31 | 167 _≈⟨⟩_ : { a b : Obj A } ( x : Hom A a b ) → { y : Hom A a b } → x IsRelatedTo y → x IsRelatedTo y |
| 168 _ ≈⟨⟩ x∼y = x∼y | |
| 169 | |
| 170 _∎ : { a b : Obj A } ( x : Hom A a b ) → x IsRelatedTo x | |
| 171 _∎ _ = relTo refl-hom | |
| 172 | |
| 948 | 173 |
| 174 --- | |
| 175 -- to avoid assoc storm, flatten composition according to the template | |
| 176 -- | |
| 177 | |
| 178 data MP : { a b : Obj A } ( x : Hom A a b ) → Set (c₁ ⊔ c₂ ⊔ ℓ ) where | |
| 179 am : { a b : Obj A } → (x : Hom A a b ) → MP x | |
| 180 _repl_by_ : { a b : Obj A } → (x y : Hom A a b ) → x ≈ y → MP y | |
| 181 _∙_ : { a b c : Obj A } {x : Hom A b c } { y : Hom A a b } → MP x → MP y → MP ( x o y ) | |
| 182 | |
| 183 open import Relation.Binary.HeterogeneousEquality as HE using (_≅_ ) | |
| 184 | |
| 185 mp-before : { a b : Obj A } { f : Hom A a b } → MP f → Hom A a b | |
| 186 mp-before (am x) = x | |
| 187 mp-before (x repl y by x₁) = x | |
| 188 mp-before (m ∙ m₁) = mp-before m o mp-before m₁ | |
| 189 | |
| 190 mp-after : { a b : Obj A } { f : Hom A a b } → MP f → Hom A a b | |
| 191 mp-after (am x) = x | |
| 192 mp-after (x repl y by x₁) = y | |
| 193 mp-after (m ∙ m₁) = mp-before m o mp-before m₁ | |
| 194 | |
| 195 mp≈ : { a b : Obj A } { f g : Hom A a b } → (m : MP f ) → mp-before m ≈ mp-after m | |
| 196 mp≈ {a} {b} {f} {g} (am x) = refl-hom | |
| 197 mp≈ {a} {b} {f} {g} (x repl y by x=y ) = x=y | |
| 198 mp≈ {a} {b} {f} {g} (m ∙ m₁) = resp refl-hom refl-hom | |
| 199 | |
| 200 mpf : {a b c : Obj A } {y : Hom A b c } → (m : MP y ) → Hom A a b → Hom A a c | |
| 201 mpf (am x) y = x o y | |
| 202 mpf (x repl y by eq ) z = y o z | |
| 203 mpf (m ∙ m₁) y = mpf m ( mpf m₁ y ) | |
| 204 | |
| 205 mp-flatten : {a b : Obj A } {x : Hom A a b } → (m : MP x ) → Hom A a b | |
| 206 mp-flatten m = mpf m (id _) | |
| 207 | |
| 208 mpl1 : {a b c : Obj A } → Hom A b c → {y : Hom A a b } → MP y → Hom A a c | |
| 209 mpl1 x (am y) = x o y | |
| 210 mpl1 x (z repl y by eq ) = x o y | |
| 211 mpl1 x (y ∙ y1) = mpl1 ( mpl1 x y ) y1 | |
| 212 | |
| 213 mpl : {a b c : Obj A } {x : Hom A b c } {z : Hom A a b } → MP x → MP z → Hom A a c | |
| 214 mpl (am x) m = mpl1 x m | |
| 215 mpl (y repl x by eq ) m = mpl1 x m | |
| 216 mpl (m ∙ m1) m2 = mpl m (m1 ∙ m2) | |
| 217 | |
| 218 mp-flattenl : {a b : Obj A } {x : Hom A a b } → (m : MP x ) → Hom A a b | |
| 219 mp-flattenl m = mpl m (am (id _)) | |
| 220 | |
| 221 _⁻¹ : {a b : Obj A } ( f : Hom A a b ) → Set c₂ | |
| 222 _⁻¹ {a} {b} f = Hom A b a | |
| 223 | |
| 224 test1 : {a b c : Obj A } ( f : Hom A b c ) ( g : Hom A a b ) → ( _⁻¹ : {a b : Obj A } ( f : Hom A a b ) → Hom A b a ) → Hom A c a | |
| 225 test1 f g _⁻¹ = mp-flattenl ((am (g ⁻¹) ∙ am (f ⁻¹) ) ∙ ( (am f ∙ am g) ∙ am ((f o g) ⁻¹ ))) | |
| 226 | |
| 227 test2 : {a b c : Obj A } ( f : Hom A b c ) ( g : Hom A a b ) → ( _⁻¹ : {a b : Obj A } ( f : Hom A a b ) → Hom A b a ) → test1 f g _⁻¹ ≈ ((((g ⁻¹ o f ⁻¹ )o f ) o g ) o (f o g) ⁻¹ ) o id _ | |
| 228 test2 f g _⁻¹ = refl-hom | |
| 229 | |
| 230 test3 : {a b c : Obj A } ( f : Hom A b c ) ( g : Hom A a b ) → ( _⁻¹ : {a b : Obj A } ( f : Hom A a b ) → Hom A b a ) → Hom A c a | |
| 231 test3 f g _⁻¹ = mp-flatten ((am (g ⁻¹) ∙ am (f ⁻¹) ) ∙ ( (am f ∙ am g) ∙ am ((f o g) ⁻¹ ))) | |
| 232 | |
| 233 test4 : {a b c : Obj A } ( f : Hom A b c ) ( g : Hom A a b ) → ( _⁻¹ : {a b : Obj A } ( f : Hom A a b ) → Hom A b a ) → test3 f g _⁻¹ ≈ g ⁻¹ o (f ⁻¹ o (f o (g o ((f o g) ⁻¹ o id _)))) | |
| 234 test4 f g _⁻¹ = refl-hom | |
| 235 | |
| 236 o-flatten : {a b : Obj A } {x : Hom A a b } → (m : MP x ) → x ≈ mp-flatten m | |
| 237 o-flatten (am y) = sym-hom (idR ) | |
| 238 o-flatten (y repl x by eq) = sym-hom (idR ) | |
| 239 o-flatten (am x ∙ q) = resp ( o-flatten q ) refl-hom | |
| 240 o-flatten ((y repl x by eq) ∙ q) = resp ( o-flatten q ) refl-hom | |
| 241 -- d <- c <- b <- a ( p ∙ q ) ∙ r , ( x o y ) o z | |
| 242 o-flatten {a} {d} (_∙_ {a} {b} {d} {xy} {z} (_∙_ {b} {c} {d} {x} {y} p q) r) = | |
| 243 lemma9 _ _ _ ( o-flatten {b} {d} {x o y } (p ∙ q )) ( o-flatten {a} {b} {z} r ) where | |
| 244 mp-cong : { a b c : Obj A } → {p : Hom A b c} {q r : Hom A a b} → (P : MP p) → q ≈ r → mpf P q ≈ mpf P r | |
| 245 mp-cong (am x) q=r = resp q=r refl-hom | |
| 246 mp-cong (y repl x by eq) q=r = resp q=r refl-hom | |
| 247 mp-cong (P ∙ P₁) q=r = mp-cong P ( mp-cong P₁ q=r ) | |
| 248 mp-assoc : {a b c d : Obj A } {p : Hom A c d} {q : Hom A b c} {r : Hom A a b} → (P : MP p) → mpf P q o r ≈ mpf P (q o r ) | |
| 249 mp-assoc (am x) = sym-hom assoc | |
| 250 mp-assoc (y repl x by eq ) = sym-hom assoc | |
| 251 mp-assoc {_} {_} {_} {_} {p} {q} {r} (P ∙ P₁) = begin | |
| 252 mpf P (mpf P₁ q) o r ≈⟨ mp-assoc P ⟩ | |
| 253 mpf P (mpf P₁ q o r) ≈⟨ mp-cong P (mp-assoc P₁) ⟩ mpf P ((mpf P₁) (q o r)) | |
| 254 ∎ | |
| 255 lemma9 : (x : Hom A c d) (y : Hom A b c) (z : Hom A a b) → x o y ≈ mpf p (mpf q (id _)) | |
| 256 → z ≈ mpf r (id _) | |
| 257 → (x o y) o z ≈ mp-flatten ((p ∙ q) ∙ r) | |
| 258 lemma9 x y z t s = begin | |
| 259 (x o y) o z ≈⟨ resp refl-hom t ⟩ | |
| 260 mpf p (mpf q (id _)) o z ≈⟨ mp-assoc p ⟩ | |
| 261 mpf p (mpf q (id _) o z) ≈⟨ mp-cong p (mp-assoc q ) ⟩ | |
| 262 mpf p (mpf q ((id _) o z)) ≈⟨ mp-cong p (mp-cong q idL) ⟩ | |
| 263 mpf p (mpf q z) ≈⟨ mp-cong p (mp-cong q s) ⟩ | |
| 264 mpf p (mpf q (mpf r (id _))) | |
| 265 ∎ | |
| 266 | |
| 56 | 267 -- an example |
| 268 | |
| 269 Lemma61 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → | |
| 270 { a : Obj A } ( b : Obj A ) → | |
| 271 ( f : Hom A a b ) | |
| 272 → A [ A [ (Id {_} {_} {_} {A} b) o f ] ≈ f ] | |
| 273 Lemma61 c b g = -- IsCategory.identityL (Category.isCategory c) | |
| 606 | 274 let open ≈-Reasoning (c) in begin |
| 275 c [ ( Id {_} {_} {_} {c} b ) o g ] | |
| 56 | 276 ≈⟨ IsCategory.identityL (Category.isCategory c) ⟩ |
| 277 g | |
| 278 ∎ | |
| 253 | 279 |
| 280 Lemma62 : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) → | |
| 281 { a b : Obj A } → | |
| 282 ( f g : Hom A a b ) | |
| 283 → A [ A [ (Id {_} {_} {_} {A} b) o f ] ≈ A [ (Id {_} {_} {_} {A} b) o g ] ] | |
| 284 → A [ g ≈ f ] | |
| 285 Lemma62 A {a} {b} f g 1g=1f = let open ≈-Reasoning A in | |
| 286 begin | |
| 287 g | |
| 288 ≈↑⟨ idL ⟩ | |
| 289 id b o g | |
| 290 ≈↑⟨ 1g=1f ⟩ | |
| 291 id b o f | |
| 292 ≈⟨ idL ⟩ | |
| 293 f | |
| 294 ∎ |