Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 216:0135419f375c
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 03 Sep 2013 13:29:21 +0900 |
| parents | 637b5f58ed28 |
| children | 306f07bece85 |
| rev | line source |
|---|---|
| 205 | 1 --- |
| 2 -- | |
| 3 -- Equalizer | |
| 4 -- | |
| 208 | 5 -- e f |
| 205 | 6 -- c --------> a ----------> b |
| 208 | 7 -- ^ . ----------> |
| 205 | 8 -- | . g |
| 208 | 9 -- |k . |
| 10 -- | . h | |
| 205 | 11 -- d |
| 12 -- | |
| 13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 14 ---- | |
| 15 | |
| 16 open import Category -- https://github.com/konn/category-agda | |
| 17 open import Level | |
| 18 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where | |
| 19 | |
| 20 open import HomReasoning | |
| 21 open import cat-utility | |
| 22 | |
| 209 | 23 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 205 | 24 field |
| 209 | 25 e : Hom A c a |
| 215 | 26 ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] |
| 209 | 27 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 215 | 28 ek=h : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ e o k {d} h eq ] ≈ h ] |
| 29 uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → | |
| 214 | 30 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
| 209 | 31 equalizer : Hom A c a |
| 32 equalizer = e | |
| 206 | 33 |
| 209 | 34 record EqEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 206 | 35 field |
| 212 | 36 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
| 214 | 37 γ : {a b c d : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
| 212 | 38 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
| 213 | 39 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
| 214 | 40 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α f g) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] |
| 213 | 41 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
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Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
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42 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
| 215 | 43 b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ] |
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44 -- A [ α f g o β f g h ] ≈ h |
| 214 | 45 β : { d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
| 46 β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] | |
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Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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47 |
| 209 | 48 open Equalizer |
| 49 open EqEqualizer | |
| 50 | |
| 215 | 51 |
| 211 | 52 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
| 53 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
| 209 | 54 lemma-equ1 A {a} {b} {c} f g eqa = record { |
| 216 | 55 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 56 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (e ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d |
| 213 | 57 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 211 | 58 b1 = ef=eg (eqa f g) ; |
| 212 | 59 b2 = lemma-equ5 ; |
| 60 b3 = lemma-equ3 ; | |
| 215 | 61 b4 = lemma-equ6 |
| 211 | 62 } where |
| 216 | 63 -- |
| 64 -- e eqa f g f | |
| 65 -- c ----------> a ------->b | |
| 66 -- ^ g | |
| 67 -- | | |
| 68 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
| 69 -- | | |
| 70 -- d | |
| 71 -- | |
| 72 -- | |
| 73 -- e o id1 ≈ e → k e ≈ id | |
| 74 ff-equal4 : A [ A [ e (eqa f g ) o (e (eqa (A [ f o e (eqa f g) ] ) (A [ g o e (eqa f g) ] ))) ] ≈ | |
| 75 e (eqa f g ) | |
| 76 ] → | |
| 77 A [ k (eqa f g ) (e (eqa f g)) (ef=eg (eqa f g)) ≈ e (eqa (A [ f o e (eqa f g) ] ) (A [ g o e (eqa f g) ] )) ] | |
| 78 ff-equal4 eq = uniqueness (eqa f g) eq | |
| 79 | |
| 80 ff-equal3 : A [ e (eqa (A [ f o e (eqa f g) ] ) (A [ g o e (eqa f g) ] ) ) ≈ k (eqa f g ) (e (eqa f g)) (ef=eg (eqa f g)) ] | |
| 81 ff-equal3 = let open ≈-Reasoning (A) in | |
| 82 begin | |
| 83 e (eqa (A [ f o e (eqa f g) ] ) (A [ g o e (eqa f g) ] ) ) | |
| 84 ≈↑⟨ uniqueness (eqa f g) {!!} ⟩ | |
| 85 k (eqa f g ) (e (eqa f g)) (ef=eg (eqa f g)) | |
| 86 ∎ | |
| 87 ff-equal2 : A [ k (eqa f g) (e (eqa f g)) (ef=eg (eqa f g)) ≈ id1 A a ] | |
| 88 ff-equal2 = let open ≈-Reasoning (A) in | |
| 89 begin | |
| 90 k (eqa f g) (e (eqa f g)) (ef=eg (eqa f g)) | |
| 91 ≈⟨ uniqueness (eqa f g) idR ⟩ | |
| 92 id1 A a | |
| 93 ∎ | |
| 94 ff-equal1 : A [ e (eqa (A [ f o e (eqa f g) ] ) (A [ g o e (eqa f g) ] ) ) ≈ id1 A a ] | |
| 95 ff-equal1 = let open ≈-Reasoning (A) in | |
| 96 begin | |
| 97 e (eqa (f o e (eqa f g) ) (g o e (eqa f g) )) | |
| 98 ≈⟨ {!!} ⟩ | |
| 99 id1 A a | |
| 100 ∎ | |
| 101 ff-equal : {d : Obj A} {k₁ : Hom A d c} → A [ e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ] ) (A [ f o A [ e (eqa f g) o k₁ ] ] ) ) ≈ id1 A d ] | |
| 102 ff-equal {d} {k₁} = let open ≈-Reasoning (A) in | |
| 103 begin | |
| 104 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) | |
| 105 ≈⟨ {!!} ⟩ | |
| 106 id1 A d | |
| 107 ∎ | |
| 108 fg-equal : {d : Obj A} {k₁ : Hom A d c} → A [ e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ] ) (A [ g o A [ e (eqa f g) o k₁ ] ] ) ) ≈ id1 A d ] | |
| 109 fg-equal = {!!} | |
| 211 | 110 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 111 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 112 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 113 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 114 begin |
| 115 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 116 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 117 id1 A a |
| 118 ∎ | |
| 214 | 119 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 120 A [ A [ f o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
| 214 | 121 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 122 begin |
| 123 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 124 ≈⟨ assoc ⟩ | |
| 125 (f o h) o e (eqa (f o h) ( g o h )) | |
| 126 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
| 127 (g o h) o e (eqa (f o h) ( g o h )) | |
| 128 ≈↑⟨ assoc ⟩ | |
| 129 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 130 ∎ | |
| 131 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 132 A [ e (eqa f g) o k (eqa f g) (A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
| 212 | 133 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 134 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 135 begin | |
| 215 | 136 e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
| 137 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 138 h o e (eqa (f o h ) ( g o h )) |
| 139 ∎ | |
| 215 | 140 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
| 141 A [ k (eqa f g) (A [ A [ e (eqa f g) o k₁ ] o e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ g o A [ e (eqa f g) o k₁ ] ])) ]) | |
| 142 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
| 143 k (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ f o A [ e (eqa f g) o k₁ ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o k₁ ] ])) ] | |
| 144 ≈ k₁ ] | |
| 145 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
| 146 begin | |
| 147 ( k (eqa f g) (( ( e (eqa f g) o k₁ ) o e (eqa (( f o ( e (eqa f g) o k₁ ) )) (( g o ( e (eqa f g) o k₁ ) ))) )) | |
| 148 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
| 149 k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) ) | |
| 150 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
| 151 e (eqa f g) o k₁ | |
| 152 ≈⟨ {!!} ⟩ | |
| 153 (e (eqa f g) o k₁) o e (eqa (f o e (eqa f g) o k₁) (g o e (eqa f g) o k₁)) | |
| 154 ∎ )) ⟩ | |
| 155 k₁ o k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) | |
| 156 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
| 157 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
| 158 ≈⟨ {!!} ⟩ | |
| 159 id1 A d | |
| 160 ∎ )) ⟩ | |
| 161 k₁ o id1 A d | |
| 162 ≈⟨ idR ⟩ | |
| 163 k₁ | |
| 164 ∎ | |
| 211 | 165 |
| 166 | |
| 212 | 167 |
| 168 | |
| 169 | |
| 215 | 170 |
| 171 |