Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 257:99751fb809e0
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 14 Sep 2013 11:21:42 +0900 |
| parents | 80dfdeb3e4e7 |
| children | 281b8962abbe |
| rev | line source |
|---|---|
| 205 | 1 --- |
| 2 -- | |
| 3 -- Equalizer | |
| 4 -- | |
| 208 | 5 -- e f |
| 205 | 6 -- c --------> a ----------> b |
| 208 | 7 -- ^ . ----------> |
| 205 | 8 -- | . g |
| 230 | 9 -- |k . |
| 10 -- | . h | |
| 11 -- d | |
| 205 | 12 -- |
| 13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 14 ---- | |
| 15 | |
| 230 | 16 open import Category -- https://github.com/konn/category-agda |
| 205 | 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 | |
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23 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 205 | 24 field |
| 221 | 25 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] |
| 209 | 26 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 215 | 27 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 ] |
| 230 | 28 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 | 29 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
| 209 | 30 equalizer : Hom A c a |
| 31 equalizer = e | |
| 206 | 32 |
| 253 | 33 |
| 230 | 34 -- |
| 251 | 35 -- Burroni's Flat Equational Definition of Equalizer |
| 230 | 36 -- |
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37 record Burroni { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) (e : Hom A c a) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 206 | 38 field |
| 245 | 39 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (e : Hom A c a ) → Hom A c a |
| 214 | 40 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 |
| 245 | 41 δ : {a b c : Obj A } → (e : Hom A c a ) → (f : Hom A a b) → Hom A a c |
| 242 | 42 cong-α : {a b c : Obj A } → { e : Hom A c a } |
| 245 | 43 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] |
| 242 | 44 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] |
| 243 | 45 → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] |
| 245 | 46 cong-δ : {a b c : Obj A } → {e : Hom A c a} → {f f' : Hom A a b} → A [ f ≈ f' ] → A [ δ e f ≈ δ e f' ] |
| 47 b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] | |
| 48 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α {a} {b} {c} f g e ) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) (id1 A d) ] ] | |
| 49 b3 : {a b d : Obj A} → (f : Hom A a b ) → {h : Hom A d a } → A [ A [ α {a} {b} {d} f f h o δ {a} {b} {d} h f ] ≈ id1 A a ] | |
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50 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
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51 b4 : {d : Obj A } {k : Hom A d c} → |
| 245 | 52 A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g e o k ] ) o ( δ {d} {b} {d} (id1 A d) (A [ f o A [ α {a} {b} {c} f g e o k ] ] ) )] ≈ k ] |
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53 -- A [ α f g o β f g h ] ≈ h |
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54 β : { d a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
| 245 | 55 β {d} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ {d} {b} {d} (id1 A d) (A [ f o h ]) ] |
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56 |
| 209 | 57 open Equalizer |
| 225 | 58 open Burroni |
| 209 | 59 |
| 225 | 60 -- |
| 61 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 62 -- | |
| 219 | 63 |
| 224 | 64 f1=g1 : { a b c : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → (h : Hom A c a) → A [ A [ f o h ] ≈ A [ g o h ] ] |
| 65 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 66 | |
| 226 | 67 f1=f1 : { a b : Obj A } (f : Hom A a b ) → A [ A [ f o (id1 A a) ] ≈ A [ f o (id1 A a) ] ] |
| 230 | 68 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 69 |
| 224 | 70 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 71 (eq : A [ A [ f o e ] ≈ A [ g o e ] ] ) → A [ A [ f o A [ e o h ] ] ≈ A [ g o A [ e o h ] ] ] | |
| 230 | 72 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 73 begin |
| 74 f o ( e o h ) | |
| 75 ≈⟨ assoc ⟩ | |
| 230 | 76 (f o e ) o h |
| 224 | 77 ≈⟨ car eq ⟩ |
| 230 | 78 (g o e ) o h |
| 224 | 79 ≈↑⟨ assoc ⟩ |
| 80 g o ( e o h ) | |
| 81 ∎ | |
| 219 | 82 |
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83 ------------------------------- |
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84 -- |
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85 -- Every equalizer is monic |
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86 -- |
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87 -- e i = e j → i = j |
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88 -- |
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89 |
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90 monoic : { c a b d : Obj A } {f g : Hom A a b } → {e : Hom A c a } ( eqa : Equalizer A e f g) |
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91 → { i j : Hom A d c } |
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92 → A [ A [ equalizer eqa o i ] ≈ A [ equalizer eqa o j ] ] → A [ i ≈ j ] |
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93 monoic {c} {a} {b} {d} {f} {g} {e} eqa {i} {j} ei=ej = let open ≈-Reasoning (A) in begin |
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94 i |
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95 ≈↑⟨ uniqueness eqa ( begin |
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96 equalizer eqa o i |
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97 ≈⟨ ei=ej ⟩ |
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98 equalizer eqa o j |
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99 ∎ )⟩ |
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100 k eqa (equalizer eqa o j) ( f1=gh (fe=ge eqa ) ) |
| 257 | 101 ≈⟨ uniqueness eqa ( begin |
| 102 equalizer eqa o j | |
| 103 ≈⟨⟩ | |
| 104 equalizer eqa o j | |
| 105 ∎ )⟩ | |
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106 j |
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107 ∎ |
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108 |
| 251 | 109 -------------------------------- |
| 225 | 110 -- |
| 111 -- | |
| 249 | 112 -- An isomorphic arrow c' to c makes another equalizer |
| 225 | 113 -- |
| 230 | 114 -- e eqa f g f |
| 222 | 115 -- c ----------> a ------->b |
| 230 | 116 -- |^ |
| 117 -- || | |
| 222 | 118 -- h || h-1 |
| 230 | 119 -- v| |
| 120 -- c' | |
| 222 | 121 |
| 234 | 122 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
| 123 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 124 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
| 125 ( eqa : Equalizer A e f g ) | |
| 126 → Equalizer A (A [ e o h-1 ] ) f g | |
| 254 | 127 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 eqa = record { |
| 222 | 128 fe=ge = fe=ge1 ; |
| 129 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 130 ek=h = ek=h1 ; |
| 222 | 131 uniqueness = uniqueness1 |
| 132 } where | |
| 234 | 133 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 254 | 134 fe=ge1 = f1=gh ( fe=ge eqa ) |
| 222 | 135 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 234 | 136 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 137 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 138 begin | |
| 234 | 139 ( e o h-1 ) o ( h o k eqa j eq ) |
| 140 ≈↑⟨ assoc ⟩ | |
| 141 e o ( h-1 o ( h o k eqa j eq ) ) | |
| 142 ≈⟨ cdr assoc ⟩ | |
| 143 e o (( h-1 o h) o k eqa j eq ) | |
| 144 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
| 253 | 145 e o (id c o k eqa j eq ) |
| 234 | 146 ≈⟨ cdr idL ⟩ |
| 147 e o k eqa j eq | |
| 222 | 148 ≈⟨ ek=h eqa ⟩ |
| 149 j | |
| 150 ∎ | |
| 151 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 234 | 152 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
| 222 | 153 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 154 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 155 begin | |
| 156 h o k eqa h' eq | |
| 234 | 157 ≈⟨ cdr (uniqueness eqa ( begin |
| 158 e o ( h-1 o j ) | |
| 159 ≈⟨ assoc ⟩ | |
| 160 (e o h-1 ) o j | |
| 161 ≈⟨ ej=h ⟩ | |
| 162 h' | |
| 163 ∎ )) ⟩ | |
| 164 h o ( h-1 o j ) | |
| 165 ≈⟨ assoc ⟩ | |
| 166 (h o h-1 ) o j | |
| 167 ≈⟨ car hh-1=1 ⟩ | |
| 253 | 168 id c' o j |
| 234 | 169 ≈⟨ idL ⟩ |
| 222 | 170 j |
| 171 ∎ | |
| 172 | |
| 251 | 173 -------------------------------- |
| 225 | 174 -- |
| 175 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 176 -- | |
| 177 -- h : c → c' h' : c' → c | |
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178 -- e' = h o e |
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179 -- e = h' o e' |
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180 -- we assume equalizer on fe,ge and fe',ge' |
| 225 | 181 |
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182 c-iso-l : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } |
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183 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 184 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 185 → Hom A c c' -- should be e' = c-sio-l o e |
| 234 | 186 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
| 226 | 187 |
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188 c-iso-r : { c c' a b : Obj A } {f g : Hom A a b } {e : Hom A c a } {e' : Hom A c' a} → ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 189 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 190 → Hom A c' c -- e = c-sio-r o e' |
| 230 | 191 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
| 223 | 192 |
| 234 | 193 -- e' f |
| 230 | 194 -- c'----------> a ------->b f e j = g e j |
| 195 -- ^ g | |
| 196 -- |k h | |
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197 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 198 -- | |
| 199 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 200 -- | |
| 201 -- h j e f = h j e g → h = 'j e f | |
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202 -- h = j e f -> j = j' |
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203 -- |
| 228 | 204 |
| 251 | 205 e←e' : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 250 | 206 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 207 → A [ A [ e' o c-iso-l eqa eqa' keqa ] ≈ e ] | |
| 251 | 208 e←e' {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 209 e' o c-iso-l eqa eqa' keqa |
| 210 ≈⟨⟩ | |
| 211 e' o k eqa' e (fe=ge eqa) | |
| 212 ≈⟨⟩ | |
| 213 equalizer eqa' o k eqa' e (fe=ge eqa) | |
| 214 ≈⟨ ek=h eqa' ⟩ | |
| 215 e | |
| 216 ∎ | |
| 217 | |
| 251 | 218 e'←e : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 250 | 219 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 220 → A [ A [ e o c-iso-r eqa eqa' keqa ] ≈ e' ] | |
| 251 | 221 e'←e {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 222 e o c-iso-r eqa eqa' keqa |
| 223 ≈⟨⟩ | |
| 253 | 224 e o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 225 ≈↑⟨ car (ek=h eqa' ) ⟩ |
| 253 | 226 ( equalizer eqa' o k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 227 ≈⟨⟩ |
| 253 | 228 ( e' o k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 229 ≈↑⟨ assoc ⟩ |
| 253 | 230 e' o (( k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) ) |
| 250 | 231 ≈⟨⟩ |
| 253 | 232 e' o (equalizer keqa o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) ) |
| 250 | 233 ≈⟨ cdr ( ek=h keqa ) ⟩ |
| 253 | 234 e' o id c' |
| 250 | 235 ≈⟨ idR ⟩ |
| 236 e' | |
| 237 ∎ | |
| 238 | |
| 253 | 239 -- e←e' e'←e e = e |
| 240 -- e'←e e←e e' = e' is enough for isomorphism but we can prove l o r = id also. | |
| 252 | 241 |
| 234 | 242 c-iso→ : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 243 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
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244 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
| 234 | 245 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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246 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
| 234 | 247 ≈⟨⟩ |
| 253 | 248 equalizer keqa o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 234 | 249 ≈⟨ ek=h keqa ⟩ |
| 253 | 250 id c' |
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251 ∎ |
| 226 | 252 |
| 234 | 253 c-iso← : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 254 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
| 255 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
| 256 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] | |
| 250 | 257 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' = let open ≈-Reasoning (A) in begin |
| 234 | 258 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
| 259 ≈⟨⟩ | |
| 253 | 260 k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) o k eqa' e (fe=ge eqa ) |
| 234 | 261 ≈⟨⟩ |
| 262 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
| 263 ≈⟨ cdr ( begin | |
| 264 k eqa' e (fe=ge eqa ) | |
| 265 ≈⟨ uniqueness eqa' ( begin | |
| 253 | 266 e' o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c)) |
| 251 | 267 ≈↑⟨ car (e'←e eqa eqa' keqa ) ⟩ |
| 253 | 268 ( e o equalizer keqa' ) o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c)) |
| 234 | 269 ≈↑⟨ assoc ⟩ |
| 253 | 270 e o ( equalizer keqa' o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c))) |
| 234 | 271 ≈⟨ cdr ( ek=h keqa' ) ⟩ |
| 253 | 272 e o id c |
| 234 | 273 ≈⟨ idR ⟩ |
| 274 e | |
| 275 ∎ )⟩ | |
| 253 | 276 k keqa' (id c) ( f1=g1 (fe=ge eqa) (id c) ) |
| 234 | 277 ∎ )⟩ |
| 254 | 278 equalizer keqa' o k keqa' (id c) ( f1=g1 (fe=ge eqa) (id c) ) |
| 279 ≈⟨ ek=h keqa' ⟩ | |
| 253 | 280 id c |
| 234 | 281 ∎ |
| 282 | |
| 283 | |
| 230 | 284 |
| 251 | 285 -------------------------------- |
| 225 | 286 ---- |
| 287 -- | |
| 254 | 288 -- Existence of equalizer satisfies Burroni equations |
| 225 | 289 -- |
| 290 ---- | |
| 291 | |
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292 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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293 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
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294 → Burroni A {c} {a} {b} f g e |
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295 lemma-equ1 {a} {b} {c} f g e eqa = record { |
| 245 | 296 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
| 242 | 297 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) |
| 298 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
| 249 | 299 δ = λ {a} {b} {c} e f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f); -- Hom A a c |
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300 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
| 247 | 301 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
| 245 | 302 cong-δ = λ {a b c e f f'} f=f' → cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' ; |
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303 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
| 226 | 304 b2 = lemma-b2 ; |
| 305 b3 = lemma-b3 ; | |
| 230 | 306 b4 = lemma-b4 |
| 211 | 307 } where |
| 216 | 308 -- |
| 309 -- e eqa f g f | |
| 310 -- c ----------> a ------->b | |
| 230 | 311 -- ^ g |
| 312 -- | | |
| 216 | 313 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 314 -- | |
| 216 | 315 -- d |
| 230 | 316 -- |
| 317 -- | |
| 216 | 318 -- e o id1 ≈ e → k e ≈ id |
| 319 | |
| 249 | 320 lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (f1=f1 f) ] ≈ id1 A a ] |
| 240 | 321 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in |
| 230 | 322 begin |
| 253 | 323 equalizer (eqa f f) o k (eqa f f) (id a) (f1=f1 f) |
| 215 | 324 ≈⟨ ek=h (eqa f f ) ⟩ |
| 253 | 325 id a |
| 211 | 326 ∎ |
| 230 | 327 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
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328 A [ A [ f o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
| 214 | 329 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 330 begin |
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331 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 332 ≈⟨ assoc ⟩ |
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333 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 334 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
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335 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 336 ≈↑⟨ assoc ⟩ |
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337 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 338 ∎ |
| 245 | 339 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
| 340 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer (eqa {a} {b} {c} f g {e} )≈ equalizer (eqa {a} {b} {c} f g' {e} ) ] | |
| 341 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
| 247 | 342 cong-γ1 : {a c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → { e : Hom A c a} → |
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343 A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) |
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344 ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] |
| 247 | 345 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
| 245 | 346 k (eqa f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) |
| 347 ≈⟨ uniqueness (eqa f g) ( begin | |
| 248 | 348 e o k (eqa f g ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) |
| 349 ≈⟨ ek=h (eqa f g ) ⟩ | |
| 350 h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 351 ≈↑⟨ car h=h' ⟩ | |
| 352 h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 245 | 353 ∎ )⟩ |
| 354 k (eqa f g ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) | |
| 355 ∎ | |
| 249 | 356 cong-δ1 : {a b c : Obj A} {e : Hom A c a } {f f' : Hom A a b} → A [ f ≈ f' ] → A [ k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f) ≈ |
| 357 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] | |
| 247 | 358 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
| 359 begin | |
| 253 | 360 k (eqa {a} {b} {c} f f {e} ) (id a) (f1=f1 f) |
| 247 | 361 ≈⟨ uniqueness (eqa f f) ( begin |
| 253 | 362 e o k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
| 247 | 363 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ |
| 253 | 364 id a |
| 247 | 365 ∎ )⟩ |
| 253 | 366 k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
| 247 | 367 ∎ |
| 368 | |
| 230 | 369 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
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370 A [ equalizer (eqa f g) o k (eqa f g) (A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
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371 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 372 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 373 begin |
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374 equalizer (eqa f g) o k (eqa f g) (h o equalizer (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
| 215 | 375 ≈⟨ ek=h (eqa f g) ⟩ |
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376 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 377 ∎ |
| 230 | 378 |
| 379 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
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380 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
| 254 | 381 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
| 382 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) | |
| 383 o k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) | |
| 384 (id1 A d) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] | |
| 222 | 385 ≈ j ] |
| 230 | 386 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 387 begin |
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388 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g {e}) o j ) )) (( g o ( equalizer (eqa f g {e}) o j ) ))) )) |
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389 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
| 249 | 390 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) ) |
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391 ≈⟨ car ((uniqueness (eqa f g) ( begin |
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392 equalizer (eqa f g) o j |
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393 ≈↑⟨ idR ⟩ |
| 253 | 394 (equalizer (eqa f g) o j ) o id d |
| 395 ≈⟨⟩ -- here we decide e (fej) (gej)' is id d | |
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396 ((equalizer (eqa f g) o j) o equalizer (eqa (f o equalizer (eqa f g {e}) o j) (g o equalizer (eqa f g {e}) o j))) |
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397 ∎ ))) ⟩ |
| 249 | 398 j o k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (f1=f1 (( f o ( equalizer (eqa f g) o j ) ))) |
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399 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
| 253 | 400 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id d |
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401 ≈⟨ idR ⟩ |
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402 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
| 253 | 403 ≈⟨⟩ -- here we decide e (fej) (fej)' is id d |
| 404 id d | |
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405 ∎ ))) ⟩ |
| 253 | 406 j o id d |
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407 ≈⟨ idR ⟩ |
| 222 | 408 j |
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409 ∎ |
| 211 | 410 |
| 251 | 411 -------------------------------- |
| 412 -- | |
| 413 -- Bourroni equations gives an Equalizer | |
| 414 -- | |
| 211 | 415 |
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416 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
| 245 | 417 → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g e) f g |
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418 lemma-equ2 {a} {b} {c} f g e bur = record { |
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419 fe=ge = fe=ge1 ; |
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420 k = k1 ; |
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421 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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422 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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423 } where |
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424 k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
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425 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
| 245 | 426 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
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427 fe=ge1 = b1 bur |
| 245 | 428 ek=h1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ (α bur f g e) o k1 {d} h eq ] ≈ h ] |
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429 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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430 begin |
| 245 | 431 α bur f g e o k1 h eq |
| 239 | 432 ≈⟨⟩ |
| 253 | 433 α bur f g e o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) ) |
| 239 | 434 ≈⟨ assoc ⟩ |
| 253 | 435 ( α bur f g e o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} (id d) (f o h) |
| 239 | 436 ≈⟨ car (b2 bur) ⟩ |
| 253 | 437 ( h o ( α bur ( f o h ) ( g o h ) (id d))) o δ bur {d} {b} {d} (id d) (f o h) |
| 239 | 438 ≈↑⟨ assoc ⟩ |
| 253 | 439 h o ((( α bur ( f o h ) ( g o h ) (id d) )) o δ bur {d} {b} {d} (id d) (f o h) ) |
| 240 | 440 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
| 253 | 441 h o ((( α bur ( f o h ) ( f o h ) (id d)))o δ bur {d} {b} {d} (id d) (f o h) ) |
| 442 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id d} ) ⟩ | |
| 443 h o id d | |
| 240 | 444 ≈⟨ idR ⟩ |
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445 h |
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446 ∎ |
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447 uniqueness1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
| 245 | 448 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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449 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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450 begin |
| 240 | 451 k1 {d} h eq |
| 239 | 452 ≈⟨⟩ |
| 253 | 453 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) |
| 240 | 454 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
| 253 | 455 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) (f o h) |
| 245 | 456 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ |
| 253 | 457 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) ( f o ( α bur f g e o k') ) |
| 240 | 458 ≈⟨ b4 bur ⟩ |
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459 k' |
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460 ∎ |
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461 |
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462 |
| 225 | 463 -- end |
| 212 | 464 |
| 465 | |
| 466 | |
| 215 | 467 |
| 468 |