Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 256:80dfdeb3e4e7
simpler proof
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 14 Sep 2013 11:20:18 +0900 |
| parents | 7e7b2c38dee1 |
| children | 99751fb809e0 |
| 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 ) ) |
| 256 | 101 ≈⟨ uniqueness eqa refl-hom ⟩ |
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102 j |
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103 ∎ |
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104 |
| 251 | 105 -------------------------------- |
| 225 | 106 -- |
| 107 -- | |
| 249 | 108 -- An isomorphic arrow c' to c makes another equalizer |
| 225 | 109 -- |
| 230 | 110 -- e eqa f g f |
| 222 | 111 -- c ----------> a ------->b |
| 230 | 112 -- |^ |
| 113 -- || | |
| 222 | 114 -- h || h-1 |
| 230 | 115 -- v| |
| 116 -- c' | |
| 222 | 117 |
| 234 | 118 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
| 119 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 120 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
| 121 ( eqa : Equalizer A e f g ) | |
| 122 → Equalizer A (A [ e o h-1 ] ) f g | |
| 254 | 123 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 eqa = record { |
| 222 | 124 fe=ge = fe=ge1 ; |
| 125 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 126 ek=h = ek=h1 ; |
| 222 | 127 uniqueness = uniqueness1 |
| 128 } where | |
| 234 | 129 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 254 | 130 fe=ge1 = f1=gh ( fe=ge eqa ) |
| 222 | 131 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 234 | 132 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 133 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 134 begin | |
| 234 | 135 ( e o h-1 ) o ( h o k eqa j eq ) |
| 136 ≈↑⟨ assoc ⟩ | |
| 137 e o ( h-1 o ( h o k eqa j eq ) ) | |
| 138 ≈⟨ cdr assoc ⟩ | |
| 139 e o (( h-1 o h) o k eqa j eq ) | |
| 140 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
| 253 | 141 e o (id c o k eqa j eq ) |
| 234 | 142 ≈⟨ cdr idL ⟩ |
| 143 e o k eqa j eq | |
| 222 | 144 ≈⟨ ek=h eqa ⟩ |
| 145 j | |
| 146 ∎ | |
| 147 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 | 148 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
| 222 | 149 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 150 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 151 begin | |
| 152 h o k eqa h' eq | |
| 234 | 153 ≈⟨ cdr (uniqueness eqa ( begin |
| 154 e o ( h-1 o j ) | |
| 155 ≈⟨ assoc ⟩ | |
| 156 (e o h-1 ) o j | |
| 157 ≈⟨ ej=h ⟩ | |
| 158 h' | |
| 159 ∎ )) ⟩ | |
| 160 h o ( h-1 o j ) | |
| 161 ≈⟨ assoc ⟩ | |
| 162 (h o h-1 ) o j | |
| 163 ≈⟨ car hh-1=1 ⟩ | |
| 253 | 164 id c' o j |
| 234 | 165 ≈⟨ idL ⟩ |
| 222 | 166 j |
| 167 ∎ | |
| 168 | |
| 251 | 169 -------------------------------- |
| 225 | 170 -- |
| 171 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 172 -- | |
| 173 -- h : c → c' h' : c' → c | |
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174 -- e' = h o e |
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175 -- e = h' o e' |
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176 -- we assume equalizer on fe,ge and fe',ge' |
| 225 | 177 |
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178 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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179 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 180 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 181 → Hom A c c' -- should be e' = c-sio-l o e |
| 234 | 182 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
| 226 | 183 |
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184 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 | 185 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 186 → Hom A c' c -- e = c-sio-r o e' |
| 230 | 187 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
| 223 | 188 |
| 234 | 189 -- e' f |
| 230 | 190 -- c'----------> a ------->b f e j = g e j |
| 191 -- ^ g | |
| 192 -- |k h | |
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193 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 194 -- | |
| 195 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 196 -- | |
| 197 -- h j e f = h j e g → h = 'j e f | |
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198 -- h = j e f -> j = j' |
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199 -- |
| 228 | 200 |
| 251 | 201 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 | 202 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 203 → A [ A [ e' o c-iso-l eqa eqa' keqa ] ≈ e ] | |
| 251 | 204 e←e' {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 205 e' o c-iso-l eqa eqa' keqa |
| 206 ≈⟨⟩ | |
| 207 e' o k eqa' e (fe=ge eqa) | |
| 208 ≈⟨⟩ | |
| 209 equalizer eqa' o k eqa' e (fe=ge eqa) | |
| 210 ≈⟨ ek=h eqa' ⟩ | |
| 211 e | |
| 212 ∎ | |
| 213 | |
| 251 | 214 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 | 215 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 216 → A [ A [ e o c-iso-r eqa eqa' keqa ] ≈ e' ] | |
| 251 | 217 e'←e {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 218 e o c-iso-r eqa eqa' keqa |
| 219 ≈⟨⟩ | |
| 253 | 220 e o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 221 ≈↑⟨ car (ek=h eqa' ) ⟩ |
| 253 | 222 ( equalizer eqa' o k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 223 ≈⟨⟩ |
| 253 | 224 ( e' o k eqa' e (fe=ge eqa) ) o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 250 | 225 ≈↑⟨ assoc ⟩ |
| 253 | 226 e' 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 (equalizer keqa o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) ) |
| 250 | 229 ≈⟨ cdr ( ek=h keqa ) ⟩ |
| 253 | 230 e' o id c' |
| 250 | 231 ≈⟨ idR ⟩ |
| 232 e' | |
| 233 ∎ | |
| 234 | |
| 253 | 235 -- e←e' e'←e e = e |
| 236 -- e'←e e←e e' = e' is enough for isomorphism but we can prove l o r = id also. | |
| 252 | 237 |
| 234 | 238 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 ) |
| 239 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
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240 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
| 234 | 241 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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242 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
| 234 | 243 ≈⟨⟩ |
| 253 | 244 equalizer keqa o k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) |
| 234 | 245 ≈⟨ ek=h keqa ⟩ |
| 253 | 246 id c' |
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247 ∎ |
| 226 | 248 |
| 234 | 249 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 ) |
| 250 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
| 251 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
| 252 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] | |
| 250 | 253 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' = let open ≈-Reasoning (A) in begin |
| 234 | 254 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
| 255 ≈⟨⟩ | |
| 253 | 256 k keqa (id c') ( f1=g1 (fe=ge eqa') (id c') ) o k eqa' e (fe=ge eqa ) |
| 234 | 257 ≈⟨⟩ |
| 258 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
| 259 ≈⟨ cdr ( begin | |
| 260 k eqa' e (fe=ge eqa ) | |
| 261 ≈⟨ uniqueness eqa' ( begin | |
| 253 | 262 e' o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c)) |
| 251 | 263 ≈↑⟨ car (e'←e eqa eqa' keqa ) ⟩ |
| 253 | 264 ( e o equalizer keqa' ) o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c)) |
| 234 | 265 ≈↑⟨ assoc ⟩ |
| 253 | 266 e o ( equalizer keqa' o k keqa' (id c) (f1=g1 (fe=ge eqa) (id c))) |
| 234 | 267 ≈⟨ cdr ( ek=h keqa' ) ⟩ |
| 253 | 268 e o id c |
| 234 | 269 ≈⟨ idR ⟩ |
| 270 e | |
| 271 ∎ )⟩ | |
| 253 | 272 k keqa' (id c) ( f1=g1 (fe=ge eqa) (id c) ) |
| 234 | 273 ∎ )⟩ |
| 254 | 274 equalizer keqa' o k keqa' (id c) ( f1=g1 (fe=ge eqa) (id c) ) |
| 275 ≈⟨ ek=h keqa' ⟩ | |
| 253 | 276 id c |
| 234 | 277 ∎ |
| 278 | |
| 279 | |
| 230 | 280 |
| 251 | 281 -------------------------------- |
| 225 | 282 ---- |
| 283 -- | |
| 254 | 284 -- Existence of equalizer satisfies Burroni equations |
| 225 | 285 -- |
| 286 ---- | |
| 287 | |
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288 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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289 ( 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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290 → Burroni A {c} {a} {b} f g e |
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291 lemma-equ1 {a} {b} {c} f g e eqa = record { |
| 245 | 292 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
| 242 | 293 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 ] ))) ] ) |
| 294 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
| 249 | 295 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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296 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
| 247 | 297 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
| 245 | 298 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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299 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
| 226 | 300 b2 = lemma-b2 ; |
| 301 b3 = lemma-b3 ; | |
| 230 | 302 b4 = lemma-b4 |
| 211 | 303 } where |
| 216 | 304 -- |
| 305 -- e eqa f g f | |
| 306 -- c ----------> a ------->b | |
| 230 | 307 -- ^ g |
| 308 -- | | |
| 216 | 309 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 310 -- | |
| 216 | 311 -- d |
| 230 | 312 -- |
| 313 -- | |
| 216 | 314 -- e o id1 ≈ e → k e ≈ id |
| 315 | |
| 249 | 316 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 | 317 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in |
| 230 | 318 begin |
| 253 | 319 equalizer (eqa f f) o k (eqa f f) (id a) (f1=f1 f) |
| 215 | 320 ≈⟨ ek=h (eqa f f ) ⟩ |
| 253 | 321 id a |
| 211 | 322 ∎ |
| 230 | 323 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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324 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 | 325 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 326 begin |
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327 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 328 ≈⟨ assoc ⟩ |
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329 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 330 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
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331 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 332 ≈↑⟨ assoc ⟩ |
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333 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 334 ∎ |
| 245 | 335 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
| 336 → {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} ) ] | |
| 337 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
| 247 | 338 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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339 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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340 ≈ 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 | 341 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
| 245 | 342 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 ) |
| 343 ≈⟨ uniqueness (eqa f g) ( begin | |
| 248 | 344 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' ) |
| 345 ≈⟨ ek=h (eqa f g ) ⟩ | |
| 346 h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 347 ≈↑⟨ car h=h' ⟩ | |
| 348 h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 245 | 349 ∎ )⟩ |
| 350 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' ) | |
| 351 ∎ | |
| 249 | 352 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) ≈ |
| 353 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] | |
| 247 | 354 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
| 355 begin | |
| 253 | 356 k (eqa {a} {b} {c} f f {e} ) (id a) (f1=f1 f) |
| 247 | 357 ≈⟨ uniqueness (eqa f f) ( begin |
| 253 | 358 e o k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
| 247 | 359 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ |
| 253 | 360 id a |
| 247 | 361 ∎ )⟩ |
| 253 | 362 k (eqa {a} {b} {c} f' f' {e} ) (id a) (f1=f1 f') |
| 247 | 363 ∎ |
| 364 | |
| 230 | 365 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
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366 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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367 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 368 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 369 begin |
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370 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 | 371 ≈⟨ ek=h (eqa f g) ⟩ |
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372 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 373 ∎ |
| 230 | 374 |
| 375 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
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376 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
| 254 | 377 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
| 378 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) | |
| 379 o k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) | |
| 380 (id1 A d) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] | |
| 222 | 381 ≈ j ] |
| 230 | 382 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 383 begin |
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384 ( 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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385 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
| 249 | 386 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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387 ≈⟨ car ((uniqueness (eqa f g) ( begin |
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388 equalizer (eqa f g) o j |
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389 ≈↑⟨ idR ⟩ |
| 253 | 390 (equalizer (eqa f g) o j ) o id d |
| 391 ≈⟨⟩ -- here we decide e (fej) (gej)' is id d | |
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392 ((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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393 ∎ ))) ⟩ |
| 249 | 394 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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395 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
| 253 | 396 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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397 ≈⟨ idR ⟩ |
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398 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
| 253 | 399 ≈⟨⟩ -- here we decide e (fej) (fej)' is id d |
| 400 id d | |
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401 ∎ ))) ⟩ |
| 253 | 402 j o id d |
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403 ≈⟨ idR ⟩ |
| 222 | 404 j |
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405 ∎ |
| 211 | 406 |
| 251 | 407 -------------------------------- |
| 408 -- | |
| 409 -- Bourroni equations gives an Equalizer | |
| 410 -- | |
| 211 | 411 |
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412 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
| 245 | 413 → ( 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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414 lemma-equ2 {a} {b} {c} f g e bur = record { |
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415 fe=ge = fe=ge1 ; |
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416 k = k1 ; |
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417 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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418 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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419 } where |
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420 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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421 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
| 245 | 422 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
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423 fe=ge1 = b1 bur |
| 245 | 424 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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425 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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426 begin |
| 245 | 427 α bur f g e o k1 h eq |
| 239 | 428 ≈⟨⟩ |
| 253 | 429 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 | 430 ≈⟨ assoc ⟩ |
| 253 | 431 ( 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 | 432 ≈⟨ car (b2 bur) ⟩ |
| 253 | 433 ( h o ( α bur ( f o h ) ( g o h ) (id d))) o δ bur {d} {b} {d} (id d) (f o h) |
| 239 | 434 ≈↑⟨ assoc ⟩ |
| 253 | 435 h o ((( α bur ( f o h ) ( g o h ) (id d) )) o δ bur {d} {b} {d} (id d) (f o h) ) |
| 240 | 436 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
| 253 | 437 h o ((( α bur ( f o h ) ( f o h ) (id d)))o δ bur {d} {b} {d} (id d) (f o h) ) |
| 438 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id d} ) ⟩ | |
| 439 h o id d | |
| 240 | 440 ≈⟨ idR ⟩ |
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441 h |
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442 ∎ |
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443 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 | 444 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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445 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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446 begin |
| 240 | 447 k1 {d} h eq |
| 239 | 448 ≈⟨⟩ |
| 253 | 449 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id d) (f o h) |
| 240 | 450 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
| 253 | 451 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id d) (f o h) |
| 245 | 452 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ |
| 253 | 453 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 | 454 ≈⟨ b4 bur ⟩ |
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455 k' |
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456 ∎ |
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457 |
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458 |
| 225 | 459 -- end |
| 212 | 460 |
| 461 | |
| 462 | |
| 215 | 463 |
| 464 |