Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 251:40947f08bab6
comment
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 09 Sep 2013 16:03:14 +0900 |
| parents | a1e2228c2a6b |
| children | e0835b8dd51b |
| 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 |
| 230 | 33 -- |
| 251 | 34 -- Burroni's Flat Equational Definition of Equalizer |
| 230 | 35 -- |
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36 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 | 37 field |
| 245 | 38 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 | 39 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 | 40 δ : {a b c : Obj A } → (e : Hom A c a ) → (f : Hom A a b) → Hom A a c |
| 242 | 41 cong-α : {a b c : Obj A } → { e : Hom A c a } |
| 245 | 42 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] |
| 242 | 43 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] |
| 243 | 44 → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] |
| 245 | 45 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' ] |
| 46 b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] | |
| 47 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) ] ] | |
| 48 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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49 -- 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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50 b4 : {d : Obj A } {k : Hom A d c} → |
| 245 | 51 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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52 -- A [ α f g o β f g h ] ≈ h |
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53 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 | 54 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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55 |
| 209 | 56 open Equalizer |
| 225 | 57 open Burroni |
| 209 | 58 |
| 225 | 59 -- |
| 60 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 61 -- | |
| 219 | 62 |
| 224 | 63 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 ] ] |
| 64 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 65 | |
| 226 | 66 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 | 67 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 68 |
| 224 | 69 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 70 (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 | 71 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 72 begin |
| 73 f o ( e o h ) | |
| 74 ≈⟨ assoc ⟩ | |
| 230 | 75 (f o e ) o h |
| 224 | 76 ≈⟨ car eq ⟩ |
| 230 | 77 (g o e ) o h |
| 224 | 78 ≈↑⟨ assoc ⟩ |
| 79 g o ( e o h ) | |
| 80 ∎ | |
| 219 | 81 |
| 251 | 82 -------------------------------- |
| 225 | 83 -- |
| 84 -- | |
| 249 | 85 -- An isomorphic arrow c' to c makes another equalizer |
| 225 | 86 -- |
| 230 | 87 -- e eqa f g f |
| 222 | 88 -- c ----------> a ------->b |
| 230 | 89 -- |^ |
| 90 -- || | |
| 222 | 91 -- h || h-1 |
| 230 | 92 -- v| |
| 93 -- c' | |
| 222 | 94 |
| 234 | 95 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
| 96 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 97 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
| 98 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
| 99 ( eqa : Equalizer A e f g ) | |
| 100 → Equalizer A (A [ e o h-1 ] ) f g | |
| 101 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
| 222 | 102 fe=ge = fe=ge1 ; |
| 103 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 104 ek=h = ek=h1 ; |
| 222 | 105 uniqueness = uniqueness1 |
| 106 } where | |
| 234 | 107 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 108 fe=ge1 = fe=ge' | |
| 222 | 109 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 234 | 110 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 111 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 112 begin | |
| 234 | 113 ( e o h-1 ) o ( h o k eqa j eq ) |
| 114 ≈↑⟨ assoc ⟩ | |
| 115 e o ( h-1 o ( h o k eqa j eq ) ) | |
| 116 ≈⟨ cdr assoc ⟩ | |
| 117 e o (( h-1 o h) o k eqa j eq ) | |
| 118 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
| 119 e o (id1 A c o k eqa j eq ) | |
| 120 ≈⟨ cdr idL ⟩ | |
| 121 e o k eqa j eq | |
| 222 | 122 ≈⟨ ek=h eqa ⟩ |
| 123 j | |
| 124 ∎ | |
| 125 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 | 126 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
| 222 | 127 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 128 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 129 begin | |
| 130 h o k eqa h' eq | |
| 234 | 131 ≈⟨ cdr (uniqueness eqa ( begin |
| 132 e o ( h-1 o j ) | |
| 133 ≈⟨ assoc ⟩ | |
| 134 (e o h-1 ) o j | |
| 135 ≈⟨ ej=h ⟩ | |
| 136 h' | |
| 137 ∎ )) ⟩ | |
| 138 h o ( h-1 o j ) | |
| 139 ≈⟨ assoc ⟩ | |
| 140 (h o h-1 ) o j | |
| 141 ≈⟨ car hh-1=1 ⟩ | |
| 142 id1 A c' o j | |
| 143 ≈⟨ idL ⟩ | |
| 222 | 144 j |
| 145 ∎ | |
| 146 | |
| 251 | 147 -------------------------------- |
| 225 | 148 -- |
| 149 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 150 -- | |
| 151 -- h : c → c' h' : c' → c | |
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152 -- e' = h o e |
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153 -- e = h' o e' |
| 225 | 154 |
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155 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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156 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 157 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 158 → Hom A c c' -- should be e' = c-sio-l o e |
| 234 | 159 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
| 226 | 160 |
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161 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 | 162 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 163 → Hom A c' c -- e = c-sio-r o e' |
| 230 | 164 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
| 223 | 165 |
| 234 | 166 -- e' f |
| 230 | 167 -- c'----------> a ------->b f e j = g e j |
| 168 -- ^ g | |
| 169 -- |k h | |
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170 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 171 -- | |
| 172 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 173 -- | |
| 174 -- h j e f = h j e g → h = 'j e f | |
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175 -- h = j e f -> j = j' |
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176 -- |
| 228 | 177 |
| 251 | 178 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 | 179 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 180 → A [ A [ e' o c-iso-l eqa eqa' keqa ] ≈ e ] | |
| 251 | 181 e←e' {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 182 e' o c-iso-l eqa eqa' keqa |
| 183 ≈⟨⟩ | |
| 184 e' o k eqa' e (fe=ge eqa) | |
| 185 ≈⟨⟩ | |
| 186 equalizer eqa' o k eqa' e (fe=ge eqa) | |
| 187 ≈⟨ ek=h eqa' ⟩ | |
| 188 e | |
| 189 ∎ | |
| 190 | |
| 251 | 191 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 | 192 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 193 → A [ A [ e o c-iso-r eqa eqa' keqa ] ≈ e' ] | |
| 251 | 194 e'←e {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
| 250 | 195 e o c-iso-r eqa eqa' keqa |
| 196 ≈⟨⟩ | |
| 197 e o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 198 ≈↑⟨ car (ek=h eqa' ) ⟩ | |
| 199 ( equalizer eqa' o k eqa' e (fe=ge eqa) ) o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 200 ≈⟨⟩ | |
| 201 ( e' o k eqa' e (fe=ge eqa) ) o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 202 ≈↑⟨ assoc ⟩ | |
| 203 e' o (( k eqa' e (fe=ge eqa) ) o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) ) | |
| 204 ≈⟨⟩ | |
| 205 e' o (equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) ) | |
| 206 ≈⟨ cdr ( ek=h keqa ) ⟩ | |
| 207 e' o id1 A c' | |
| 208 ≈⟨ idR ⟩ | |
| 209 e' | |
| 210 ∎ | |
| 211 | |
| 234 | 212 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 ) |
| 213 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
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214 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
| 234 | 215 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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216 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
| 234 | 217 ≈⟨⟩ |
| 218 equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 219 ≈⟨ ek=h keqa ⟩ | |
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220 id1 A c' |
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221 ∎ |
| 226 | 222 |
| 234 | 223 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 ) |
| 224 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
| 225 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
| 226 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] | |
| 250 | 227 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' = let open ≈-Reasoning (A) in begin |
| 234 | 228 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
| 229 ≈⟨⟩ | |
| 230 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) | |
| 231 ≈⟨⟩ | |
| 232 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
| 233 ≈⟨ cdr ( begin | |
| 234 k eqa' e (fe=ge eqa ) | |
| 235 ≈⟨ uniqueness eqa' ( begin | |
| 236 e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
| 251 | 237 ≈↑⟨ car (e'←e eqa eqa' keqa ) ⟩ |
| 234 | 238 ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) |
| 239 ≈↑⟨ assoc ⟩ | |
| 240 e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) | |
| 241 ≈⟨ cdr ( ek=h keqa' ) ⟩ | |
| 242 e o id1 A c | |
| 243 ≈⟨ idR ⟩ | |
| 244 e | |
| 245 ∎ )⟩ | |
| 246 k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
| 247 ∎ )⟩ | |
| 250 | 248 equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) ≈⟨ ek=h keqa' ⟩ |
| 234 | 249 id1 A c |
| 250 ∎ | |
| 251 | |
| 252 | |
| 230 | 253 |
| 251 | 254 -------------------------------- |
| 225 | 255 ---- |
| 256 -- | |
| 257 -- An equalizer satisfies Burroni equations | |
| 258 -- | |
| 259 ---- | |
| 260 | |
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261 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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262 ( 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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263 → Burroni A {c} {a} {b} f g e |
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264 lemma-equ1 {a} {b} {c} f g e eqa = record { |
| 245 | 265 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
| 242 | 266 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 ] ))) ] ) |
| 267 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
| 249 | 268 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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269 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
| 247 | 270 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
| 245 | 271 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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272 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
| 226 | 273 b2 = lemma-b2 ; |
| 274 b3 = lemma-b3 ; | |
| 230 | 275 b4 = lemma-b4 |
| 211 | 276 } where |
| 216 | 277 -- |
| 278 -- e eqa f g f | |
| 279 -- c ----------> a ------->b | |
| 230 | 280 -- ^ g |
| 281 -- | | |
| 216 | 282 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 283 -- | |
| 216 | 284 -- d |
| 230 | 285 -- |
| 286 -- | |
| 216 | 287 -- e o id1 ≈ e → k e ≈ id |
| 288 | |
| 249 | 289 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 | 290 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in |
| 230 | 291 begin |
| 249 | 292 equalizer (eqa f f) o k (eqa f f) (id1 A a) (f1=f1 f) |
| 215 | 293 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 294 id1 A a |
| 295 ∎ | |
| 230 | 296 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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297 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 | 298 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 299 begin |
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300 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 301 ≈⟨ assoc ⟩ |
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302 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 303 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
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304 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 305 ≈↑⟨ assoc ⟩ |
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306 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 307 ∎ |
| 245 | 308 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
| 309 → {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} ) ] | |
| 310 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
| 247 | 311 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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312 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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313 ≈ 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 | 314 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
| 245 | 315 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 ) |
| 316 ≈⟨ uniqueness (eqa f g) ( begin | |
| 248 | 317 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' ) |
| 318 ≈⟨ ek=h (eqa f g ) ⟩ | |
| 319 h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 320 ≈↑⟨ car h=h' ⟩ | |
| 321 h o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ))) | |
| 245 | 322 ∎ )⟩ |
| 323 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' ) | |
| 324 ∎ | |
| 249 | 325 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) ≈ |
| 326 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') ] | |
| 247 | 327 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
| 328 begin | |
| 249 | 329 k (eqa {a} {b} {c} f f {e} ) (id1 A a) (f1=f1 f) |
| 247 | 330 ≈⟨ uniqueness (eqa f f) ( begin |
| 249 | 331 e o k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') |
| 247 | 332 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ |
| 333 id1 A a | |
| 334 ∎ )⟩ | |
| 249 | 335 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (f1=f1 f') |
| 247 | 336 ∎ |
| 337 | |
| 230 | 338 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
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339 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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340 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 341 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 342 begin |
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343 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 | 344 ≈⟨ ek=h (eqa f g) ⟩ |
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345 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 346 ∎ |
| 230 | 347 |
| 348 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
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349 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
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350 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
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351 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
| 249 | 352 k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) (id1 A d) (f1=f1 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] |
| 222 | 353 ≈ j ] |
| 230 | 354 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 355 begin |
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356 ( 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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357 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
| 249 | 358 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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359 ≈⟨ car ((uniqueness (eqa f g) ( begin |
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360 equalizer (eqa f g) o j |
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361 ≈↑⟨ idR ⟩ |
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362 (equalizer (eqa f g) o j ) o id1 A d |
| 241 | 363 ≈⟨⟩ -- here we decide e (fej) (gej)' is id1 A d |
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364 ((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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365 ∎ ))) ⟩ |
| 249 | 366 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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367 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
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368 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id1 A d |
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369 ≈⟨ idR ⟩ |
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370 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
| 241 | 371 ≈⟨⟩ -- here we decide e (fej) (fej)' is id1 A d |
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372 id1 A d |
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373 ∎ ))) ⟩ |
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374 j o id1 A d |
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375 ≈⟨ idR ⟩ |
| 222 | 376 j |
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377 ∎ |
| 211 | 378 |
| 251 | 379 -------------------------------- |
| 380 -- | |
| 381 -- Bourroni equations gives an Equalizer | |
| 382 -- | |
| 211 | 383 |
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384 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
| 245 | 385 → ( 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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386 lemma-equ2 {a} {b} {c} f g e bur = record { |
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387 fe=ge = fe=ge1 ; |
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388 k = k1 ; |
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389 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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390 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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391 } where |
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392 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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393 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
| 245 | 394 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
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395 fe=ge1 = b1 bur |
| 245 | 396 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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397 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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398 begin |
| 245 | 399 α bur f g e o k1 h eq |
| 239 | 400 ≈⟨⟩ |
| 245 | 401 α bur f g e o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
| 239 | 402 ≈⟨ assoc ⟩ |
| 245 | 403 ( α bur f g e o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 239 | 404 ≈⟨ car (b2 bur) ⟩ |
| 245 | 405 ( h o ( α bur ( f o h ) ( g o h ) (id1 A d))) o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 239 | 406 ≈↑⟨ assoc ⟩ |
| 245 | 407 h o ((( α bur ( f o h ) ( g o h ) (id1 A d) )) o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
| 240 | 408 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
| 245 | 409 h o ((( α bur ( f o h ) ( f o h ) (id1 A d)))o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
| 240 | 410 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id1 A d} ) ⟩ |
| 239 | 411 h o id1 A d |
| 240 | 412 ≈⟨ idR ⟩ |
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413 h |
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414 ∎ |
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415 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 | 416 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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417 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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418 begin |
| 240 | 419 k1 {d} h eq |
| 239 | 420 ≈⟨⟩ |
| 245 | 421 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 240 | 422 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
| 245 | 423 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 424 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ | |
| 425 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id1 A d) ( f o ( α bur f g e o k') ) | |
| 240 | 426 ≈⟨ b4 bur ⟩ |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
427 k' |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
428 ∎ |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
429 |
|
c8db99cdf72a
Burrnoi to Equalizer problem written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
237
diff
changeset
|
430 |
| 225 | 431 -- end |
| 212 | 432 |
| 433 | |
| 434 | |
| 215 | 435 |
| 436 |