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annotate src/equalizer.agda @ 1036:b836c3dc7a29
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Fri, 02 Apr 2021 11:26:44 +0900 |
| parents | 4b517d46e987 |
| children | 849f85e543f1 |
| rev | line source |
|---|---|
| 205 | 1 --- |
| 2 -- | |
| 3 -- Equalizer | |
| 4 -- | |
| 208 | 5 -- e f |
| 300 | 6 -- c -------→ a ---------→ b |
| 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 | |
| 230 | 23 -- |
| 225 | 24 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) |
| 25 -- | |
| 219 | 26 |
| 224 | 27 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 ] ] |
| 28 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 29 | |
| 226 | 30 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 | 31 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 32 |
| 224 | 33 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 34 (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 | 35 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 36 begin |
| 37 f o ( e o h ) | |
| 38 ≈⟨ assoc ⟩ | |
| 230 | 39 (f o e ) o h |
| 224 | 40 ≈⟨ car eq ⟩ |
| 230 | 41 (g o e ) o h |
| 224 | 42 ≈↑⟨ assoc ⟩ |
| 43 g o ( e o h ) | |
| 44 ∎ | |
| 219 | 45 |
| 956 | 46 -- |
| 47 -- Burroni's Flat Equational Definition of Equalizer | |
| 48 -- | |
| 49 | |
| 958 | 50 record Burroni : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 956 | 51 field |
| 52 equ : {a b : Obj A } → (f g : Hom A a b) → Obj A | |
| 53 α : {a b : Obj A } → (f g : Hom A a b) → Hom A (equ f g) a | |
| 54 γ : {a b d : Obj A } → (f g : Hom A a b) → (h : Hom A d a ) → Hom A (equ (A [ f o h ]) (A [ g o h ])) (equ f g) | |
| 55 δ : {a b : Obj A } → (f g : Hom A a b) → A [ f ≈ g ] → Hom A a (equ f g) | |
| 958 | 56 b1 : {a b : Obj A } → (f g : Hom A a b) → A [ A [ f o α f g ] ≈ A [ g o α f g ] ] |
| 57 b1k : {a b : Obj A } → (f g : Hom A a b) → {d : Obj A } {k : Hom A d (equ f g)} → A [ A [ f o A [ α f g o k ] ] ≈ A [ g o A [ α f g o k ] ] ] | |
| 58 b1k f g {d} {k} = ≈-Reasoning.trans-hom A (≈-Reasoning.assoc A) (≈-Reasoning.trans-hom A (≈-Reasoning.car A (b1 f g)) (≈-Reasoning.sym A (≈-Reasoning.assoc A))) | |
| 956 | 59 field |
| 958 | 60 b2 : {a b d : Obj A} {h : Hom A d a } → (f g : Hom A a b) → A [ A [ ( α f g ) o (γ f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] |
| 61 b3 : {a b : Obj A} (f g : Hom A a b) → (f=g : A [ f ≈ g ]) → A [ A [ α f g o δ f g f=g ] ≈ id1 A a ] | |
| 62 b4 : {a b d : Obj A} (f g : Hom A a b) → {k : Hom A d (equ f g)} → | |
| 63 A [ A [ γ f g ( A [ α f g o k ] ) o ( δ (A [ f o A [ α f g o k ] ] ) (A [ g o A [ α f g o k ] ] ) (f1=gh (b1 f g) ) )] ≈ k ] | |
| 956 | 64 β : { d a b : Obj A} → (f g : Hom A a b) → (h : Hom A d a ) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d (equ f g) |
| 65 β {d} {a} {b} f g h eq = A [ γ f g h o δ (A [ f o h ]) (A [ g o h ]) eq ] | |
| 66 | |
| 67 open Equalizer | |
| 68 open IsEqualizer | |
| 69 open Burroni | |
| 70 | |
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71 ------------------------------- |
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72 -- |
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73 -- Every equalizer is monic |
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74 -- |
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75 -- e i = e j → i = j |
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76 -- |
| 259 | 77 -- e eqa f g f |
| 300 | 78 -- c ---------→ a ------→b |
| 259 | 79 -- ^^ |
| 80 -- || | |
| 81 -- i||j | |
| 82 -- || | |
| 83 -- d | |
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84 |
| 963 | 85 monic : { a b d : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A f g) |
| 954 | 86 → { i j : Hom A d (equalizer-c eqa) } |
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87 → A [ A [ equalizer eqa o i ] ≈ A [ equalizer eqa o j ] ] → A [ i ≈ j ] |
| 963 | 88 monic {a} {b} {d} {f} {g} eqa {i} {j} ei=ej = let open ≈-Reasoning (A) in begin |
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89 i |
| 443 | 90 ≈↑⟨ uniqueness (isEqualizer eqa) ( begin |
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91 equalizer eqa o i |
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92 ≈⟨ ei=ej ⟩ |
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93 equalizer eqa o j |
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94 ∎ )⟩ |
| 443 | 95 k (isEqualizer eqa) (equalizer eqa o j) ( f1=gh (fe=ge (isEqualizer eqa) ) ) |
| 96 ≈⟨ uniqueness (isEqualizer eqa) ( begin | |
| 257 | 97 equalizer eqa o j |
| 98 ≈⟨⟩ | |
| 99 equalizer eqa o j | |
| 100 ∎ )⟩ | |
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101 j |
| 443 | 102 ∎ |
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103 |
| 251 | 104 -------------------------------- |
| 225 | 105 -- |
| 106 -- | |
| 259 | 107 -- Isomorphic arrows from c' to c makes another equalizer |
| 225 | 108 -- |
| 230 | 109 -- e eqa f g f |
| 300 | 110 -- c ---------→ a ------→b |
| 230 | 111 -- |^ |
| 112 -- || | |
| 222 | 113 -- h || h-1 |
| 230 | 114 -- v| |
| 115 -- c' | |
| 222 | 116 |
| 443 | 117 equalizer+iso : {a b c' : Obj A } {f g : Hom A a b } → |
| 118 ( eqa : Equalizer A f g ) → | |
| 119 (h-1 : Hom A c' (equalizer-c eqa) ) → (h : Hom A (equalizer-c eqa) c' ) → | |
| 120 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A (equalizer-c eqa) ] | |
| 121 → IsEqualizer A (A [ equalizer eqa o h-1 ] ) f g | |
| 122 equalizer+iso {a} {b} {c'} {f} {g} eqa h-1 h hh-1=1 h-1h=1 = record { | |
| 222 | 123 fe=ge = fe=ge1 ; |
| 443 | 124 k = λ j eq → A [ h o k (isEqualizer eqa) j eq ] ; |
| 230 | 125 ek=h = ek=h1 ; |
| 222 | 126 uniqueness = uniqueness1 |
| 127 } where | |
| 443 | 128 e = equalizer eqa |
| 234 | 129 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 443 | 130 fe=ge1 = f1=gh ( fe=ge (isEqualizer eqa) ) |
| 222 | 131 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 443 | 132 A [ A [ A [ e o h-1 ] o A [ h o k (isEqualizer eqa) j eq ] ] ≈ j ] |
| 222 | 133 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 134 begin | |
| 443 | 135 ( e o h-1 ) o ( h o k (isEqualizer eqa) j eq ) |
| 234 | 136 ≈↑⟨ assoc ⟩ |
| 443 | 137 e o ( h-1 o ( h o k (isEqualizer eqa) j eq ) ) |
| 234 | 138 ≈⟨ cdr assoc ⟩ |
| 443 | 139 e o (( h-1 o h) o k (isEqualizer eqa) j eq ) |
| 234 | 140 ≈⟨ cdr (car h-1h=1 ) ⟩ |
| 443 | 141 e o (id1 A (equalizer-c eqa) o k (isEqualizer eqa) j eq ) |
| 234 | 142 ≈⟨ cdr idL ⟩ |
| 443 | 143 e o k (isEqualizer eqa) j eq |
| 144 ≈⟨ ek=h (isEqualizer eqa) ⟩ | |
| 222 | 145 j |
| 443 | 146 ∎ |
| 222 | 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' ] → |
| 443 | 149 A [ A [ h o k (isEqualizer eqa) h' eq ] ≈ j ] |
| 222 | 150 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in |
| 151 begin | |
| 443 | 152 h o k (isEqualizer eqa) h' eq |
| 153 ≈⟨ cdr (uniqueness (isEqualizer eqa) ( begin | |
| 234 | 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 | |
| 1018 | 169 |
| 170 equalizerIso : {a b c : Obj A} → (f g : Hom A a b ) → (equ : Equalizer A f g ) | |
| 171 → (m : Hom A c a) | |
| 172 → ( ker-iso : IsoL A m (equalizer equ) ) | |
| 173 → IsEqualizer A m f g | |
| 174 equalizerIso {a} {b} {c} f g equ m ker-iso = record { | |
| 175 fe=ge = fe-ge | |
| 176 ; k = λ {d} h eq → A [ Iso.≅← (IsoL.iso-L ker-iso) o IsEqualizer.k (Equalizer.isEqualizer equ) h eq ] | |
| 177 ; ek=h = ek=h1 | |
| 178 ; uniqueness = uniqueness1 } where | |
| 179 ker : Hom A ( equalizer-c equ ) a | |
| 180 ker = equalizer equ | |
| 181 mm : A [ A [ equalizer equ o Iso.≅→ (IsoL.iso-L ker-iso ) ] ≈ m ] | |
| 182 mm = IsoL.L≈iso ker-iso | |
| 183 fe-ge : A [ A [ f o m ] ≈ A [ g o m ] ] | |
| 184 fe-ge = begin | |
| 185 f o m ≈↑⟨ cdr mm ⟩ | |
| 186 f o (equalizer equ o Iso.≅→ (IsoL.iso-L ker-iso)) ≈⟨ assoc ⟩ | |
| 187 (f o equalizer equ) o Iso.≅→ (IsoL.iso-L ker-iso) ≈⟨ car ( IsEqualizer.fe=ge (Equalizer.isEqualizer equ) ) ⟩ | |
| 188 (g o equalizer equ ) o Iso.≅→ (IsoL.iso-L ker-iso) ≈↑⟨ assoc ⟩ | |
| 189 g o (equalizer equ o Iso.≅→ (IsoL.iso-L ker-iso)) ≈⟨ cdr mm ⟩ | |
| 190 g o m ∎ where open ≈-Reasoning A | |
| 191 ek=h1 : {d : Obj A} {h : Hom A d a} | |
| 192 {eq : A [ A [ f o h ] ≈ A [ g o h ] ]} → | |
| 193 A [ A [ m o (A Category.o Iso.≅← (IsoL.iso-L ker-iso)) (IsEqualizer.k (isEqualizer equ) h eq) ] ≈ h ] | |
| 194 ek=h1 {d} {h} {eq} = begin | |
| 195 m o ( Iso.≅← (IsoL.iso-L ker-iso) o IsEqualizer.k (Equalizer.isEqualizer equ) h eq ) ≈↑⟨ car mm ⟩ | |
| 196 (equalizer equ o Iso.≅→ (IsoL.iso-L ker-iso)) o ( Iso.≅← (IsoL.iso-L ker-iso) o IsEqualizer.k (Equalizer.isEqualizer equ) h eq ) ≈↑⟨ assoc ⟩ | |
| 197 _ o (Iso.≅→ (IsoL.iso-L ker-iso) o ( Iso.≅← (IsoL.iso-L ker-iso) o IsEqualizer.k (Equalizer.isEqualizer equ) h eq )) ≈⟨ cdr assoc ⟩ | |
| 198 equalizer equ o ((Iso.≅→ (IsoL.iso-L ker-iso) o Iso.≅← (IsoL.iso-L ker-iso)) o IsEqualizer.k (Equalizer.isEqualizer equ) h eq ) ≈⟨ cdr (car (Iso.iso← (IsoL.iso-L ker-iso))) ⟩ | |
| 199 equalizer equ o (id1 A _ o IsEqualizer.k (Equalizer.isEqualizer equ) h eq ) ≈⟨ cdr idL ⟩ | |
| 200 equalizer equ o IsEqualizer.k (Equalizer.isEqualizer equ) h eq ≈⟨ IsEqualizer.ek=h (isEqualizer equ) ⟩ | |
| 201 h ∎ where open ≈-Reasoning A | |
| 202 uniqueness1 : {d : Obj A} {h : Hom A d a} | |
| 203 {eq : A [ A [ f o h ] ≈ A [ g o h ] ]} | |
| 204 {k' : Hom A d c} → A [ A [ m o k' ] ≈ h ] | |
| 205 → A [ (A Category.o Iso.≅← (IsoL.iso-L ker-iso)) (IsEqualizer.k (isEqualizer equ) h eq) ≈ k' ] | |
| 206 uniqueness1 {d} {h} {eq} {k'} eqk = begin | |
| 207 Iso.≅← (IsoL.iso-L ker-iso) o (IsEqualizer.k (isEqualizer equ) h eq) ≈⟨ cdr ( IsEqualizer.uniqueness (Equalizer.isEqualizer equ) ( begin | |
| 208 equalizer equ o ((Iso.≅→ (IsoL.iso-L ker-iso)) o k' ) ≈⟨ assoc ⟩ | |
| 209 (equalizer equ o Iso.≅→ (IsoL.iso-L ker-iso)) o k' ≈⟨ car mm ⟩ | |
| 210 m o k' ≈⟨ eqk ⟩ | |
| 211 h ∎ )) ⟩ | |
| 212 Iso.≅← (IsoL.iso-L ker-iso) o ( Iso.≅→ (IsoL.iso-L ker-iso) o k' ) ≈⟨ assoc ⟩ | |
| 213 (Iso.≅← (IsoL.iso-L ker-iso) o Iso.≅→ (IsoL.iso-L ker-iso) ) o k' ≈⟨ car (Iso.iso→ (IsoL.iso-L ker-iso) )⟩ | |
| 214 id1 A _ o k' ≈⟨ idL ⟩ | |
| 215 k' ∎ where open ≈-Reasoning A | |
| 216 mequ : Equalizer A f g | |
| 217 mequ = record { equalizer-c = c ; equalizer = m ; isEqualizer = record { | |
| 218 fe=ge = fe-ge | |
| 219 ; k = λ {d} h fh=gh → A [ Iso.≅← (IsoL.iso-L ker-iso) o IsEqualizer.k (Equalizer.isEqualizer equ) h fh=gh ] | |
| 220 ; ek=h = ek=h1 | |
| 221 ; uniqueness = uniqueness1 | |
| 222 } } | |
| 223 | |
| 251 | 224 -------------------------------- |
| 225 | 225 -- |
| 226 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 227 -- | |
| 228 -- h : c → c' h' : c' → c | |
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229 -- e' = h o e |
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230 -- e = h' o e' |
| 259 | 231 -- |
| 232 -- | |
| 233 -- | |
| 234 -- e eqa f g f | |
| 300 | 235 -- c ---------→a ------→b |
| 259 | 236 -- ^ ^ g |
| 237 -- | | | |
| 238 -- |k = id c' | | |
| 239 -- v | | |
| 240 -- c'-----------+ | |
| 241 -- e eqa' f g | |
| 225 | 242 |
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243 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 } |
| 443 | 244 ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) |
| 258 | 245 → Hom A c c' |
| 443 | 246 c-iso-l {c} {c'} {a} {b} {f} {g} {e} eqa eqa' = k eqa' e ( fe=ge eqa ) |
| 223 | 247 |
| 258 | 248 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 } |
| 443 | 249 ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) |
| 258 | 250 → Hom A c' c |
| 443 | 251 c-iso-r {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = k eqa e' ( fe=ge eqa' ) |
| 228 | 252 |
| 258 | 253 c-iso-lr : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } |
| 443 | 254 ( eqa : IsEqualizer A e f g) → ( eqa' : IsEqualizer A e' f g ) → |
| 258 | 255 A [ A [ c-iso-l eqa eqa' o c-iso-r eqa eqa' ] ≈ id1 A c' ] |
| 256 c-iso-lr {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' = let open ≈-Reasoning (A) in begin | |
| 257 c-iso-l eqa eqa' o c-iso-r eqa eqa' | |
| 250 | 258 ≈⟨⟩ |
| 443 | 259 k eqa' e ( fe=ge eqa ) o k eqa e' ( fe=ge eqa' ) |
| 258 | 260 ≈↑⟨ uniqueness eqa' ( begin |
| 443 | 261 e' o ( k eqa' e (fe=ge eqa) o k eqa e' (fe=ge eqa') ) |
| 258 | 262 ≈⟨ assoc ⟩ |
| 443 | 263 ( e' o k eqa' e (fe=ge eqa) ) o k eqa e' (fe=ge eqa') |
| 258 | 264 ≈⟨ car (ek=h eqa') ⟩ |
| 443 | 265 e o k eqa e' (fe=ge eqa') |
| 258 | 266 ≈⟨ ek=h eqa ⟩ |
| 267 e' | |
| 268 ∎ )⟩ | |
| 269 k eqa' e' ( fe=ge eqa' ) | |
| 270 ≈⟨ uniqueness eqa' ( begin | |
| 271 e' o id c' | |
| 250 | 272 ≈⟨ idR ⟩ |
| 258 | 273 e' |
| 274 ∎ )⟩ | |
| 253 | 275 id c' |
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276 ∎ |
| 226 | 277 |
| 258 | 278 -- c-iso-rl is obvious from the symmetry |
| 234 | 279 |
| 954 | 280 -- |
| 281 -- we cannot have equalizer ≈ id. we only have Iso A (equalizer-c equ) a | |
| 282 -- | |
| 283 equ-ff : {a b : Obj A} → (f : Hom A a b ) → IsEqualizer A (id1 A a) f f | |
| 284 equ-ff {a} {b} f = record { | |
| 285 fe=ge = ≈-Reasoning.refl-hom A ; | |
| 286 k = λ {d} h eq → h ; | |
| 287 ek=h = λ {d} {h} {eq} → ≈-Reasoning.idL A ; | |
| 288 uniqueness = λ {d} {h} {eq} {k'} ek=h → begin | |
| 289 h | |
| 290 ≈↑⟨ ek=h ⟩ | |
| 291 id1 A a o k' | |
| 292 ≈⟨ idL ⟩ | |
| 293 k' | |
| 294 ∎ | |
| 295 } where open ≈-Reasoning A | |
| 230 | 296 |
| 443 | 297 |
| 251 | 298 -------------------------------- |
| 225 | 299 ---- |
| 300 -- | |
| 254 | 301 -- Existence of equalizer satisfies Burroni equations |
| 225 | 302 -- |
| 303 ---- | |
| 304 | |
| 958 | 305 lemma-equ1 : ({a b : Obj A} (f g : Hom A a b) → Equalizer A f g ) → Burroni |
| 306 lemma-equ1 eqa = record { | |
| 955 | 307 equ = λ f g → equalizer-c (eqa f g) |
| 308 ; α = λ f g → equalizer (eqa f g) | |
| 309 ; γ = λ f g h → k (isEqualizer (eqa f g )) ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) | |
| 956 | 310 (lemma-equ4 f g h) |
| 311 ; δ = λ {a} {b} f g f=g → k (isEqualizer (eqa {a} {b} f g )) {a} (id1 A a) (f1=g1 f=g _ ) | |
| 958 | 312 ; b1 = λ f g → fe=ge (isEqualizer (eqa f g )) |
| 956 | 313 ; b2 = lemma-b2 |
| 958 | 314 ; b3 = λ {a } {b} f g f=g → lemma-b3 f g f=g |
| 956 | 315 ; b4 = lemma-b4 |
| 316 } where | |
| 317 ieqa : {a b : Obj A} (f g : Hom A a b) → IsEqualizer A ( equalizer (eqa f g )) f g | |
| 318 ieqa f g = isEqualizer (eqa f g) | |
| 958 | 319 lemma-b3 : {a b : Obj A} (f g : Hom A a b ) |
| 956 | 320 → (f=g : A [ f ≈ g ] ) → A [ A [ equalizer (eqa f g ) o k (isEqualizer (eqa f g)) (id1 A a) (f1=g1 f=g _ ) ] ≈ id1 A a ] |
| 321 lemma-b3 {a} f g f=g = let open ≈-Reasoning (A) in | |
| 322 begin | |
| 323 equalizer (eqa f g) o k (isEqualizer (eqa f g)) (id a) (f1=g1 f=g _ ) | |
| 324 ≈⟨ ek=h (isEqualizer (eqa f g )) ⟩ | |
| 325 id a | |
| 326 ∎ | |
| 327 lemma-equ4 : {a b d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → | |
| 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 ])) ] ] ] | |
| 329 lemma-equ4 {a} {b} {d} f g h = let open ≈-Reasoning (A) in | |
| 330 begin | |
| 331 f o ( h o equalizer (eqa (f o h) ( g o h ))) | |
| 332 ≈⟨ assoc ⟩ | |
| 333 (f o h) o equalizer (eqa (f o h) ( g o h )) | |
| 334 ≈⟨ fe=ge (isEqualizer (eqa (A [ f o h ]) (A [ g o h ]))) ⟩ | |
| 335 (g o h) o equalizer (eqa (f o h) ( g o h )) | |
| 336 ≈↑⟨ assoc ⟩ | |
| 337 g o ( h o equalizer (eqa (f o h) ( g o h ))) | |
| 338 ∎ | |
| 958 | 339 lemma-b2 : {a b d : Obj A} {h : Hom A d a} → (f g : Hom A a b) → A [ |
| 956 | 340 A [ equalizer (eqa f g) o k (isEqualizer (eqa f g)) (A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} f g h) ] |
| 341 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] | |
| 958 | 342 lemma-b2 {a} {b} {d} {h} f g = let open ≈-Reasoning (A) in |
| 956 | 343 begin |
| 344 equalizer (eqa f g) o k (isEqualizer (eqa f g)) (h o equalizer (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} f g h) | |
| 345 ≈⟨ ek=h (isEqualizer (eqa f g)) ⟩ | |
| 346 h o equalizer (eqa (f o h ) ( g o h )) | |
| 347 ∎ | |
| 958 | 348 lemma-b4 : {a b d : Obj A} (f g : Hom A a b) → {j : Hom A d (equalizer-c (eqa f g))} → A [ |
| 956 | 349 A [ k (ieqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
| 350 equalizer (eqa (A [ f o A [ equalizer (eqa f g ) o j ] ]) (A [ g o A [ equalizer (eqa f g ) o j ] ])) ]) | |
| 351 (lemma-equ4 {a} {b} {d} f g (A [ equalizer (eqa f g) o j ])) | |
| 352 o k (ieqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ g o A [ equalizer (eqa f g) o j ] ])) (id1 A _) | |
| 353 (f1=g1 (f1=gh (fe=ge (ieqa f g))) (id1 A _))] ≈ j ] | |
| 957 | 354 -- h = equalizer (eqa f g) o j |
| 958 | 355 lemma-b4 {a} {b} {d} f g {j} = |
| 956 | 356 begin |
| 958 | 357 k (ieqa f g) ( h o equalizer (eqa ( f o h ) ( g o h )) ) (lemma-equ4 {a} {b} {d} f g h) |
| 957 | 358 o k (ieqa (f o h) ( g o h)) (id1 A _) (f1=g1 (f1=gh (fe=ge (ieqa f g))) (id1 A _)) |
| 359 ≈↑⟨ uniqueness (ieqa f g) ( begin | |
| 360 equalizer (eqa f g) o ( k (ieqa f g) (( h o equalizer (eqa ( f o h ) ( g o h )) )) (lemma-equ4 {a} {b} {d} f g h) | |
| 361 o k (ieqa (f o h) ( g o h)) (id1 A _) (f1=g1 (f1=gh (fe=ge (ieqa f g))) (id1 A _)) ) | |
| 362 ≈⟨ assoc ⟩ | |
| 363 (equalizer (eqa f g) o ( k (ieqa f g) (( h o equalizer (eqa ( f o h ) ( g o h )) )) (lemma-equ4 {a} {b} {d} f g h))) | |
| 364 o k (ieqa (f o h) ( g o h)) (id1 A _) (f1=g1 (f1=gh (fe=ge (ieqa f g))) (id1 A _)) | |
| 365 ≈⟨ car (ek=h (ieqa f g) ) ⟩ | |
| 366 (( h o equalizer (eqa ( f o h ) ( g o h )) )) | |
| 367 o k (ieqa (f o h) ( g o h)) (id1 A _) (f1=g1 (f1=gh (fe=ge (ieqa f g))) (id1 A _)) | |
| 368 ≈↑⟨ assoc ⟩ | |
| 369 h o (equalizer (eqa ( f o h ) ( g o h )) o k (ieqa (f o h) ( g o h)) (id1 A _) (f1=g1 (f1=gh (fe=ge (ieqa f g))) (id1 A _))) | |
| 370 ≈⟨ cdr (ek=h (ieqa (f o h) ( g o h))) ⟩ | |
| 371 h o id1 A _ | |
| 372 ≈⟨ idR ⟩ | |
| 373 h | |
| 374 ∎ | |
| 375 ) ⟩ | |
| 376 k (ieqa f g) h (f1=gh (fe=ge (ieqa f g)) ) | |
| 377 ≈⟨ uniqueness (ieqa f g) refl-hom ⟩ | |
| 956 | 378 j |
| 957 | 379 ∎ where |
| 380 open ≈-Reasoning A | |
| 381 h : Hom A d a | |
| 382 h = equalizer (eqa f g) o j | |
| 1018 | 383 -- cong-γ1 : {a b c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → |
| 384 -- A [ k (ieqa f g ) {_} ( A [ h o (equalizer1 ( ieqa (A [ f o h ] ) (A [ g o h ] ) )) ] ) (lemma-equ4 {a} {b} {d} f g h ) | |
| 385 -- ≈ A [ k (ieqa f g ) {_} ( A [ h' o (equalizer1 ( ieqa (A [ f o h' ] ) (A [ g o h' ] ) )) ] ) (lemma-equ4 {a} {b} {d} f g h' ) o {!!} ] ] | |
| 386 -- cong-γ1 {a} {b} {c} {d} {f} {g} {h} {h'} h=h' = let open ≈-Reasoning (A) in begin | |
| 387 -- k (ieqa f g ) {_} ( A [ h o (equalizer1 ( ieqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {d} f g h ) | |
| 388 -- ≈⟨ uniqueness (ieqa f g) {!!} ⟩ | |
| 389 -- {!!} -- k (ieqa f g ) {_} ( A [ h' o (equalizer1 ( ieqa (A [ f o h' ] ) (A [ g o h' ] ))) ] ) (lemma-equ4 {a} {b} {d} f g h' ) | |
| 390 -- ∎ | |
| 391 -- cong-δ1 : {a b c : Obj A} {e : Hom A c a } {f f' : Hom A a b} → A [ f ≈ f' ] → A [ k (ieqa f f ) (id1 A a) (f1=f1 f) ≈ | |
| 392 -- A [ {!!} o k (ieqa f' f' ) (id1 A a) (f1=f1 f') ] ] | |
| 393 -- cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in | |
| 394 -- begin | |
| 395 -- k (ieqa f f ) (id a) (f1=f1 f) | |
| 396 -- ≈⟨ uniqueness (ieqa f f) {!!} ⟩ | |
| 397 -- {!!} -- k (ieqa f' f' ) (id a) (f1=f1 f') | |
| 398 -- ∎ | |
| 399 | |
| 956 | 400 |
| 211 | 401 |
| 251 | 402 -------------------------------- |
| 403 -- | |
| 404 -- Bourroni equations gives an Equalizer | |
| 405 -- | |
| 211 | 406 |
| 963 | 407 lemma-equ2 : {a b : Obj A} (f g : Hom A a b) → ( bur : Burroni ) → IsEqualizer A {equ bur f g} {a} {b} (α bur f g ) f g |
| 962 | 408 lemma-equ2 {a} {b} f g bur = record { |
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409 fe=ge = fe=ge1 ; |
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410 k = k1 ; |
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411 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
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412 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
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413 } where |
| 955 | 414 c : Obj A |
| 415 c = equ bur f g | |
| 416 e : Hom A c a | |
| 417 e = α bur f g | |
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418 k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 955 | 419 k1 {d} h fh=gh = β bur {d} {a} {b} f g h fh=gh |
| 420 fe=ge1 : A [ A [ f o (α bur f g ) ] ≈ A [ g o (α bur f g ) ] ] | |
| 958 | 421 fe=ge1 = b1 bur f g |
| 955 | 422 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 ) o k1 {d} h eq ] ≈ h ] |
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423 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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424 begin |
| 955 | 425 α bur f g o k1 h eq |
| 958 | 426 ≈⟨ assoc ⟩ |
| 427 (α bur f g o γ bur f g h) o δ bur (f o h) (g o h) eq | |
| 428 ≈⟨ car (b2 bur f g) ⟩ | |
| 429 ( h o α bur ( f o h ) ( g o h ) ) o δ bur (f o h) (g o h) eq | |
| 430 ≈↑⟨ assoc ⟩ | |
| 431 h o α bur (f o h) (g o h) o δ bur (f o h) (g o h) eq | |
| 432 ≈⟨ cdr ( b3 bur (f o h) (g o h) eq ) ⟩ | |
| 253 | 433 h o id d |
| 240 | 434 ≈⟨ idR ⟩ |
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435 h |
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436 ∎ |
| 961 | 437 |
| 438 -- e f | |
| 439 -- c -------→ a ---------→ b | |
| 440 -- ^ . ---------→ | |
| 441 -- | . g | |
| 442 -- |k . | |
| 443 -- | . h | |
| 444 -- d | |
| 445 | |
| 962 | 446 postulate |
| 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 } → | |
| 448 A [ A [ (α bur f g ) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] | |
| 449 -- uniqueness1 {d} {h} {eq} {k'} ek=h = | |
| 450 -- begin | |
| 451 -- k1 {d} h eq | |
| 452 -- ≈⟨⟩ | |
| 453 -- γ bur f g h o δ bur (f o h) (g o h) eq | |
| 454 -- ≈⟨ ? ⟩ -- without locality, we cannot simply replace h with (α bur f g o k' | |
| 455 -- γ bur f g (α bur f g o k' ) o (δ bur ( f o ( α bur f g o k' )) ( g o ( α bur f g o k' )) (f1=gh (b1 bur f g ))) | |
| 456 -- ≈⟨ b4 bur f g ⟩ | |
| 457 -- k' | |
| 458 -- ∎ | |
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459 |
| 225 | 460 -- end |
| 212 | 461 |
| 462 | |
| 463 | |
| 215 | 464 |
| 465 |