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annotate equalizer.agda @ 230:1ef8c70c7054
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 07 Sep 2013 18:56:46 +0900 |
| parents | 68b2681ea9df |
| children | 1dc1c697145f |
| 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 | |
| 209 | 23 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 205 | 24 field |
| 230 | 25 e : Hom A c a |
| 221 | 26 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] |
| 209 | 27 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 215 | 28 ek=h : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ e o k {d} h eq ] ≈ h ] |
| 230 | 29 uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
| 214 | 30 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
| 209 | 31 equalizer : Hom A c a |
| 32 equalizer = e | |
| 206 | 33 |
| 230 | 34 -- |
| 225 | 35 -- Flat Equational Definition of Equalizer |
| 230 | 36 -- |
| 225 | 37 record Burroni { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 206 | 38 field |
| 212 | 39 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → 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 |
| 230 | 41 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
| 213 | 42 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
| 214 | 43 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α f g) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] |
| 213 | 44 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
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45 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
| 230 | 46 b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ k ] |
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47 -- A [ α f g o β f g h ] ≈ h |
| 214 | 48 β : { d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
| 230 | 49 β {d} {e} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ (A [ f o h ]) ] |
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50 |
| 209 | 51 open Equalizer |
| 227 | 52 |
| 225 | 53 open Burroni |
| 209 | 54 |
| 225 | 55 -- |
| 56 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 57 -- | |
| 219 | 58 |
| 224 | 59 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 ] ] |
| 60 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 61 | |
| 226 | 62 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 | 63 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 64 |
| 224 | 65 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 66 (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 | 67 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 68 begin |
| 69 f o ( e o h ) | |
| 70 ≈⟨ assoc ⟩ | |
| 230 | 71 (f o e ) o h |
| 224 | 72 ≈⟨ car eq ⟩ |
| 230 | 73 (g o e ) o h |
| 224 | 74 ≈↑⟨ assoc ⟩ |
| 75 g o ( e o h ) | |
| 76 ∎ | |
| 219 | 77 |
| 225 | 78 -- |
| 79 -- For e f f, we need e eqa = id1 A a, but it is equal to say k eqa (id a) is id | |
| 80 -- | |
| 81 -- Equalizer has free choice of c up to isomorphism, we cannot prove eqa = id a | |
| 219 | 82 |
| 230 | 83 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → |
| 219 | 84 A [ e eqa ≈ id1 A a ] → |
| 230 | 85 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] |
| 219 | 86 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in |
| 87 begin | |
| 230 | 88 k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 219 | 89 ≈⟨ uniqueness eqa ( begin |
| 90 e eqa o id1 A a | |
| 91 ≈⟨ idR ⟩ | |
| 230 | 92 e eqa |
| 219 | 93 ≈⟨ e=1 ⟩ |
| 94 id1 A a | |
| 95 ∎ )⟩ | |
| 96 id1 A a | |
| 97 ∎ | |
| 98 | |
| 230 | 99 equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) → |
| 224 | 100 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] → |
| 230 | 101 A [ e eqa ≈ id1 A a ] |
| 222 | 102 equalizer-eq-e {a} {b} {f} {g} eqa eq k=1 = let open ≈-Reasoning (A) in |
| 103 begin | |
| 230 | 104 e eqa |
| 222 | 105 ≈↑⟨ idR ⟩ |
| 106 e eqa o id1 A a | |
| 107 ≈↑⟨ cdr k=1 ⟩ | |
| 224 | 108 e eqa o k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 222 | 109 ≈⟨ ek=h eqa ⟩ |
| 110 id1 A a | |
| 111 ∎ | |
| 112 | |
| 225 | 113 -- |
| 114 -- | |
| 115 -- An isomorphic element c' of c makes another equalizer | |
| 116 -- | |
| 230 | 117 -- e eqa f g f |
| 222 | 118 -- c ----------> a ------->b |
| 230 | 119 -- |^ |
| 120 -- || | |
| 222 | 121 -- h || h-1 |
| 230 | 122 -- v| |
| 123 -- c' | |
| 222 | 124 |
| 125 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) → (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 126 A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] | |
| 230 | 127 → Equalizer A {c'} f g |
| 222 | 128 equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { |
| 129 e = A [ e eqa o h-1 ] ; | |
| 130 fe=ge = fe=ge1 ; | |
| 131 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 132 ek=h = ek=h1 ; |
| 222 | 133 uniqueness = uniqueness1 |
| 134 } where | |
| 135 fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] | |
| 230 | 136 fe=ge1 = let open ≈-Reasoning (A) in |
| 222 | 137 begin |
| 138 f o ( e eqa o h-1 ) | |
| 139 ≈⟨ assoc ⟩ | |
| 140 (f o e eqa ) o h-1 | |
| 141 ≈⟨ car (fe=ge eqa) ⟩ | |
| 142 (g o e eqa ) o h-1 | |
| 143 ≈↑⟨ assoc ⟩ | |
| 144 g o ( e eqa o h-1 ) | |
| 145 ∎ | |
| 146 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → | |
| 147 A [ A [ A [ e eqa o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] | |
| 148 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in | |
| 149 begin | |
| 150 (e eqa o h-1 ) o ( h o k eqa j eq ) | |
| 151 ≈↑⟨ assoc ⟩ | |
| 152 e eqa o ( h-1 o ( h o k eqa j eq )) | |
| 153 ≈⟨ cdr assoc ⟩ | |
| 154 e eqa o (( h-1 o h ) o k eqa j eq ) | |
| 155 ≈⟨ cdr (car (h-1-id )) ⟩ | |
| 156 e eqa o (id1 A c o k eqa j eq ) | |
| 157 ≈⟨ cdr idL ⟩ | |
| 158 e eqa o (k eqa j eq ) | |
| 159 ≈⟨ ek=h eqa ⟩ | |
| 160 j | |
| 161 ∎ | |
| 162 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 163 A [ A [ A [ e eqa o h-1 ] o j ] ≈ h' ] → | |
| 164 A [ A [ h o k eqa h' eq ] ≈ j ] | |
| 165 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 166 begin | |
| 167 h o k eqa h' eq | |
| 168 ≈⟨ cdr (uniqueness eqa ( | |
| 169 begin | |
| 170 e eqa o ( h-1 o j ) | |
| 171 ≈⟨ assoc ⟩ | |
| 230 | 172 (e eqa o h-1 ) o j |
| 222 | 173 ≈⟨ ej=h ⟩ |
| 174 h' | |
| 175 ∎ | |
| 176 )) ⟩ | |
| 177 h o ( h-1 o j ) | |
| 178 ≈⟨ assoc ⟩ | |
| 230 | 179 (h o h-1 ) o j |
| 222 | 180 ≈⟨ car h-id ⟩ |
| 230 | 181 id1 A c' o j |
| 222 | 182 ≈⟨ idL ⟩ |
| 183 j | |
| 184 ∎ | |
| 185 | |
| 225 | 186 -- If we have equalizer f g, e fh gh is also equalizer if we have isomorphic pair (same as above) |
| 187 -- | |
| 217 | 188 -- e eqa f g f |
| 189 -- c ----------> a ------->b | |
| 230 | 190 -- ^ ---> d ---> |
| 218 | 191 -- | i h |
| 192 -- j| k' (d' → d) | |
| 193 -- | k (d' → a) | |
| 230 | 194 -- d' |
| 217 | 195 |
| 230 | 196 equalizer+h : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) (i : Hom A c d ) → (h : Hom A d a ) → (h-1 : Hom A a d ) |
| 218 | 197 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] |
| 230 | 198 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
| 218 | 199 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
| 200 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 221 | 201 fe=ge = fe=ge1 ; |
| 217 | 202 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 230 | 203 ek=h = ek=h1 ; |
| 217 | 204 uniqueness = uniqueness1 |
| 205 } where | |
| 230 | 206 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → |
| 217 | 207 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → |
| 230 | 208 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] |
| 217 | 209 fhj=ghj j eq' = let open ≈-Reasoning (A) in |
| 210 begin | |
| 211 f o ( h o j ) | |
| 212 ≈⟨ assoc ⟩ | |
| 230 | 213 (f o h ) o j |
| 217 | 214 ≈⟨ eq' ⟩ |
| 230 | 215 (g o h ) o j |
| 217 | 216 ≈↑⟨ assoc ⟩ |
| 217 g o ( h o j ) | |
| 218 ∎ | |
| 221 | 219 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 230 | 220 fe=ge1 = let open ≈-Reasoning (A) in |
| 217 | 221 begin |
| 222 ( f o h ) o i | |
| 223 ≈↑⟨ assoc ⟩ | |
| 224 f o (h o i ) | |
| 225 ≈⟨ cdr eq ⟩ | |
| 226 f o (e eqa) | |
| 221 | 227 ≈⟨ fe=ge eqa ⟩ |
| 217 | 228 g o (e eqa) |
| 229 ≈↑⟨ cdr eq ⟩ | |
| 230 g o (h o i ) | |
| 231 ≈⟨ assoc ⟩ | |
| 232 ( g o h ) o i | |
| 233 ∎ | |
| 218 | 234 ek=h1 : {d' : Obj A} {k' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o k' ] ≈ A [ A [ g o h ] o k' ] ]} → |
| 235 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 236 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 217 | 237 begin |
| 218 | 238 i o k eqa (h o k' ) (fhj=ghj k' eq') -- h-1 (h o i ) o k eqa (h o k' ) = h-1 (h o k') |
| 239 ≈↑⟨ idL ⟩ | |
| 230 | 240 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) |
| 218 | 241 ≈↑⟨ car h-1-id ⟩ |
| 230 | 242 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) |
| 218 | 243 ≈↑⟨ assoc ⟩ |
| 244 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
| 245 ≈⟨ cdr assoc ⟩ | |
| 246 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 247 ≈⟨ cdr (car eq ) ⟩ | |
| 248 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 249 ≈⟨ cdr (ek=h eqa) ⟩ | |
| 250 h-1 o ( h o k' ) | |
| 251 ≈⟨ assoc ⟩ | |
| 230 | 252 ( h-1 o h ) o k' |
| 218 | 253 ≈⟨ car h-1-id ⟩ |
| 230 | 254 id1 A d o k' |
| 218 | 255 ≈⟨ idL ⟩ |
| 256 k' | |
| 217 | 257 ∎ |
| 258 uniqueness1 : {d' : Obj A} {h' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} {k' : Hom A d' c} → | |
| 259 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 260 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 261 begin | |
| 262 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 263 ≈⟨ uniqueness eqa ( begin | |
| 264 e eqa o k' | |
| 265 ≈↑⟨ car eq ⟩ | |
| 266 (h o i ) o k' | |
| 267 ≈↑⟨ assoc ⟩ | |
| 268 h o (i o k') | |
| 269 ≈⟨ cdr ik=h ⟩ | |
| 230 | 270 h o h' |
| 217 | 271 ∎ ) ⟩ |
| 272 k' | |
| 273 ∎ | |
| 215 | 274 |
| 230 | 275 -- If we have equalizer f g, e hf hg is also equalizer if we have isomorphic pair |
| 225 | 276 |
| 230 | 277 h+equalizer : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) (h : Hom A b d ) |
| 218 | 278 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] |
| 230 | 279 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) |
| 218 | 280 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { |
| 230 | 281 e = e eqa ; |
| 282 fe=ge = fe=ge1 ; | |
| 218 | 283 k = λ j eq' → k eqa j (fj=gj j eq') ; |
| 230 | 284 ek=h = ek=h1 ; |
| 218 | 285 uniqueness = uniqueness1 |
| 286 } where | |
| 287 fj=gj : {e : Obj A} → (j : Hom A e a ) → A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ] → A [ A [ f o j ] ≈ A [ g o j ] ] | |
| 288 fj=gj j eq = let open ≈-Reasoning (A) in | |
| 289 begin | |
| 290 f o j | |
| 291 ≈↑⟨ idL ⟩ | |
| 292 id1 A b o ( f o j ) | |
| 293 ≈↑⟨ car h-1-id ⟩ | |
| 294 (h-1 o h ) o ( f o j ) | |
| 295 ≈↑⟨ assoc ⟩ | |
| 296 h-1 o (h o ( f o j )) | |
| 297 ≈⟨ cdr assoc ⟩ | |
| 298 h-1 o ((h o f) o j ) | |
| 299 ≈⟨ cdr eq ⟩ | |
| 300 h-1 o ((h o g) o j ) | |
| 301 ≈↑⟨ cdr assoc ⟩ | |
| 302 h-1 o (h o ( g o j )) | |
| 303 ≈⟨ assoc ⟩ | |
| 304 (h-1 o h) o ( g o j ) | |
| 305 ≈⟨ car h-1-id ⟩ | |
| 306 id1 A b o ( g o j ) | |
| 307 ≈⟨ idL ⟩ | |
| 308 g o j | |
| 309 ∎ | |
| 221 | 310 fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] |
| 311 fe=ge1 = let open ≈-Reasoning (A) in | |
| 218 | 312 begin |
| 313 ( h o f ) o e eqa | |
| 314 ≈↑⟨ assoc ⟩ | |
| 315 h o (f o e eqa ) | |
| 221 | 316 ≈⟨ cdr (fe=ge eqa) ⟩ |
| 218 | 317 h o (g o e eqa ) |
| 318 ≈⟨ assoc ⟩ | |
| 230 | 319 ( h o g ) o e eqa |
| 218 | 320 ∎ |
| 321 ek=h1 : {d₁ : Obj A} {j : Hom A d₁ a} {eq : A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ]} → | |
| 322 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
| 323 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
| 324 uniqueness1 : {d₁ : Obj A} {j : Hom A d₁ a} {eq : A [ A [ A [ h o f ] o j ] ≈ A [ A [ h o g ] o j ] ]} {k' : Hom A d₁ c} → | |
| 325 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
| 326 uniqueness1 = uniqueness eqa | |
| 230 | 327 |
| 225 | 328 -- If we have equalizer f g, e (ef) (eg) is also an equalizer and e = id c |
| 329 | |
| 230 | 330 eefeg : {a b c : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
| 331 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
| 227 | 332 eefeg {a} {b} {c} {f} {g} eqa = record { |
| 220 | 333 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d |
| 221 | 334 fe=ge = fe=ge1 ; |
| 220 | 335 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 230 | 336 ek=h = ek=h1 ; |
| 220 | 337 uniqueness = uniqueness1 |
| 338 } where | |
| 339 i = id1 A c | |
| 340 h = e eqa | |
| 230 | 341 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → |
| 220 | 342 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → |
| 230 | 343 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] |
| 220 | 344 fhj=ghj j eq' = let open ≈-Reasoning (A) in |
| 345 begin | |
| 346 f o ( h o j ) | |
| 347 ≈⟨ assoc ⟩ | |
| 230 | 348 (f o h ) o j |
| 220 | 349 ≈⟨ eq' ⟩ |
| 230 | 350 (g o h ) o j |
| 220 | 351 ≈↑⟨ assoc ⟩ |
| 352 g o ( h o j ) | |
| 353 ∎ | |
| 221 | 354 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 230 | 355 fe=ge1 = let open ≈-Reasoning (A) in |
| 220 | 356 begin |
| 357 ( f o h ) o i | |
| 221 | 358 ≈⟨ car ( fe=ge eqa ) ⟩ |
| 220 | 359 ( g o h ) o i |
| 360 ∎ | |
| 361 ek=h1 : {d' : Obj A} {k' : Hom A d' c} {eq' : A [ A [ A [ f o h ] o k' ] ≈ A [ A [ g o h ] o k' ] ]} → | |
| 362 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 363 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 364 begin | |
| 365 i o k eqa (h o k' ) (fhj=ghj k' eq') -- h-1 (h o i ) o k eqa (h o k' ) = h-1 (h o k') | |
| 366 ≈⟨ idL ⟩ | |
| 367 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
| 368 ≈⟨ uniqueness eqa refl-hom ⟩ | |
| 369 k' | |
| 370 ∎ | |
| 371 uniqueness1 : {d' : Obj A} {h' : Hom A d' c} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} {k' : Hom A d' c} → | |
| 372 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 373 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 374 begin | |
| 375 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
| 376 ≈⟨ uniqueness eqa ( begin | |
| 377 e eqa o k' | |
| 378 ≈↑⟨ cdr idL ⟩ | |
| 379 e eqa o (id1 A c o k' ) | |
| 380 ≈⟨ cdr ik=h ⟩ | |
| 230 | 381 e eqa o h' |
| 220 | 382 ∎ ) ⟩ |
| 383 k' | |
| 384 ∎ | |
| 385 | |
| 225 | 386 -- |
| 387 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 388 -- | |
| 389 -- h : c → c' h' : c' → c | |
| 226 | 390 -- h h' = 1 and h' h = 1 |
| 225 | 391 -- not yet done |
| 392 | |
| 393 | |
| 230 | 394 c-iso-l : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) |
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229
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395 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) |
| 230 | 396 → Hom A c c' |
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397 c-iso-l {c} {c'} eqa eqa' keqa = e keqa |
| 226 | 398 |
| 230 | 399 c-iso-r : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) |
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229
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400 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) |
| 230 | 401 → Hom A c' c |
| 402 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 223 | 403 |
| 227 | 404 |
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405 -- e(eqa') f |
| 230 | 406 -- c'----------> a ------->b f e j = g e j |
| 407 -- ^ g | |
| 408 -- |k h | |
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409 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 410 -- | |
| 411 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 412 -- | |
| 413 -- h j e f = h j e g → h = 'j e f | |
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414 -- h = j e f -> j = j' |
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415 -- |
| 228 | 416 |
| 230 | 417 c-iso : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) |
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418 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) |
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419 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
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420 c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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421 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
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422 ≈⟨ ek=h keqa ⟩ |
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423 id1 A c' |
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424 ∎ |
| 226 | 425 |
| 230 | 426 -- To prove c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
| 221 | 427 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 |
| 428 -- ke = e' | |
| 429 -- k'ke = k'e' = e kk' = 1 | |
| 430 -- x e = e -> x = id? | |
| 218 | 431 |
| 230 | 432 ----- |
| 433 -- reverse arrow of e (eqa f g) | |
| 434 -- | |
| 435 -- e eqa f g f | |
| 436 -- c ----------> a ------->b | |
| 437 -- <--------- | |
| 438 -- | |
| 439 reverse-e : {a b c : Obj A} (f g : Hom A a b) → | |
| 440 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → | |
| 441 A [ A [ k (eqa (A [ f o e (eqa f g) ]) (A [ g o e (eqa f g) ]) ) (id1 A a) | |
| 442 (f1=g1 (fe=ge (eqa f g) ) (id1 A a) ) o e (eqa f g ) ] ≈ id1 A c ] | |
| 443 reverse-e {a} {b} {c} f g eqa = ? | |
| 444 | |
| 225 | 445 ---- |
| 446 -- | |
| 447 -- An equalizer satisfies Burroni equations | |
| 448 -- | |
| 230 | 449 -- b4 is not yet done |
| 225 | 450 ---- |
| 451 | |
| 230 | 452 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → |
| 225 | 453 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → Burroni A {c} f g |
| 222 | 454 lemma-equ1 {a} {b} {c} f g eqa = record { |
| 216 | 455 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 456 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (e ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d |
| 213 | 457 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 221 | 458 b1 = fe=ge (eqa f g) ; |
| 226 | 459 b2 = lemma-b2 ; |
| 460 b3 = lemma-b3 ; | |
| 230 | 461 b4 = lemma-b4 |
| 211 | 462 } where |
| 216 | 463 -- |
| 464 -- e eqa f g f | |
| 465 -- c ----------> a ------->b | |
| 230 | 466 -- ^ g |
| 467 -- | | |
| 216 | 468 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 469 -- | |
| 216 | 470 -- d |
| 230 | 471 -- |
| 472 -- | |
| 216 | 473 -- e o id1 ≈ e → k e ≈ id |
| 474 | |
| 211 | 475 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 476 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 226 | 477 lemma-b3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 478 lemma-b3 = let open ≈-Reasoning (A) in | |
| 230 | 479 begin |
| 211 | 480 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
| 215 | 481 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 482 id1 A a |
| 483 ∎ | |
| 230 | 484 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 485 A [ A [ f o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
| 214 | 486 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 487 begin |
| 488 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 489 ≈⟨ assoc ⟩ | |
| 490 (f o h) o e (eqa (f o h) ( g o h )) | |
| 221 | 491 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
| 212 | 492 (g o h) o e (eqa (f o h) ( g o h )) |
| 493 ≈↑⟨ assoc ⟩ | |
| 494 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 495 ∎ | |
| 230 | 496 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
| 214 | 497 A [ e (eqa f g) o k (eqa f g) (A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
| 212 | 498 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 499 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 500 begin |
| 230 | 501 e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
| 215 | 502 ≈⟨ ek=h (eqa f g) ⟩ |
| 212 | 503 h o e (eqa (f o h ) ( g o h )) |
| 504 ∎ | |
| 230 | 505 |
| 506 ------- α(f,g)j id d = α(f,g)j | |
| 507 ------- α(f,g)j id d = α(f,g)j | |
| 508 ------- α(f,g)j α(fα(f,g)j,fα(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 509 ------ fα = gα | |
| 510 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 511 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 512 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j | |
| 513 | |
| 514 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
| 222 | 515 A [ k (eqa f g) (A [ A [ e (eqa f g) o j ] o e (eqa (A [ f o A [ e (eqa f g) o j ] ]) (A [ g o A [ e (eqa f g) o j ] ])) ]) |
| 230 | 516 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o |
| 222 | 517 k (eqa (A [ f o A [ e (eqa f g) o j ] ]) (A [ f o A [ e (eqa f g) o j ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o j ] ])) ] |
| 518 ≈ j ] | |
| 230 | 519 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 520 begin |
| 222 | 521 ( k (eqa f g) (( ( e (eqa f g) o j ) o e (eqa (( f o ( e (eqa f g) o j ) )) (( g o ( e (eqa f g) o j ) ))) )) |
| 522 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o | |
| 523 k (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o j ) ))) ) | |
| 230 | 524 ≈⟨ {!!} ⟩ |
| 222 | 525 j |
| 215 | 526 ∎ |
| 211 | 527 |
| 528 | |
| 225 | 529 -- end |
| 212 | 530 |
| 531 | |
| 532 | |
| 215 | 533 |
| 534 |