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annotate equalizer.agda @ 232:b0fe61882014
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 07 Sep 2013 23:29:13 +0900 |
| parents | 1dc1c697145f |
| children | 4bba19bc71be |
| 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 |
| 225 | 52 open Burroni |
| 209 | 53 |
| 225 | 54 -- |
| 55 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 56 -- | |
| 219 | 57 |
| 224 | 58 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 ] ] |
| 59 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 60 | |
| 226 | 61 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 | 62 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 63 |
| 224 | 64 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 65 (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 | 66 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 67 begin |
| 68 f o ( e o h ) | |
| 69 ≈⟨ assoc ⟩ | |
| 230 | 70 (f o e ) o h |
| 224 | 71 ≈⟨ car eq ⟩ |
| 230 | 72 (g o e ) o h |
| 224 | 73 ≈↑⟨ assoc ⟩ |
| 74 g o ( e o h ) | |
| 75 ∎ | |
| 219 | 76 |
| 225 | 77 -- |
| 78 -- For e f f, we need e eqa = id1 A a, but it is equal to say k eqa (id a) is id | |
| 79 -- | |
| 80 -- Equalizer has free choice of c up to isomorphism, we cannot prove eqa = id a | |
| 219 | 81 |
| 230 | 82 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → |
| 219 | 83 A [ e eqa ≈ id1 A a ] → |
| 230 | 84 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] |
| 219 | 85 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in |
| 86 begin | |
| 230 | 87 k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 219 | 88 ≈⟨ uniqueness eqa ( begin |
| 89 e eqa o id1 A a | |
| 90 ≈⟨ idR ⟩ | |
| 230 | 91 e eqa |
| 219 | 92 ≈⟨ e=1 ⟩ |
| 93 id1 A a | |
| 94 ∎ )⟩ | |
| 95 id1 A a | |
| 96 ∎ | |
| 97 | |
| 230 | 98 equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) → |
| 224 | 99 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] → |
| 230 | 100 A [ e eqa ≈ id1 A a ] |
| 222 | 101 equalizer-eq-e {a} {b} {f} {g} eqa eq k=1 = let open ≈-Reasoning (A) in |
| 102 begin | |
| 230 | 103 e eqa |
| 222 | 104 ≈↑⟨ idR ⟩ |
| 105 e eqa o id1 A a | |
| 106 ≈↑⟨ cdr k=1 ⟩ | |
| 224 | 107 e eqa o k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 222 | 108 ≈⟨ ek=h eqa ⟩ |
| 109 id1 A a | |
| 110 ∎ | |
| 111 | |
| 225 | 112 -- |
| 113 -- | |
| 114 -- An isomorphic element c' of c makes another equalizer | |
| 115 -- | |
| 230 | 116 -- e eqa f g f |
| 222 | 117 -- c ----------> a ------->b |
| 230 | 118 -- |^ |
| 119 -- || | |
| 222 | 120 -- h || h-1 |
| 230 | 121 -- v| |
| 122 -- c' | |
| 222 | 123 |
| 124 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' ) → | |
| 125 A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] | |
| 230 | 126 → Equalizer A {c'} f g |
| 222 | 127 equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { |
| 128 e = A [ e eqa o h-1 ] ; | |
| 129 fe=ge = fe=ge1 ; | |
| 130 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 131 ek=h = ek=h1 ; |
| 222 | 132 uniqueness = uniqueness1 |
| 133 } where | |
| 134 fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] | |
| 230 | 135 fe=ge1 = let open ≈-Reasoning (A) in |
| 222 | 136 begin |
| 137 f o ( e eqa o h-1 ) | |
| 138 ≈⟨ assoc ⟩ | |
| 139 (f o e eqa ) o h-1 | |
| 140 ≈⟨ car (fe=ge eqa) ⟩ | |
| 141 (g o e eqa ) o h-1 | |
| 142 ≈↑⟨ assoc ⟩ | |
| 143 g o ( e eqa o h-1 ) | |
| 144 ∎ | |
| 145 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → | |
| 146 A [ A [ A [ e eqa o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] | |
| 147 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in | |
| 148 begin | |
| 149 (e eqa o h-1 ) o ( h o k eqa j eq ) | |
| 150 ≈↑⟨ assoc ⟩ | |
| 151 e eqa o ( h-1 o ( h o k eqa j eq )) | |
| 152 ≈⟨ cdr assoc ⟩ | |
| 153 e eqa o (( h-1 o h ) o k eqa j eq ) | |
| 154 ≈⟨ cdr (car (h-1-id )) ⟩ | |
| 155 e eqa o (id1 A c o k eqa j eq ) | |
| 156 ≈⟨ cdr idL ⟩ | |
| 157 e eqa o (k eqa j eq ) | |
| 158 ≈⟨ ek=h eqa ⟩ | |
| 159 j | |
| 160 ∎ | |
| 161 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 162 A [ A [ A [ e eqa o h-1 ] o j ] ≈ h' ] → | |
| 163 A [ A [ h o k eqa h' eq ] ≈ j ] | |
| 164 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 165 begin | |
| 166 h o k eqa h' eq | |
| 167 ≈⟨ cdr (uniqueness eqa ( | |
| 168 begin | |
| 169 e eqa o ( h-1 o j ) | |
| 170 ≈⟨ assoc ⟩ | |
| 230 | 171 (e eqa o h-1 ) o j |
| 222 | 172 ≈⟨ ej=h ⟩ |
| 173 h' | |
| 174 ∎ | |
| 175 )) ⟩ | |
| 176 h o ( h-1 o j ) | |
| 177 ≈⟨ assoc ⟩ | |
| 230 | 178 (h o h-1 ) o j |
| 222 | 179 ≈⟨ car h-id ⟩ |
| 230 | 180 id1 A c' o j |
| 222 | 181 ≈⟨ idL ⟩ |
| 182 j | |
| 183 ∎ | |
| 184 | |
| 225 | 185 -- If we have equalizer f g, e fh gh is also equalizer if we have isomorphic pair (same as above) |
| 186 -- | |
| 217 | 187 -- e eqa f g f |
| 188 -- c ----------> a ------->b | |
| 230 | 189 -- ^ ---> d ---> |
| 218 | 190 -- | i h |
| 191 -- j| k' (d' → d) | |
| 192 -- | k (d' → a) | |
| 230 | 193 -- d' |
| 217 | 194 |
| 230 | 195 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 | 196 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] |
| 230 | 197 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
| 218 | 198 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
| 199 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 221 | 200 fe=ge = fe=ge1 ; |
| 217 | 201 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 230 | 202 ek=h = ek=h1 ; |
| 217 | 203 uniqueness = uniqueness1 |
| 204 } where | |
| 230 | 205 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → |
| 217 | 206 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → |
| 230 | 207 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] |
| 217 | 208 fhj=ghj j eq' = let open ≈-Reasoning (A) in |
| 209 begin | |
| 210 f o ( h o j ) | |
| 211 ≈⟨ assoc ⟩ | |
| 230 | 212 (f o h ) o j |
| 217 | 213 ≈⟨ eq' ⟩ |
| 230 | 214 (g o h ) o j |
| 217 | 215 ≈↑⟨ assoc ⟩ |
| 216 g o ( h o j ) | |
| 217 ∎ | |
| 221 | 218 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 230 | 219 fe=ge1 = let open ≈-Reasoning (A) in |
| 217 | 220 begin |
| 221 ( f o h ) o i | |
| 222 ≈↑⟨ assoc ⟩ | |
| 223 f o (h o i ) | |
| 224 ≈⟨ cdr eq ⟩ | |
| 225 f o (e eqa) | |
| 221 | 226 ≈⟨ fe=ge eqa ⟩ |
| 217 | 227 g o (e eqa) |
| 228 ≈↑⟨ cdr eq ⟩ | |
| 229 g o (h o i ) | |
| 230 ≈⟨ assoc ⟩ | |
| 231 ( g o h ) o i | |
| 232 ∎ | |
| 218 | 233 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' ] ]} → |
| 234 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 235 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 217 | 236 begin |
| 218 | 237 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') |
| 238 ≈↑⟨ idL ⟩ | |
| 230 | 239 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) |
| 218 | 240 ≈↑⟨ car h-1-id ⟩ |
| 230 | 241 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) |
| 218 | 242 ≈↑⟨ assoc ⟩ |
| 243 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
| 244 ≈⟨ cdr assoc ⟩ | |
| 245 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 246 ≈⟨ cdr (car eq ) ⟩ | |
| 247 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 248 ≈⟨ cdr (ek=h eqa) ⟩ | |
| 249 h-1 o ( h o k' ) | |
| 250 ≈⟨ assoc ⟩ | |
| 230 | 251 ( h-1 o h ) o k' |
| 218 | 252 ≈⟨ car h-1-id ⟩ |
| 230 | 253 id1 A d o k' |
| 218 | 254 ≈⟨ idL ⟩ |
| 255 k' | |
| 217 | 256 ∎ |
| 257 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} → | |
| 258 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 259 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 260 begin | |
| 261 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 262 ≈⟨ uniqueness eqa ( begin | |
| 263 e eqa o k' | |
| 264 ≈↑⟨ car eq ⟩ | |
| 265 (h o i ) o k' | |
| 266 ≈↑⟨ assoc ⟩ | |
| 267 h o (i o k') | |
| 268 ≈⟨ cdr ik=h ⟩ | |
| 230 | 269 h o h' |
| 217 | 270 ∎ ) ⟩ |
| 271 k' | |
| 272 ∎ | |
| 215 | 273 |
| 230 | 274 -- If we have equalizer f g, e hf hg is also equalizer if we have isomorphic pair |
| 225 | 275 |
| 230 | 276 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 | 277 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] |
| 230 | 278 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) |
| 218 | 279 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { |
| 230 | 280 e = e eqa ; |
| 281 fe=ge = fe=ge1 ; | |
| 218 | 282 k = λ j eq' → k eqa j (fj=gj j eq') ; |
| 230 | 283 ek=h = ek=h1 ; |
| 218 | 284 uniqueness = uniqueness1 |
| 285 } where | |
| 286 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 ] ] | |
| 287 fj=gj j eq = let open ≈-Reasoning (A) in | |
| 288 begin | |
| 289 f o j | |
| 290 ≈↑⟨ idL ⟩ | |
| 291 id1 A b o ( f o j ) | |
| 292 ≈↑⟨ car h-1-id ⟩ | |
| 293 (h-1 o h ) o ( f o j ) | |
| 294 ≈↑⟨ assoc ⟩ | |
| 295 h-1 o (h o ( f o j )) | |
| 296 ≈⟨ cdr assoc ⟩ | |
| 297 h-1 o ((h o f) o j ) | |
| 298 ≈⟨ cdr eq ⟩ | |
| 299 h-1 o ((h o g) o j ) | |
| 300 ≈↑⟨ cdr assoc ⟩ | |
| 301 h-1 o (h o ( g o j )) | |
| 302 ≈⟨ assoc ⟩ | |
| 303 (h-1 o h) o ( g o j ) | |
| 304 ≈⟨ car h-1-id ⟩ | |
| 305 id1 A b o ( g o j ) | |
| 306 ≈⟨ idL ⟩ | |
| 307 g o j | |
| 308 ∎ | |
| 221 | 309 fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] |
| 310 fe=ge1 = let open ≈-Reasoning (A) in | |
| 218 | 311 begin |
| 312 ( h o f ) o e eqa | |
| 313 ≈↑⟨ assoc ⟩ | |
| 314 h o (f o e eqa ) | |
| 221 | 315 ≈⟨ cdr (fe=ge eqa) ⟩ |
| 218 | 316 h o (g o e eqa ) |
| 317 ≈⟨ assoc ⟩ | |
| 230 | 318 ( h o g ) o e eqa |
| 218 | 319 ∎ |
| 320 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 ] ]} → | |
| 321 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
| 322 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
| 323 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} → | |
| 324 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
| 325 uniqueness1 = uniqueness eqa | |
| 230 | 326 |
| 225 | 327 -- If we have equalizer f g, e (ef) (eg) is also an equalizer and e = id c |
| 328 | |
| 230 | 329 eefeg : {a b c : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
| 330 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
| 227 | 331 eefeg {a} {b} {c} {f} {g} eqa = record { |
| 220 | 332 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d |
| 221 | 333 fe=ge = fe=ge1 ; |
| 220 | 334 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 230 | 335 ek=h = ek=h1 ; |
| 220 | 336 uniqueness = uniqueness1 |
| 337 } where | |
| 338 i = id1 A c | |
| 339 h = e eqa | |
| 230 | 340 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → |
| 220 | 341 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → |
| 230 | 342 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] |
| 220 | 343 fhj=ghj j eq' = let open ≈-Reasoning (A) in |
| 344 begin | |
| 345 f o ( h o j ) | |
| 346 ≈⟨ assoc ⟩ | |
| 230 | 347 (f o h ) o j |
| 220 | 348 ≈⟨ eq' ⟩ |
| 230 | 349 (g o h ) o j |
| 220 | 350 ≈↑⟨ assoc ⟩ |
| 351 g o ( h o j ) | |
| 352 ∎ | |
| 221 | 353 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 230 | 354 fe=ge1 = let open ≈-Reasoning (A) in |
| 220 | 355 begin |
| 356 ( f o h ) o i | |
| 221 | 357 ≈⟨ car ( fe=ge eqa ) ⟩ |
| 220 | 358 ( g o h ) o i |
| 359 ∎ | |
| 360 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' ] ]} → | |
| 361 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 362 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 363 begin | |
| 364 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') | |
| 365 ≈⟨ idL ⟩ | |
| 366 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
| 367 ≈⟨ uniqueness eqa refl-hom ⟩ | |
| 368 k' | |
| 369 ∎ | |
| 370 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} → | |
| 371 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 372 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 373 begin | |
| 374 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
| 375 ≈⟨ uniqueness eqa ( begin | |
| 376 e eqa o k' | |
| 377 ≈↑⟨ cdr idL ⟩ | |
| 378 e eqa o (id1 A c o k' ) | |
| 379 ≈⟨ cdr ik=h ⟩ | |
| 230 | 380 e eqa o h' |
| 220 | 381 ∎ ) ⟩ |
| 382 k' | |
| 383 ∎ | |
| 384 | |
| 225 | 385 -- |
| 386 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 387 -- | |
| 388 -- h : c → c' h' : c' → c | |
| 226 | 389 -- h h' = 1 and h' h = 1 |
| 225 | 390 -- not yet done |
| 391 | |
| 392 | |
| 230 | 393 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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394 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) |
| 230 | 395 → Hom A c c' |
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396 c-iso-l {c} {c'} eqa eqa' keqa = e keqa |
| 226 | 397 |
| 230 | 398 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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399 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) |
| 230 | 400 → Hom A c' c |
| 401 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 223 | 402 |
| 227 | 403 |
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404 -- e(eqa') f |
| 230 | 405 -- c'----------> a ------->b f e j = g e j |
| 406 -- ^ g | |
| 407 -- |k h | |
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408 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 409 -- | |
| 410 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 411 -- | |
| 412 -- h j e f = h j e g → h = 'j e f | |
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413 -- h = j e f -> j = j' |
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414 -- |
| 228 | 415 |
| 230 | 416 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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417 → ( keqa : Equalizer A {c} (A [ f o e eqa' ]) (A [ g o e eqa' ]) ) |
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418 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
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419 c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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420 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
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421 ≈⟨ ek=h keqa ⟩ |
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422 id1 A c' |
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423 ∎ |
| 226 | 424 |
| 230 | 425 -- To prove c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa |
| 221 | 426 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 |
| 427 -- ke = e' | |
| 428 -- k'ke = k'e' = e kk' = 1 | |
| 429 -- x e = e -> x = id? | |
| 218 | 430 |
| 230 | 431 ----- |
| 432 -- reverse arrow of e (eqa f g) | |
| 433 -- | |
| 434 -- e eqa f g f | |
| 435 -- c ----------> a ------->b | |
| 436 -- <--------- | |
| 231 | 437 -- k (eff) id1a |
| 438 -- (e eqa f g) o k (eff) id1 A a = id1 A a | |
| 439 -- | |
| 440 -- eqa (f (e eqa f g) ) (g (e eqa f g) ) | |
| 441 -- e (eqa (f (e eqa f g) ) (g (e eqa f g) ) ) = k (eff) id1 a | |
| 442 -- | |
| 443 -- (e α) o k α (id1 A c) = id1 A c | |
| 444 -- c a c | |
| 445 -- ((k (eff) id1a )) o k α e = id1 A c | |
| 446 | |
| 447 | |
| 232 | 448 reverse-e' : {a b c : Obj A} (f g : Hom A a b) → (h i : Hom A c b ) → |
| 449 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → | |
| 450 A [ k (eqa f f ) (id1 A a ) ( f1=f1 f ) ≈ (e (eqa (A [ f o e (eqa f g) ]) (A [ g o e (eqa f g) ]))) ] | |
| 451 reverse-e' = ? | |
| 452 | |
| 231 | 453 reverse-e : {a b c : Obj A} (f g : Hom A a b) → (h i : Hom A c b ) → |
| 230 | 454 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → |
| 231 | 455 A [ |
| 232 | 456 A [ k (eqa f f ) (id1 A a ) ( f1=f1 f ) o k (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) (id1 A c) (f1=g1 (fe=ge (eqa f g)) (id1 A c)) ] |
| 231 | 457 ≈ id1 A c ] |
| 458 reverse-e {a} {b} {c} f g h i eqa = let open ≈-Reasoning (A) in | |
| 459 begin | |
| 232 | 460 k (eqa f f ) (id1 A a ) (f1=f1 f) o k (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) (id1 A c) {!!} |
| 461 ≈⟨ car {!!} ⟩ | |
| 462 e (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) o k (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) (id1 A c) (f1=g1 (fe=ge (eqa f g)) (id1 A c)) | |
| 231 | 463 ≈⟨ ek=h (eqa ( A [ f o (e (eqa f g)) ] ) (A [ g o (e (eqa f g )) ] )) ⟩ |
| 464 id1 A c | |
| 465 ∎ | |
| 230 | 466 |
| 225 | 467 ---- |
| 468 -- | |
| 469 -- An equalizer satisfies Burroni equations | |
| 470 -- | |
| 230 | 471 -- b4 is not yet done |
| 225 | 472 ---- |
| 473 | |
| 230 | 474 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → |
| 225 | 475 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → Burroni A {c} f g |
| 222 | 476 lemma-equ1 {a} {b} {c} f g eqa = record { |
| 216 | 477 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 478 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 | 479 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 221 | 480 b1 = fe=ge (eqa f g) ; |
| 226 | 481 b2 = lemma-b2 ; |
| 482 b3 = lemma-b3 ; | |
| 230 | 483 b4 = lemma-b4 |
| 211 | 484 } where |
| 216 | 485 -- |
| 486 -- e eqa f g f | |
| 487 -- c ----------> a ------->b | |
| 230 | 488 -- ^ g |
| 489 -- | | |
| 216 | 490 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 491 -- | |
| 216 | 492 -- d |
| 230 | 493 -- |
| 494 -- | |
| 216 | 495 -- e o id1 ≈ e → k e ≈ id |
| 496 | |
| 211 | 497 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 498 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 226 | 499 lemma-b3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 500 lemma-b3 = let open ≈-Reasoning (A) in | |
| 230 | 501 begin |
| 211 | 502 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
| 215 | 503 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 504 id1 A a |
| 505 ∎ | |
| 230 | 506 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 507 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 | 508 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 509 begin |
| 510 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 511 ≈⟨ assoc ⟩ | |
| 512 (f o h) o e (eqa (f o h) ( g o h )) | |
| 221 | 513 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
| 212 | 514 (g o h) o e (eqa (f o h) ( g o h )) |
| 515 ≈↑⟨ assoc ⟩ | |
| 516 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 517 ∎ | |
| 230 | 518 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
| 214 | 519 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 | 520 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 521 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 522 begin |
| 230 | 523 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 | 524 ≈⟨ ek=h (eqa f g) ⟩ |
| 212 | 525 h o e (eqa (f o h ) ( g o h )) |
| 526 ∎ | |
| 230 | 527 |
| 528 ------- α(f,g)j id d = α(f,g)j | |
| 529 ------- α(f,g)j id d = α(f,g)j | |
| 530 ------- α(f,g)j α(fα(f,g)j,fα(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 531 ------ fα = gα | |
| 532 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 533 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 534 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j | |
| 535 | |
| 536 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
| 222 | 537 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 | 538 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o |
| 222 | 539 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 ] ])) ] |
| 540 ≈ j ] | |
| 230 | 541 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 542 begin |
| 222 | 543 ( 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 ) ))) )) |
| 544 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o | |
| 545 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 | 546 ≈⟨ {!!} ⟩ |
| 222 | 547 j |
| 215 | 548 ∎ |
| 211 | 549 |
| 550 | |
| 225 | 551 -- end |
| 212 | 552 |
| 553 | |
| 554 | |
| 215 | 555 |
| 556 |