Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 223:8b3aeba14b5e
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 05 Sep 2013 04:35:22 +0900 |
| parents | 0bc85361b7d0 |
| children | a9d311cea336 |
| rev | line source |
|---|---|
| 205 | 1 --- |
| 2 -- | |
| 3 -- Equalizer | |
| 4 -- | |
| 208 | 5 -- e f |
| 205 | 6 -- c --------> a ----------> b |
| 208 | 7 -- ^ . ----------> |
| 205 | 8 -- | . g |
| 208 | 9 -- |k . |
| 10 -- | . h | |
| 205 | 11 -- d |
| 12 -- | |
| 13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> | |
| 14 ---- | |
| 15 | |
| 16 open import Category -- https://github.com/konn/category-agda | |
| 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 |
| 209 | 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 ] |
| 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 |
| 209 | 34 record EqEqualizer { 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 | 35 field |
| 212 | 36 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
| 214 | 37 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 |
| 212 | 38 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
| 213 | 39 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
| 214 | 40 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 | 41 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
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42 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
| 215 | 43 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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44 -- A [ α f g o β f g h ] ≈ h |
| 214 | 45 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 |
| 46 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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47 |
| 209 | 48 open Equalizer |
| 49 open EqEqualizer | |
| 50 | |
| 219 | 51 |
| 52 f1=g1 : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → A [ A [ f o id1 A a ] ≈ A [ g o id1 A a ] ] | |
| 53 f1=g1 eq = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 54 | |
| 55 | |
| 56 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → | |
| 57 A [ e eqa ≈ id1 A a ] → | |
| 58 A [ k eqa (id1 A a) (f1=g1 eq) ≈ id1 A a ] | |
| 59 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in | |
| 60 begin | |
| 61 k eqa (id1 A a) (f1=g1 eq) | |
| 62 ≈⟨ uniqueness eqa ( begin | |
| 63 e eqa o id1 A a | |
| 64 ≈⟨ idR ⟩ | |
| 65 e eqa | |
| 66 ≈⟨ e=1 ⟩ | |
| 67 id1 A a | |
| 68 ∎ )⟩ | |
| 69 id1 A a | |
| 70 ∎ | |
| 71 | |
| 222 | 72 equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) → |
| 73 A [ k eqa (id1 A a) (f1=g1 eq) ≈ id1 A a ] → | |
| 74 A [ e eqa ≈ id1 A a ] | |
| 75 equalizer-eq-e {a} {b} {f} {g} eqa eq k=1 = let open ≈-Reasoning (A) in | |
| 76 begin | |
| 77 e eqa | |
| 78 ≈↑⟨ idR ⟩ | |
| 79 e eqa o id1 A a | |
| 80 ≈↑⟨ cdr k=1 ⟩ | |
| 81 e eqa o k eqa (id1 A a) (f1=g1 eq) | |
| 82 ≈⟨ ek=h eqa ⟩ | |
| 83 id1 A a | |
| 84 ∎ | |
| 85 | |
| 86 -- e eqa f g f | |
| 87 -- c ----------> a ------->b | |
| 88 -- |^ | |
| 89 -- || | |
| 90 -- h || h-1 | |
| 91 -- v| | |
| 92 -- c' | |
| 93 | |
| 94 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' ) → | |
| 95 A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] | |
| 96 → Equalizer A {c'} f g | |
| 97 equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { | |
| 98 e = A [ e eqa o h-1 ] ; | |
| 99 fe=ge = fe=ge1 ; | |
| 100 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 101 ek=h = ek=h1 ; | |
| 102 uniqueness = uniqueness1 | |
| 103 } where | |
| 104 fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] | |
| 105 fe=ge1 = let open ≈-Reasoning (A) in | |
| 106 begin | |
| 107 f o ( e eqa o h-1 ) | |
| 108 ≈⟨ assoc ⟩ | |
| 109 (f o e eqa ) o h-1 | |
| 110 ≈⟨ car (fe=ge eqa) ⟩ | |
| 111 (g o e eqa ) o h-1 | |
| 112 ≈↑⟨ assoc ⟩ | |
| 113 g o ( e eqa o h-1 ) | |
| 114 ∎ | |
| 115 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → | |
| 116 A [ A [ A [ e eqa o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] | |
| 117 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in | |
| 118 begin | |
| 119 (e eqa o h-1 ) o ( h o k eqa j eq ) | |
| 120 ≈↑⟨ assoc ⟩ | |
| 121 e eqa o ( h-1 o ( h o k eqa j eq )) | |
| 122 ≈⟨ cdr assoc ⟩ | |
| 123 e eqa o (( h-1 o h ) o k eqa j eq ) | |
| 124 ≈⟨ cdr (car (h-1-id )) ⟩ | |
| 125 e eqa o (id1 A c o k eqa j eq ) | |
| 126 ≈⟨ cdr idL ⟩ | |
| 127 e eqa o (k eqa j eq ) | |
| 128 ≈⟨ ek=h eqa ⟩ | |
| 129 j | |
| 130 ∎ | |
| 131 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 132 A [ A [ A [ e eqa o h-1 ] o j ] ≈ h' ] → | |
| 133 A [ A [ h o k eqa h' eq ] ≈ j ] | |
| 134 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 135 begin | |
| 136 h o k eqa h' eq | |
| 137 ≈⟨ cdr (uniqueness eqa ( | |
| 138 begin | |
| 139 e eqa o ( h-1 o j ) | |
| 140 ≈⟨ assoc ⟩ | |
| 141 (e eqa o h-1 ) o j | |
| 142 ≈⟨ ej=h ⟩ | |
| 143 h' | |
| 144 ∎ | |
| 145 )) ⟩ | |
| 146 h o ( h-1 o j ) | |
| 147 ≈⟨ assoc ⟩ | |
| 148 (h o h-1 ) o j | |
| 149 ≈⟨ car h-id ⟩ | |
| 150 id1 A c' o j | |
| 151 ≈⟨ idL ⟩ | |
| 152 j | |
| 153 ∎ | |
| 154 | |
| 217 | 155 -- e eqa f g f |
| 156 -- c ----------> a ------->b | |
| 218 | 157 -- ^ ---> d ---> |
| 158 -- | i h | |
| 159 -- j| k' (d' → d) | |
| 160 -- | k (d' → a) | |
| 161 -- d' | |
| 217 | 162 |
| 218 | 163 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 ) |
| 164 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
| 217 | 165 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
| 218 | 166 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
| 167 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 221 | 168 fe=ge = fe=ge1 ; |
| 217 | 169 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 170 ek=h = ek=h1 ; | |
| 171 uniqueness = uniqueness1 | |
| 172 } where | |
| 173 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
| 174 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 175 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 176 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 177 begin | |
| 178 f o ( h o j ) | |
| 179 ≈⟨ assoc ⟩ | |
| 180 (f o h ) o j | |
| 181 ≈⟨ eq' ⟩ | |
| 182 (g o h ) o j | |
| 183 ≈↑⟨ assoc ⟩ | |
| 184 g o ( h o j ) | |
| 185 ∎ | |
| 221 | 186 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 187 fe=ge1 = let open ≈-Reasoning (A) in | |
| 217 | 188 begin |
| 189 ( f o h ) o i | |
| 190 ≈↑⟨ assoc ⟩ | |
| 191 f o (h o i ) | |
| 192 ≈⟨ cdr eq ⟩ | |
| 193 f o (e eqa) | |
| 221 | 194 ≈⟨ fe=ge eqa ⟩ |
| 217 | 195 g o (e eqa) |
| 196 ≈↑⟨ cdr eq ⟩ | |
| 197 g o (h o i ) | |
| 198 ≈⟨ assoc ⟩ | |
| 199 ( g o h ) o i | |
| 200 ∎ | |
| 218 | 201 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' ] ]} → |
| 202 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 203 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 217 | 204 begin |
| 218 | 205 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') |
| 206 ≈↑⟨ idL ⟩ | |
| 207 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 208 ≈↑⟨ car h-1-id ⟩ | |
| 209 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 210 ≈↑⟨ assoc ⟩ | |
| 211 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
| 212 ≈⟨ cdr assoc ⟩ | |
| 213 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 214 ≈⟨ cdr (car eq ) ⟩ | |
| 215 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 216 ≈⟨ cdr (ek=h eqa) ⟩ | |
| 217 h-1 o ( h o k' ) | |
| 218 ≈⟨ assoc ⟩ | |
| 219 ( h-1 o h ) o k' | |
| 220 ≈⟨ car h-1-id ⟩ | |
| 221 id1 A d o k' | |
| 222 ≈⟨ idL ⟩ | |
| 223 k' | |
| 217 | 224 ∎ |
| 225 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} → | |
| 226 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 227 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 228 begin | |
| 229 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 230 ≈⟨ uniqueness eqa ( begin | |
| 231 e eqa o k' | |
| 232 ≈↑⟨ car eq ⟩ | |
| 233 (h o i ) o k' | |
| 234 ≈↑⟨ assoc ⟩ | |
| 235 h o (i o k') | |
| 236 ≈⟨ cdr ik=h ⟩ | |
| 237 h o h' | |
| 238 ∎ ) ⟩ | |
| 239 k' | |
| 240 ∎ | |
| 215 | 241 |
| 218 | 242 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 ) |
| 243 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
| 244 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
| 245 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
| 246 e = e eqa ; | |
| 221 | 247 fe=ge = fe=ge1 ; |
| 218 | 248 k = λ j eq' → k eqa j (fj=gj j eq') ; |
| 249 ek=h = ek=h1 ; | |
| 250 uniqueness = uniqueness1 | |
| 251 } where | |
| 252 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 ] ] | |
| 253 fj=gj j eq = let open ≈-Reasoning (A) in | |
| 254 begin | |
| 255 f o j | |
| 256 ≈↑⟨ idL ⟩ | |
| 257 id1 A b o ( f o j ) | |
| 258 ≈↑⟨ car h-1-id ⟩ | |
| 259 (h-1 o h ) o ( f o j ) | |
| 260 ≈↑⟨ assoc ⟩ | |
| 261 h-1 o (h o ( f o j )) | |
| 262 ≈⟨ cdr assoc ⟩ | |
| 263 h-1 o ((h o f) o j ) | |
| 264 ≈⟨ cdr eq ⟩ | |
| 265 h-1 o ((h o g) o j ) | |
| 266 ≈↑⟨ cdr assoc ⟩ | |
| 267 h-1 o (h o ( g o j )) | |
| 268 ≈⟨ assoc ⟩ | |
| 269 (h-1 o h) o ( g o j ) | |
| 270 ≈⟨ car h-1-id ⟩ | |
| 271 id1 A b o ( g o j ) | |
| 272 ≈⟨ idL ⟩ | |
| 273 g o j | |
| 274 ∎ | |
| 221 | 275 fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] |
| 276 fe=ge1 = let open ≈-Reasoning (A) in | |
| 218 | 277 begin |
| 278 ( h o f ) o e eqa | |
| 279 ≈↑⟨ assoc ⟩ | |
| 280 h o (f o e eqa ) | |
| 221 | 281 ≈⟨ cdr (fe=ge eqa) ⟩ |
| 218 | 282 h o (g o e eqa ) |
| 283 ≈⟨ assoc ⟩ | |
| 284 ( h o g ) o e eqa | |
| 285 ∎ | |
| 286 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 ] ]} → | |
| 287 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
| 288 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
| 289 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} → | |
| 290 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
| 291 uniqueness1 = uniqueness eqa | |
| 292 | |
| 220 | 293 eefeg : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
| 294 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
| 295 eefeg {a} {b} {c} {d} {f} {g} eqa = record { | |
| 296 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 221 | 297 fe=ge = fe=ge1 ; |
| 220 | 298 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 299 ek=h = ek=h1 ; | |
| 300 uniqueness = uniqueness1 | |
| 301 } where | |
| 302 i = id1 A c | |
| 303 h = e eqa | |
| 304 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → | |
| 305 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 306 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 307 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 308 begin | |
| 309 f o ( h o j ) | |
| 310 ≈⟨ assoc ⟩ | |
| 311 (f o h ) o j | |
| 312 ≈⟨ eq' ⟩ | |
| 313 (g o h ) o j | |
| 314 ≈↑⟨ assoc ⟩ | |
| 315 g o ( h o j ) | |
| 316 ∎ | |
| 221 | 317 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 318 fe=ge1 = let open ≈-Reasoning (A) in | |
| 220 | 319 begin |
| 320 ( f o h ) o i | |
| 221 | 321 ≈⟨ car ( fe=ge eqa ) ⟩ |
| 220 | 322 ( g o h ) o i |
| 323 ∎ | |
| 324 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' ] ]} → | |
| 325 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 326 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 327 begin | |
| 328 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') | |
| 329 ≈⟨ idL ⟩ | |
| 330 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
| 331 ≈⟨ uniqueness eqa refl-hom ⟩ | |
| 332 k' | |
| 333 ∎ | |
| 334 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} → | |
| 335 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 336 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 337 begin | |
| 338 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
| 339 ≈⟨ uniqueness eqa ( begin | |
| 340 e eqa o k' | |
| 341 ≈↑⟨ cdr idL ⟩ | |
| 342 e eqa o (id1 A c o k' ) | |
| 343 ≈⟨ cdr ik=h ⟩ | |
| 344 e eqa o h' | |
| 345 ∎ ) ⟩ | |
| 346 k' | |
| 347 ∎ | |
| 348 | |
| 223 | 349 iso-rev : { a b c : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {c} f g) → Hom A a c |
| 350 iso-rev {a} {b} {c} {f} {g} eq eqa = k eqa (id1 A a) (f1=g1 eq) | |
| 351 | |
| 352 equalizer-iso-pair : { a b c : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {c} f g) → | |
| 353 A [ A [ e eqa o iso-rev eq eqa ] ≈ id1 A a ] | |
| 354 equalizer-iso-pair {a} {b} {c} {f} {g} eq eqa = ek=h eqa | |
| 355 | |
| 221 | 356 -- Equalizer is unique up to iso |
| 357 | |
| 358 equalizer-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 ) | |
| 359 → Hom A c c' --- != id1 A c | |
| 360 equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (fe=ge eqa) | |
| 220 | 361 |
| 222 | 362 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 363 (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 ] ] ] | |
| 364 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in | |
| 365 begin | |
| 366 f o ( e o h ) | |
| 367 ≈⟨ assoc ⟩ | |
| 368 (f o e ) o h | |
| 369 ≈⟨ car eq ⟩ | |
| 370 (g o e ) o h | |
| 371 ≈↑⟨ assoc ⟩ | |
| 372 g o ( e o h ) | |
| 373 ∎ | |
| 374 | |
| 221 | 375 -- |
| 376 -- | |
| 377 -- e eqa f g f | |
| 378 -- c ----------> a ------->b | |
| 379 -- | |
| 380 equalizer-iso-eq : { c c' a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {c} f g) → ( eqa' : Equalizer A {c'} f g ) | |
| 222 | 381 { h : Hom A a c } → A [ A [ h o e eqa ] ≈ id1 A c ] → A [ A [ k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ] ≈ id1 A c ] |
| 382 equalizer-iso-eq {c} {c'} {f} {g} eqa eqa' {h} rev = let open ≈-Reasoning (A) in | |
| 221 | 383 begin |
| 222 | 384 k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) |
| 385 ≈↑⟨ idL ⟩ | |
| 386 (id1 A c) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 387 ≈↑⟨ car rev ⟩ | |
| 388 ( h o e eqa ) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 389 ≈↑⟨ assoc ⟩ | |
| 390 h o ( e eqa o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) ) | |
| 391 ≈⟨ cdr assoc ⟩ | |
| 392 h o (( e eqa o k eqa (e eqa' ) (fe=ge eqa')) o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 393 ≈⟨ cdr ( car (ek=h eqa) ) ⟩ | |
| 394 h o ( e eqa' o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 395 ≈⟨ cdr (ek=h eqa' ) ⟩ | |
| 396 h o e eqa | |
| 397 ≈⟨ rev ⟩ | |
| 398 id1 A c | |
| 221 | 399 ∎ |
| 400 | |
| 401 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 | |
| 402 -- ke = e' | |
| 403 -- k'ke = k'e' = e kk' = 1 | |
| 404 | |
| 405 -- x e = e -> x = id? | |
| 218 | 406 |
| 222 | 407 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → |
| 211 | 408 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g |
| 222 | 409 lemma-equ1 {a} {b} {c} f g eqa = record { |
| 216 | 410 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 411 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 | 412 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 221 | 413 b1 = fe=ge (eqa f g) ; |
| 212 | 414 b2 = lemma-equ5 ; |
| 415 b3 = lemma-equ3 ; | |
| 215 | 416 b4 = lemma-equ6 |
| 211 | 417 } where |
| 216 | 418 -- |
| 419 -- e eqa f g f | |
| 420 -- c ----------> a ------->b | |
| 421 -- ^ g | |
| 422 -- | | |
| 423 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
| 424 -- | | |
| 425 -- d | |
| 426 -- | |
| 427 -- | |
| 428 -- e o id1 ≈ e → k e ≈ id | |
| 429 | |
| 211 | 430 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 431 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 432 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 433 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 434 begin |
| 435 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 436 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 437 id1 A a |
| 438 ∎ | |
| 214 | 439 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 440 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 | 441 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 442 begin |
| 443 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 444 ≈⟨ assoc ⟩ | |
| 445 (f o h) o e (eqa (f o h) ( g o h )) | |
| 221 | 446 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
| 212 | 447 (g o h) o e (eqa (f o h) ( g o h )) |
| 448 ≈↑⟨ assoc ⟩ | |
| 449 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 450 ∎ | |
| 451 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 452 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 | 453 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 454 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 455 begin | |
| 215 | 456 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) |
| 457 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 458 h o e (eqa (f o h ) ( g o h )) |
| 459 ∎ | |
| 222 | 460 lemma-equ6 : {d : Obj A} {j : Hom A d c} → A [ |
| 461 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 ] ])) ]) | |
| 462 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o | |
| 463 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 ] ])) ] | |
| 464 ≈ j ] | |
| 465 lemma-equ6 {d} {j} = let open ≈-Reasoning (A) in | |
| 215 | 466 begin |
| 222 | 467 ( 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 ) ))) )) |
| 468 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o | |
| 469 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 ) ))) ) | |
| 215 | 470 ≈⟨ car ( uniqueness (eqa f g) ( begin |
| 222 | 471 e (eqa f g) o j |
| 215 | 472 ≈⟨ {!!} ⟩ |
| 222 | 473 (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)) |
| 215 | 474 ∎ )) ⟩ |
| 222 | 475 j o 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 ) ))) |
| 476 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) ( begin | |
| 477 e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) o id1 A d | |
| 478 ≈⟨ idR ⟩ | |
| 479 e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) | |
| 215 | 480 ≈⟨ {!!} ⟩ |
| 222 | 481 id1 A d |
| 215 | 482 ∎ )) ⟩ |
| 222 | 483 j o id1 A d |
| 215 | 484 ≈⟨ idR ⟩ |
| 222 | 485 j |
| 215 | 486 ∎ |
| 211 | 487 |
| 488 | |
| 212 | 489 |
| 490 | |
| 491 | |
| 215 | 492 |
| 493 |