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annotate equalizer.agda @ 224:a9d311cea336
fix
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Thu, 05 Sep 2013 11:39:06 +0900 |
| parents | 8b3aeba14b5e |
| children | 1a9f20917fbd |
| 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 |
| 224 | 52 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 ] ] |
| 53 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 54 | |
| 55 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → | |
| 56 (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 ] ] ] | |
| 57 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in | |
| 58 begin | |
| 59 f o ( e o h ) | |
| 60 ≈⟨ assoc ⟩ | |
| 61 (f o e ) o h | |
| 62 ≈⟨ car eq ⟩ | |
| 63 (g o e ) o h | |
| 64 ≈↑⟨ assoc ⟩ | |
| 65 g o ( e o h ) | |
| 66 ∎ | |
| 219 | 67 |
| 68 | |
| 69 equalizer-eq-k : { a b : Obj A } {f g : Hom A a b } → (eq : A [ f ≈ g ] ) → ( eqa : Equalizer A {a} f g) → | |
| 70 A [ e eqa ≈ id1 A a ] → | |
| 224 | 71 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] |
| 219 | 72 equalizer-eq-k {a} {b} {f} {g} eq eqa e=1 = let open ≈-Reasoning (A) in |
| 73 begin | |
| 224 | 74 k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 219 | 75 ≈⟨ uniqueness eqa ( begin |
| 76 e eqa o id1 A a | |
| 77 ≈⟨ idR ⟩ | |
| 78 e eqa | |
| 79 ≈⟨ e=1 ⟩ | |
| 80 id1 A a | |
| 81 ∎ )⟩ | |
| 82 id1 A a | |
| 83 ∎ | |
| 84 | |
| 222 | 85 equalizer-eq-e : { a b : Obj A } {f g : Hom A a b } → ( eqa : Equalizer A {a} f g) → (eq : A [ f ≈ g ] ) → |
| 224 | 86 A [ k eqa (id1 A a) (f1=g1 eq (id1 A a)) ≈ id1 A a ] → |
| 222 | 87 A [ e eqa ≈ id1 A a ] |
| 88 equalizer-eq-e {a} {b} {f} {g} eqa eq k=1 = let open ≈-Reasoning (A) in | |
| 89 begin | |
| 90 e eqa | |
| 91 ≈↑⟨ idR ⟩ | |
| 92 e eqa o id1 A a | |
| 93 ≈↑⟨ cdr k=1 ⟩ | |
| 224 | 94 e eqa o k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 222 | 95 ≈⟨ ek=h eqa ⟩ |
| 96 id1 A a | |
| 97 ∎ | |
| 98 | |
| 99 -- e eqa f g f | |
| 100 -- c ----------> a ------->b | |
| 101 -- |^ | |
| 102 -- || | |
| 103 -- h || h-1 | |
| 104 -- v| | |
| 105 -- c' | |
| 106 | |
| 107 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' ) → | |
| 108 A [ A [ h-1 o h ] ≈ id1 A c ] → A [ A [ h o h-1 ] ≈ id1 A c' ] | |
| 109 → Equalizer A {c'} f g | |
| 110 equalizer+iso {a} {b} {c} {c'} {f} {g} eqa h-1 h h-1-id h-id = record { | |
| 111 e = A [ e eqa o h-1 ] ; | |
| 112 fe=ge = fe=ge1 ; | |
| 113 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 114 ek=h = ek=h1 ; | |
| 115 uniqueness = uniqueness1 | |
| 116 } where | |
| 117 fe=ge1 : A [ A [ f o A [ e eqa o h-1 ] ] ≈ A [ g o A [ e eqa o h-1 ] ] ] | |
| 118 fe=ge1 = let open ≈-Reasoning (A) in | |
| 119 begin | |
| 120 f o ( e eqa o h-1 ) | |
| 121 ≈⟨ assoc ⟩ | |
| 122 (f o e eqa ) o h-1 | |
| 123 ≈⟨ car (fe=ge eqa) ⟩ | |
| 124 (g o e eqa ) o h-1 | |
| 125 ≈↑⟨ assoc ⟩ | |
| 126 g o ( e eqa o h-1 ) | |
| 127 ∎ | |
| 128 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → | |
| 129 A [ A [ A [ e eqa o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] | |
| 130 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in | |
| 131 begin | |
| 132 (e eqa o h-1 ) o ( h o k eqa j eq ) | |
| 133 ≈↑⟨ assoc ⟩ | |
| 134 e eqa o ( h-1 o ( h o k eqa j eq )) | |
| 135 ≈⟨ cdr assoc ⟩ | |
| 136 e eqa o (( h-1 o h ) o k eqa j eq ) | |
| 137 ≈⟨ cdr (car (h-1-id )) ⟩ | |
| 138 e eqa o (id1 A c o k eqa j eq ) | |
| 139 ≈⟨ cdr idL ⟩ | |
| 140 e eqa o (k eqa j eq ) | |
| 141 ≈⟨ ek=h eqa ⟩ | |
| 142 j | |
| 143 ∎ | |
| 144 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 145 A [ A [ A [ e eqa o h-1 ] o j ] ≈ h' ] → | |
| 146 A [ A [ h o k eqa h' eq ] ≈ j ] | |
| 147 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 148 begin | |
| 149 h o k eqa h' eq | |
| 150 ≈⟨ cdr (uniqueness eqa ( | |
| 151 begin | |
| 152 e eqa o ( h-1 o j ) | |
| 153 ≈⟨ assoc ⟩ | |
| 154 (e eqa o h-1 ) o j | |
| 155 ≈⟨ ej=h ⟩ | |
| 156 h' | |
| 157 ∎ | |
| 158 )) ⟩ | |
| 159 h o ( h-1 o j ) | |
| 160 ≈⟨ assoc ⟩ | |
| 161 (h o h-1 ) o j | |
| 162 ≈⟨ car h-id ⟩ | |
| 163 id1 A c' o j | |
| 164 ≈⟨ idL ⟩ | |
| 165 j | |
| 166 ∎ | |
| 167 | |
| 217 | 168 -- e eqa f g f |
| 169 -- c ----------> a ------->b | |
| 218 | 170 -- ^ ---> d ---> |
| 171 -- | i h | |
| 172 -- j| k' (d' → d) | |
| 173 -- | k (d' → a) | |
| 174 -- d' | |
| 217 | 175 |
| 218 | 176 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 ) |
| 177 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
| 217 | 178 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
| 218 | 179 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
| 180 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 221 | 181 fe=ge = fe=ge1 ; |
| 217 | 182 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 183 ek=h = ek=h1 ; | |
| 184 uniqueness = uniqueness1 | |
| 185 } where | |
| 186 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
| 187 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 188 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 189 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 190 begin | |
| 191 f o ( h o j ) | |
| 192 ≈⟨ assoc ⟩ | |
| 193 (f o h ) o j | |
| 194 ≈⟨ eq' ⟩ | |
| 195 (g o h ) o j | |
| 196 ≈↑⟨ assoc ⟩ | |
| 197 g o ( h o j ) | |
| 198 ∎ | |
| 221 | 199 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 200 fe=ge1 = let open ≈-Reasoning (A) in | |
| 217 | 201 begin |
| 202 ( f o h ) o i | |
| 203 ≈↑⟨ assoc ⟩ | |
| 204 f o (h o i ) | |
| 205 ≈⟨ cdr eq ⟩ | |
| 206 f o (e eqa) | |
| 221 | 207 ≈⟨ fe=ge eqa ⟩ |
| 217 | 208 g o (e eqa) |
| 209 ≈↑⟨ cdr eq ⟩ | |
| 210 g o (h o i ) | |
| 211 ≈⟨ assoc ⟩ | |
| 212 ( g o h ) o i | |
| 213 ∎ | |
| 218 | 214 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' ] ]} → |
| 215 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 216 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 217 | 217 begin |
| 218 | 218 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') |
| 219 ≈↑⟨ idL ⟩ | |
| 220 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 221 ≈↑⟨ car h-1-id ⟩ | |
| 222 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 223 ≈↑⟨ assoc ⟩ | |
| 224 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
| 225 ≈⟨ cdr assoc ⟩ | |
| 226 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 227 ≈⟨ cdr (car eq ) ⟩ | |
| 228 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 229 ≈⟨ cdr (ek=h eqa) ⟩ | |
| 230 h-1 o ( h o k' ) | |
| 231 ≈⟨ assoc ⟩ | |
| 232 ( h-1 o h ) o k' | |
| 233 ≈⟨ car h-1-id ⟩ | |
| 234 id1 A d o k' | |
| 235 ≈⟨ idL ⟩ | |
| 236 k' | |
| 217 | 237 ∎ |
| 238 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} → | |
| 239 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 240 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 241 begin | |
| 242 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 243 ≈⟨ uniqueness eqa ( begin | |
| 244 e eqa o k' | |
| 245 ≈↑⟨ car eq ⟩ | |
| 246 (h o i ) o k' | |
| 247 ≈↑⟨ assoc ⟩ | |
| 248 h o (i o k') | |
| 249 ≈⟨ cdr ik=h ⟩ | |
| 250 h o h' | |
| 251 ∎ ) ⟩ | |
| 252 k' | |
| 253 ∎ | |
| 215 | 254 |
| 218 | 255 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 ) |
| 256 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
| 257 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
| 258 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
| 259 e = e eqa ; | |
| 221 | 260 fe=ge = fe=ge1 ; |
| 218 | 261 k = λ j eq' → k eqa j (fj=gj j eq') ; |
| 262 ek=h = ek=h1 ; | |
| 263 uniqueness = uniqueness1 | |
| 264 } where | |
| 265 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 ] ] | |
| 266 fj=gj j eq = let open ≈-Reasoning (A) in | |
| 267 begin | |
| 268 f o j | |
| 269 ≈↑⟨ idL ⟩ | |
| 270 id1 A b o ( f o j ) | |
| 271 ≈↑⟨ car h-1-id ⟩ | |
| 272 (h-1 o h ) o ( f o j ) | |
| 273 ≈↑⟨ assoc ⟩ | |
| 274 h-1 o (h o ( f o j )) | |
| 275 ≈⟨ cdr assoc ⟩ | |
| 276 h-1 o ((h o f) o j ) | |
| 277 ≈⟨ cdr eq ⟩ | |
| 278 h-1 o ((h o g) o j ) | |
| 279 ≈↑⟨ cdr assoc ⟩ | |
| 280 h-1 o (h o ( g o j )) | |
| 281 ≈⟨ assoc ⟩ | |
| 282 (h-1 o h) o ( g o j ) | |
| 283 ≈⟨ car h-1-id ⟩ | |
| 284 id1 A b o ( g o j ) | |
| 285 ≈⟨ idL ⟩ | |
| 286 g o j | |
| 287 ∎ | |
| 221 | 288 fe=ge1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] |
| 289 fe=ge1 = let open ≈-Reasoning (A) in | |
| 218 | 290 begin |
| 291 ( h o f ) o e eqa | |
| 292 ≈↑⟨ assoc ⟩ | |
| 293 h o (f o e eqa ) | |
| 221 | 294 ≈⟨ cdr (fe=ge eqa) ⟩ |
| 218 | 295 h o (g o e eqa ) |
| 296 ≈⟨ assoc ⟩ | |
| 297 ( h o g ) o e eqa | |
| 298 ∎ | |
| 299 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 ] ]} → | |
| 300 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
| 301 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
| 302 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} → | |
| 303 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
| 304 uniqueness1 = uniqueness eqa | |
| 305 | |
| 220 | 306 eefeg : {a b c d : Obj A } {f g : Hom A a b } ( eqa : Equalizer A {c} f g) |
| 307 → Equalizer A {c} (A [ f o e eqa ]) (A [ g o e eqa ] ) | |
| 308 eefeg {a} {b} {c} {d} {f} {g} eqa = record { | |
| 309 e = id1 A c ; -- i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 221 | 310 fe=ge = fe=ge1 ; |
| 220 | 311 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; |
| 312 ek=h = ek=h1 ; | |
| 313 uniqueness = uniqueness1 | |
| 314 } where | |
| 315 i = id1 A c | |
| 316 h = e eqa | |
| 317 fhj=ghj : {d' : Obj A } → (j : Hom A d' c ) → | |
| 318 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 319 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 320 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 321 begin | |
| 322 f o ( h o j ) | |
| 323 ≈⟨ assoc ⟩ | |
| 324 (f o h ) o j | |
| 325 ≈⟨ eq' ⟩ | |
| 326 (g o h ) o j | |
| 327 ≈↑⟨ assoc ⟩ | |
| 328 g o ( h o j ) | |
| 329 ∎ | |
| 221 | 330 fe=ge1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] |
| 331 fe=ge1 = let open ≈-Reasoning (A) in | |
| 220 | 332 begin |
| 333 ( f o h ) o i | |
| 221 | 334 ≈⟨ car ( fe=ge eqa ) ⟩ |
| 220 | 335 ( g o h ) o i |
| 336 ∎ | |
| 337 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' ] ]} → | |
| 338 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 339 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 340 begin | |
| 341 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') | |
| 342 ≈⟨ idL ⟩ | |
| 343 k eqa (e eqa o k' ) (fhj=ghj k' eq') | |
| 344 ≈⟨ uniqueness eqa refl-hom ⟩ | |
| 345 k' | |
| 346 ∎ | |
| 347 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} → | |
| 348 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 349 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 350 begin | |
| 351 k eqa ( e eqa o h') (fhj=ghj h' eq') | |
| 352 ≈⟨ uniqueness eqa ( begin | |
| 353 e eqa o k' | |
| 354 ≈↑⟨ cdr idL ⟩ | |
| 355 e eqa o (id1 A c o k' ) | |
| 356 ≈⟨ cdr ik=h ⟩ | |
| 357 e eqa o h' | |
| 358 ∎ ) ⟩ | |
| 359 k' | |
| 360 ∎ | |
| 361 | |
| 223 | 362 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 |
| 224 | 363 iso-rev {a} {b} {c} {f} {g} eq eqa = k eqa (id1 A a) (f1=g1 eq (id1 A a)) |
| 223 | 364 |
| 224 | 365 equalizer-iso-pair : { a b c : Obj A } {f g : Hom A a b } → {eq : A [ f ≈ g ] } → ( eqa : Equalizer A {c} f g) → |
| 223 | 366 A [ A [ e eqa o iso-rev eq eqa ] ≈ id1 A a ] |
| 224 | 367 equalizer-iso-pair {a} {b} {c} {f} {g} {eq} eqa = ek=h eqa |
| 223 | 368 |
| 221 | 369 -- Equalizer is unique up to iso |
| 370 | |
| 371 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 ) | |
| 372 → Hom A c c' --- != id1 A c | |
| 373 equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (fe=ge eqa) | |
| 220 | 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 ) | |
| 224 | 381 { h : Hom A a c } → ( equalizer-iso-pair {a} {b} {c'} (eefeg eqa) ) |
| 382 → A [ A [ k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ] ≈ id1 A c ] | |
| 383 equalizer-iso-eq = ? | |
| 384 | |
| 385 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 | 386 { 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 ] |
| 224 | 387 equalizer-iso-eq' {c} {c'} {f} {g} eqa eqa' {h} rev = let open ≈-Reasoning (A) in |
| 221 | 388 begin |
| 222 | 389 k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) |
| 390 ≈↑⟨ idL ⟩ | |
| 391 (id1 A c) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 392 ≈↑⟨ car rev ⟩ | |
| 393 ( h o e eqa ) o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 394 ≈↑⟨ assoc ⟩ | |
| 395 h o ( e eqa o ( k eqa (e eqa' ) (fe=ge eqa') o k eqa' (e eqa ) (fe=ge eqa) ) ) | |
| 396 ≈⟨ cdr assoc ⟩ | |
| 397 h o (( e eqa o k eqa (e eqa' ) (fe=ge eqa')) o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 398 ≈⟨ cdr ( car (ek=h eqa) ) ⟩ | |
| 399 h o ( e eqa' o k eqa' (e eqa ) (fe=ge eqa) ) | |
| 400 ≈⟨ cdr (ek=h eqa' ) ⟩ | |
| 401 h o e eqa | |
| 402 ≈⟨ rev ⟩ | |
| 403 id1 A c | |
| 221 | 404 ∎ |
| 405 | |
| 406 -- ke = e' k'e' = e → k k' = 1 , k' k = 1 | |
| 407 -- ke = e' | |
| 408 -- k'ke = k'e' = e kk' = 1 | |
| 409 | |
| 410 -- x e = e -> x = id? | |
| 218 | 411 |
| 222 | 412 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → |
| 211 | 413 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g |
| 222 | 414 lemma-equ1 {a} {b} {c} f g eqa = record { |
| 216 | 415 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 416 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 | 417 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 221 | 418 b1 = fe=ge (eqa f g) ; |
| 212 | 419 b2 = lemma-equ5 ; |
| 420 b3 = lemma-equ3 ; | |
| 215 | 421 b4 = lemma-equ6 |
| 211 | 422 } where |
| 216 | 423 -- |
| 424 -- e eqa f g f | |
| 425 -- c ----------> a ------->b | |
| 426 -- ^ g | |
| 427 -- | | |
| 428 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
| 429 -- | | |
| 430 -- d | |
| 431 -- | |
| 432 -- | |
| 433 -- e o id1 ≈ e → k e ≈ id | |
| 434 | |
| 211 | 435 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 436 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 437 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 438 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 439 begin |
| 440 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 441 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 442 id1 A a |
| 443 ∎ | |
| 214 | 444 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 445 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 | 446 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 447 begin |
| 448 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 449 ≈⟨ assoc ⟩ | |
| 450 (f o h) o e (eqa (f o h) ( g o h )) | |
| 221 | 451 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
| 212 | 452 (g o h) o e (eqa (f o h) ( g o h )) |
| 453 ≈↑⟨ assoc ⟩ | |
| 454 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 455 ∎ | |
| 456 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 457 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 | 458 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 459 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 460 begin | |
| 215 | 461 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) |
| 462 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 463 h o e (eqa (f o h ) ( g o h )) |
| 464 ∎ | |
| 222 | 465 lemma-equ6 : {d : Obj A} {j : Hom A d c} → A [ |
| 466 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 ] ])) ]) | |
| 467 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o j ])) o | |
| 468 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 ] ])) ] | |
| 469 ≈ j ] | |
| 470 lemma-equ6 {d} {j} = let open ≈-Reasoning (A) in | |
| 215 | 471 begin |
| 222 | 472 ( 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 ) ))) )) |
| 473 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o j ))) o | |
| 474 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 | 475 ≈⟨ car ( uniqueness (eqa f g) ( begin |
| 222 | 476 e (eqa f g) o j |
| 215 | 477 ≈⟨ {!!} ⟩ |
| 222 | 478 (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 | 479 ∎ )) ⟩ |
| 222 | 480 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 ) ))) |
| 481 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o j ) )) (( f o ( e (eqa f g) o j ) ))) ( begin | |
| 482 e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) o id1 A d | |
| 483 ≈⟨ idR ⟩ | |
| 484 e (eqa (f o e (eqa f g) o j) (f o e (eqa f g) o j)) | |
| 215 | 485 ≈⟨ {!!} ⟩ |
| 222 | 486 id1 A d |
| 215 | 487 ∎ )) ⟩ |
| 222 | 488 j o id1 A d |
| 215 | 489 ≈⟨ idR ⟩ |
| 222 | 490 j |
| 215 | 491 ∎ |
| 211 | 492 |
| 493 | |
| 212 | 494 |
| 495 | |
| 496 | |
| 215 | 497 |
| 498 |