Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 218:749a1ecbc0b5
add equalizers
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 04 Sep 2013 14:51:36 +0900 |
| parents | 306f07bece85 |
| children | 2ae029454fb6 |
| 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 |
| 215 | 26 ef=eg : 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 | |
| 217 | 51 -- Equalizer is unique up to iso |
| 52 | |
| 53 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 ) | |
| 54 → Hom A c c' --- != id1 A c | |
| 55 equalizer-iso {c} eqa eqa' = k eqa' (e eqa) (ef=eg eqa) | |
| 56 | |
| 57 -- e eqa f g f | |
| 58 -- c ----------> a ------->b | |
| 218 | 59 -- ^ ---> d ---> |
| 60 -- | i h | |
| 61 -- j| k' (d' → d) | |
| 62 -- | k (d' → a) | |
| 63 -- d' | |
| 217 | 64 |
| 218 | 65 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 ) |
| 66 → A [ A [ h o i ] ≈ e eqa ] → A [ A [ h-1 o h ] ≈ id1 A d ] | |
| 217 | 67 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) |
| 218 | 68 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h h-1 eq h-1-id = record { |
| 69 e = i ; -- A [ h-1 o e eqa ] ; -- Hom A a d | |
| 217 | 70 ef=eg = ef=eg1 ; |
| 71 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; | |
| 72 ek=h = ek=h1 ; | |
| 73 uniqueness = uniqueness1 | |
| 74 } where | |
| 75 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
| 76 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 77 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 78 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 79 begin | |
| 80 f o ( h o j ) | |
| 81 ≈⟨ assoc ⟩ | |
| 82 (f o h ) o j | |
| 83 ≈⟨ eq' ⟩ | |
| 84 (g o h ) o j | |
| 85 ≈↑⟨ assoc ⟩ | |
| 86 g o ( h o j ) | |
| 87 ∎ | |
| 88 ef=eg1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] | |
| 89 ef=eg1 = let open ≈-Reasoning (A) in | |
| 90 begin | |
| 91 ( f o h ) o i | |
| 92 ≈↑⟨ assoc ⟩ | |
| 93 f o (h o i ) | |
| 94 ≈⟨ cdr eq ⟩ | |
| 95 f o (e eqa) | |
| 96 ≈⟨ ef=eg eqa ⟩ | |
| 97 g o (e eqa) | |
| 98 ≈↑⟨ cdr eq ⟩ | |
| 99 g o (h o i ) | |
| 100 ≈⟨ assoc ⟩ | |
| 101 ( g o h ) o i | |
| 102 ∎ | |
| 218 | 103 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' ] ]} → |
| 104 A [ A [ i o k eqa (A [ h o k' ]) (fhj=ghj k' eq') ] ≈ k' ] | |
| 105 ek=h1 {d'} {k'} {eq'} = let open ≈-Reasoning (A) in | |
| 217 | 106 begin |
| 218 | 107 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') |
| 108 ≈↑⟨ idL ⟩ | |
| 109 (id1 A d ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 110 ≈↑⟨ car h-1-id ⟩ | |
| 111 ( h-1 o h ) o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 112 ≈↑⟨ assoc ⟩ | |
| 113 h-1 o ( h o ( i o k eqa (h o k' ) (fhj=ghj k' eq')) ) | |
| 114 ≈⟨ cdr assoc ⟩ | |
| 115 h-1 o ( (h o i ) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 116 ≈⟨ cdr (car eq ) ⟩ | |
| 117 h-1 o ( (e eqa) o k eqa (h o k' ) (fhj=ghj k' eq')) | |
| 118 ≈⟨ cdr (ek=h eqa) ⟩ | |
| 119 h-1 o ( h o k' ) | |
| 120 ≈⟨ assoc ⟩ | |
| 121 ( h-1 o h ) o k' | |
| 122 ≈⟨ car h-1-id ⟩ | |
| 123 id1 A d o k' | |
| 124 ≈⟨ idL ⟩ | |
| 125 k' | |
| 217 | 126 ∎ |
| 127 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} → | |
| 128 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 129 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 130 begin | |
| 131 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 132 ≈⟨ uniqueness eqa ( begin | |
| 133 e eqa o k' | |
| 134 ≈↑⟨ car eq ⟩ | |
| 135 (h o i ) o k' | |
| 136 ≈↑⟨ assoc ⟩ | |
| 137 h o (i o k') | |
| 138 ≈⟨ cdr ik=h ⟩ | |
| 139 h o h' | |
| 140 ∎ ) ⟩ | |
| 141 k' | |
| 142 ∎ | |
| 215 | 143 |
| 218 | 144 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 ) |
| 145 → (h-1 : Hom A d b ) → A [ A [ h-1 o h ] ≈ id1 A b ] | |
| 146 → Equalizer A {c} (A [ h o f ]) (A [ h o g ] ) | |
| 147 h+equalizer {a} {b} {c} {d} {f} {g} eqa h h-1 h-1-id = record { | |
| 148 e = e eqa ; | |
| 149 ef=eg = ef=eg1 ; | |
| 150 k = λ j eq' → k eqa j (fj=gj j eq') ; | |
| 151 ek=h = ek=h1 ; | |
| 152 uniqueness = uniqueness1 | |
| 153 } where | |
| 154 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 ] ] | |
| 155 fj=gj j eq = let open ≈-Reasoning (A) in | |
| 156 begin | |
| 157 f o j | |
| 158 ≈↑⟨ idL ⟩ | |
| 159 id1 A b o ( f o j ) | |
| 160 ≈↑⟨ car h-1-id ⟩ | |
| 161 (h-1 o h ) o ( f o j ) | |
| 162 ≈↑⟨ assoc ⟩ | |
| 163 h-1 o (h o ( f o j )) | |
| 164 ≈⟨ cdr assoc ⟩ | |
| 165 h-1 o ((h o f) o j ) | |
| 166 ≈⟨ cdr eq ⟩ | |
| 167 h-1 o ((h o g) o j ) | |
| 168 ≈↑⟨ cdr assoc ⟩ | |
| 169 h-1 o (h o ( g o j )) | |
| 170 ≈⟨ assoc ⟩ | |
| 171 (h-1 o h) o ( g o j ) | |
| 172 ≈⟨ car h-1-id ⟩ | |
| 173 id1 A b o ( g o j ) | |
| 174 ≈⟨ idL ⟩ | |
| 175 g o j | |
| 176 ∎ | |
| 177 ef=eg1 : A [ A [ A [ h o f ] o e eqa ] ≈ A [ A [ h o g ] o e eqa ] ] | |
| 178 ef=eg1 = let open ≈-Reasoning (A) in | |
| 179 begin | |
| 180 ( h o f ) o e eqa | |
| 181 ≈↑⟨ assoc ⟩ | |
| 182 h o (f o e eqa ) | |
| 183 ≈⟨ cdr (ef=eg eqa) ⟩ | |
| 184 h o (g o e eqa ) | |
| 185 ≈⟨ assoc ⟩ | |
| 186 ( h o g ) o e eqa | |
| 187 ∎ | |
| 188 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 ] ]} → | |
| 189 A [ A [ e eqa o k eqa j (fj=gj j eq) ] ≈ j ] | |
| 190 ek=h1 {d₁} {j} {eq} = ek=h eqa | |
| 191 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} → | |
| 192 A [ A [ e eqa o k' ] ≈ j ] → A [ k eqa j (fj=gj j eq) ≈ k' ] | |
| 193 uniqueness1 = uniqueness eqa | |
| 194 | |
| 195 | |
| 211 | 196 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
| 197 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
| 209 | 198 lemma-equ1 A {a} {b} {c} f g eqa = record { |
| 216 | 199 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 200 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 | 201 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 211 | 202 b1 = ef=eg (eqa f g) ; |
| 212 | 203 b2 = lemma-equ5 ; |
| 204 b3 = lemma-equ3 ; | |
| 215 | 205 b4 = lemma-equ6 |
| 211 | 206 } where |
| 216 | 207 -- |
| 208 -- e eqa f g f | |
| 209 -- c ----------> a ------->b | |
| 210 -- ^ g | |
| 211 -- | | |
| 212 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
| 213 -- | | |
| 214 -- d | |
| 215 -- | |
| 216 -- | |
| 217 -- e o id1 ≈ e → k e ≈ id | |
| 218 | |
| 211 | 219 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 220 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 221 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 222 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 223 begin |
| 224 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 225 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 226 id1 A a |
| 227 ∎ | |
| 214 | 228 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 229 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 | 230 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 231 begin |
| 232 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 233 ≈⟨ assoc ⟩ | |
| 234 (f o h) o e (eqa (f o h) ( g o h )) | |
| 235 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
| 236 (g o h) o e (eqa (f o h) ( g o h )) | |
| 237 ≈↑⟨ assoc ⟩ | |
| 238 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 239 ∎ | |
| 240 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 241 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 | 242 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 243 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 244 begin | |
| 215 | 245 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) |
| 246 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 247 h o e (eqa (f o h ) ( g o h )) |
| 248 ∎ | |
| 215 | 249 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
| 250 A [ k (eqa f g) (A [ A [ e (eqa f g) o k₁ ] o e (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ g o A [ e (eqa f g) o k₁ ] ])) ]) | |
| 251 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
| 252 k (eqa (A [ f o A [ e (eqa f g) o k₁ ] ]) (A [ f o A [ e (eqa f g) o k₁ ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ e (eqa f g) o k₁ ] ])) ] | |
| 253 ≈ k₁ ] | |
| 254 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
| 255 begin | |
| 256 ( k (eqa f g) (( ( e (eqa f g) o k₁ ) o e (eqa (( f o ( e (eqa f g) o k₁ ) )) (( g o ( e (eqa f g) o k₁ ) ))) )) | |
| 257 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
| 258 k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) ) | |
| 259 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
| 260 e (eqa f g) o k₁ | |
| 261 ≈⟨ {!!} ⟩ | |
| 262 (e (eqa f g) o k₁) o e (eqa (f o e (eqa f g) o k₁) (g o e (eqa f g) o k₁)) | |
| 263 ∎ )) ⟩ | |
| 264 k₁ o k (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) (id1 A d) (lemma-equ2 (( f o ( e (eqa f g) o k₁ ) ))) | |
| 265 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
| 266 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
| 267 ≈⟨ {!!} ⟩ | |
| 268 id1 A d | |
| 269 ∎ )) ⟩ | |
| 270 k₁ o id1 A d | |
| 271 ≈⟨ idR ⟩ | |
| 272 k₁ | |
| 273 ∎ | |
| 211 | 274 |
| 275 | |
| 212 | 276 |
| 277 | |
| 278 | |
| 215 | 279 |
| 280 |