Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 233:4bba19bc71be
e is now explict parameter
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 08 Sep 2013 01:37:24 +0900 |
| parents | b0fe61882014 |
| children | c02287d3d2dc |
| 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 | |
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23 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 205 | 24 field |
| 221 | 25 fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] |
| 209 | 26 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
| 215 | 27 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 | 28 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 | 29 A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] |
| 209 | 30 equalizer : Hom A c a |
| 31 equalizer = e | |
| 206 | 32 |
| 230 | 33 -- |
| 225 | 34 -- Flat Equational Definition of Equalizer |
| 230 | 35 -- |
| 225 | 36 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 | 37 field |
| 212 | 38 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
| 214 | 39 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 | 40 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c |
| 213 | 41 b1 : A [ A [ f o α {a} {b} {a} f g ] ≈ A [ g o α f g ] ] |
| 214 | 42 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 | 43 b3 : A [ A [ α f f o δ {a} {b} {a} f ] ≈ id1 A a ] |
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44 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
| 230 | 45 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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46 -- A [ α f g o β f g h ] ≈ h |
| 214 | 47 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 | 48 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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49 |
| 209 | 50 open Equalizer |
| 225 | 51 open Burroni |
| 209 | 52 |
| 225 | 53 -- |
| 54 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 55 -- | |
| 219 | 56 |
| 224 | 57 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 ] ] |
| 58 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 59 | |
| 226 | 60 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 | 61 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 62 |
| 224 | 63 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 64 (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 | 65 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 66 begin |
| 67 f o ( e o h ) | |
| 68 ≈⟨ assoc ⟩ | |
| 230 | 69 (f o e ) o h |
| 224 | 70 ≈⟨ car eq ⟩ |
| 230 | 71 (g o e ) o h |
| 224 | 72 ≈↑⟨ assoc ⟩ |
| 73 g o ( e o h ) | |
| 74 ∎ | |
| 219 | 75 |
| 225 | 76 -- |
| 77 -- | |
| 78 -- An isomorphic element c' of c makes another equalizer | |
| 79 -- | |
| 230 | 80 -- e eqa f g f |
| 222 | 81 -- c ----------> a ------->b |
| 230 | 82 -- |^ |
| 83 -- || | |
| 222 | 84 -- h || h-1 |
| 230 | 85 -- v| |
| 86 -- c' | |
| 222 | 87 |
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88 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } { e' : Hom A c' a } |
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89 ( fe=ge' : A [ A [ f o e' ] ≈ A [ g o e' ] ] ) |
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90 ( eqa : Equalizer A e f g ) → (h-1 : Hom A c' c ) → (h : Hom A c c' ) → |
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91 A [ A [ e o h-1 ] ≈ e' ] → A [ A [ e' o h ] ≈ e ] |
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92 → Equalizer A e' f g |
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93 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} {e'} fe=ge' eqa h-1 h e→e' e'→e = record { |
| 222 | 94 fe=ge = fe=ge1 ; |
| 95 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 96 ek=h = ek=h1 ; |
| 222 | 97 uniqueness = uniqueness1 |
| 98 } where | |
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99 fe=ge1 : A [ A [ f o e' ] ≈ A [ g o e' ] ] |
| 230 | 100 fe=ge1 = let open ≈-Reasoning (A) in |
| 222 | 101 begin |
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102 f o e' |
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103 ≈↑⟨ cdr e→e' ⟩ |
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104 f o ( e o h-1 ) |
| 222 | 105 ≈⟨ assoc ⟩ |
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106 (f o e ) o h-1 |
| 222 | 107 ≈⟨ car (fe=ge eqa) ⟩ |
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108 (g o e ) o h-1 |
| 222 | 109 ≈↑⟨ assoc ⟩ |
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110 g o ( e o h-1 ) |
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111 ≈⟨ cdr e→e' ⟩ |
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112 g o e' |
| 222 | 113 ∎ |
| 114 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → | |
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115 A [ A [ e' o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 116 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 117 begin | |
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118 e' o ( h o k eqa j eq ) |
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119 ≈⟨ assoc ⟩ |
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120 ( e' o h) o k eqa j eq |
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121 ≈⟨ car e'→e ⟩ |
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122 e o k eqa j eq |
| 222 | 123 ≈⟨ ek=h eqa ⟩ |
| 124 j | |
| 125 ∎ | |
| 126 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
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127 A [ A [ e' o j ] ≈ h' ] → |
| 222 | 128 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 129 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 130 begin | |
| 131 h o k eqa h' eq | |
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132 ≈⟨ {!!} ⟩ |
| 222 | 133 j |
| 134 ∎ | |
| 135 | |
| 217 | 136 |
| 220 | 137 |
| 225 | 138 -- |
| 139 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 140 -- | |
| 141 -- h : c → c' h' : c' → c | |
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142 -- e' = h o e |
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143 -- e = h' o e' |
| 225 | 144 |
| 145 | |
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146 c-iso-l : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } { e' : Hom A c' a } |
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147 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
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148 → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) ) |
| 230 | 149 → Hom A c c' |
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150 c-iso-l {c} {c'} eqa eqa' eff = {!!} |
| 226 | 151 |
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152 c-iso-r : { c c' a b : Obj A } {f g : Hom A a b } {e : Hom A c a } {e' : Hom A c' a} → ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
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153 → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) ) |
| 230 | 154 → Hom A c' c |
| 155 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 223 | 156 |
| 227 | 157 |
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158 -- e(eqa') f |
| 230 | 159 -- c'----------> a ------->b f e j = g e j |
| 160 -- ^ g | |
| 161 -- |k h | |
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162 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 163 -- | |
| 164 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 165 -- | |
| 166 -- h j e f = h j e g → h = 'j e f | |
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167 -- h = j e f -> j = j' |
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168 -- |
| 228 | 169 |
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170 c-iso : { c c' a b : Obj A } {f g : Hom A a b } → {e : Hom A c a } {e' : Hom A c' a} ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
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171 → ( keqa : Equalizer A (k eqa' e {!!} ) (A [ f o e' ]) (A [ g o e' ]) ) |
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172 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
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173 c-iso {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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174 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
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175 ≈⟨ {!!} ⟩ |
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176 id1 A c' |
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177 ∎ |
| 226 | 178 |
| 230 | 179 |
| 225 | 180 ---- |
| 181 -- | |
| 182 -- An equalizer satisfies Burroni equations | |
| 183 -- | |
| 230 | 184 -- b4 is not yet done |
| 225 | 185 ---- |
| 186 | |
| 230 | 187 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → |
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188 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } { fe=ge1 : A [ A [ f o e ] ≈ A [ g o e ] ] } → Equalizer A e f g ) |
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189 → Burroni A {c} f g |
| 222 | 190 lemma-equ1 {a} {b} {c} f g eqa = record { |
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191 α = λ f g → equalizer (eqa f g ) ; -- Hom A c a |
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192 γ = λ {a} {b} {c} {d} f g h → k (eqa f g ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d |
| 213 | 193 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 221 | 194 b1 = fe=ge (eqa f g) ; |
| 226 | 195 b2 = lemma-b2 ; |
| 196 b3 = lemma-b3 ; | |
| 230 | 197 b4 = lemma-b4 |
| 211 | 198 } where |
| 216 | 199 -- |
| 200 -- e eqa f g f | |
| 201 -- c ----------> a ------->b | |
| 230 | 202 -- ^ g |
| 203 -- | | |
| 216 | 204 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 205 -- | |
| 216 | 206 -- d |
| 230 | 207 -- |
| 208 -- | |
| 216 | 209 -- e o id1 ≈ e → k e ≈ id |
| 210 | |
| 211 | 211 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 212 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
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213 lemma-b3 : A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 226 | 214 lemma-b3 = let open ≈-Reasoning (A) in |
| 230 | 215 begin |
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216 equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
| 215 | 217 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 218 id1 A a |
| 219 ∎ | |
| 230 | 220 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
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221 A [ A [ f o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ≈ A [ g o A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] ] |
| 214 | 222 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 223 begin |
|
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224 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 225 ≈⟨ assoc ⟩ |
|
233
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226 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 227 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
|
233
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228 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 229 ≈↑⟨ assoc ⟩ |
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230 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 231 ∎ |
| 230 | 232 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
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233 A [ equalizer (eqa f g) o k (eqa f g) (A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ]) (lemma-equ4 {a} {b} {c} f g h) ] |
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234 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 235 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 236 begin |
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237 equalizer (eqa f g) o k (eqa f g) (h o equalizer (eqa (f o h) (g o h))) (lemma-equ4 {a} {b} {c} f g h) |
| 215 | 238 ≈⟨ ek=h (eqa f g) ⟩ |
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239 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 240 ∎ |
| 230 | 241 |
| 242 ------- α(f,g)j id d = α(f,g)j | |
| 243 ------- α(f,g)j id d = α(f,g)j | |
| 244 ------- α(f,g)j α(fα(f,g)j,fα(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 245 ------ fα = gα | |
| 246 ------- α(f,g)j α(fα(f,g)j,gα(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 247 ------- α(f,g) γ(f,g,α(f,g)j) δ(fα(f,g)j) = α(f,g)j | |
| 248 ------- γ(f,g,α(f,g)j) δ(fα(f,g)j) = j | |
| 249 | |
| 250 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
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251 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o equalizer (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ g o A [ equalizer (eqa f g) o j ] ])) ]) |
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252 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
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253 k (eqa (A [ f o A [ equalizer (eqa f g) o j ] ]) (A [ f o A [ equalizer (eqa f g) o j ] ])) (id1 A d) (lemma-equ2 (A [ f o A [ equalizer (eqa f g) o j ] ])) ] |
| 222 | 254 ≈ j ] |
| 230 | 255 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 256 begin |
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257 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g) o j ) )) (( g o ( equalizer (eqa f g) o j ) ))) )) |
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258 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
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259 k (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) (id1 A d) (lemma-equ2 (( f o ( equalizer (eqa f g) o j ) ))) ) |
| 230 | 260 ≈⟨ {!!} ⟩ |
| 222 | 261 j |
| 215 | 262 ∎ |
| 211 | 263 |
| 264 | |
| 225 | 265 -- end |
| 212 | 266 |
| 267 | |
| 268 | |
| 215 | 269 |
| 270 |