Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 247:f6e8d6d04af8
equalizer done.
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 09 Sep 2013 12:47:53 +0900 |
| parents | 80d9ef47566b |
| children | efa2fd0e91ee |
| 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 -- |
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36 record Burroni { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (f g : Hom A a b) (e : Hom A c a) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where |
| 206 | 37 field |
| 245 | 38 α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (e : Hom A c a ) → 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 |
| 245 | 40 δ : {a b c : Obj A } → (e : Hom A c a ) → (f : Hom A a b) → Hom A a c |
| 242 | 41 cong-α : {a b c : Obj A } → { e : Hom A c a } |
| 245 | 42 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ α f g e ≈ α f g' e ] |
| 242 | 43 cong-γ : {a _ c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] |
| 243 | 44 → A [ γ {a} {b} {c} {d} f g h ≈ γ f g h' ] |
| 245 | 45 cong-δ : {a b c : Obj A } → {e : Hom A c a} → {f f' : Hom A a b} → A [ f ≈ f' ] → A [ δ e f ≈ δ e f' ] |
| 46 b1 : A [ A [ f o α {a} {b} {c} f g e ] ≈ A [ g o α {a} {b} {c} f g e ] ] | |
| 47 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α {a} {b} {c} f g e ) o (γ {a} {b} {c} f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) (id1 A d) ] ] | |
| 48 b3 : {a b d : Obj A} → (f : Hom A a b ) → {h : Hom A d a } → A [ A [ α {a} {b} {d} f f h o δ {a} {b} {d} h f ] ≈ id1 A a ] | |
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49 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
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50 b4 : {d : Obj A } {k : Hom A d c} → |
| 245 | 51 A [ A [ γ {a} {b} {c} {d} f g ( A [ α {a} {b} {c} f g e o k ] ) o ( δ {d} {b} {d} (id1 A d) (A [ f o A [ α {a} {b} {c} f g e o k ] ] ) )] ≈ k ] |
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52 -- A [ α f g o β f g h ] ≈ h |
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53 β : { d a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A d c |
| 245 | 54 β {d} {a} {b} f g h = A [ γ {a} {b} {c} f g h o δ {d} {b} {d} (id1 A d) (A [ f o h ]) ] |
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55 |
| 209 | 56 open Equalizer |
| 225 | 57 open Burroni |
| 209 | 58 |
| 225 | 59 -- |
| 60 -- Some obvious conditions for k (fe = ge) → ( fh = gh ) | |
| 61 -- | |
| 219 | 62 |
| 224 | 63 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 ] ] |
| 64 f1=g1 eq h = let open ≈-Reasoning (A) in (resp refl-hom eq ) | |
| 65 | |
| 226 | 66 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 | 67 f1=f1 f = let open ≈-Reasoning (A) in refl-hom |
| 226 | 68 |
| 224 | 69 f1=gh : { a b c d : Obj A } {f g : Hom A a b } → { e : Hom A c a } → { h : Hom A d c } → |
| 70 (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 | 71 f1=gh {a} {b} {c} {d} {f} {g} {e} {h} eq = let open ≈-Reasoning (A) in |
| 224 | 72 begin |
| 73 f o ( e o h ) | |
| 74 ≈⟨ assoc ⟩ | |
| 230 | 75 (f o e ) o h |
| 224 | 76 ≈⟨ car eq ⟩ |
| 230 | 77 (g o e ) o h |
| 224 | 78 ≈↑⟨ assoc ⟩ |
| 79 g o ( e o h ) | |
| 80 ∎ | |
| 219 | 81 |
| 225 | 82 -- |
| 83 -- | |
| 84 -- An isomorphic element c' of c makes another equalizer | |
| 85 -- | |
| 230 | 86 -- e eqa f g f |
| 222 | 87 -- c ----------> a ------->b |
| 230 | 88 -- |^ |
| 89 -- || | |
| 222 | 90 -- h || h-1 |
| 230 | 91 -- v| |
| 92 -- c' | |
| 222 | 93 |
| 234 | 94 equalizer+iso : {a b c c' : Obj A } {f g : Hom A a b } {e : Hom A c a } |
| 95 (h-1 : Hom A c' c ) → (h : Hom A c c' ) → | |
| 96 A [ A [ h o h-1 ] ≈ id1 A c' ] → A [ A [ h-1 o h ] ≈ id1 A c ] → | |
| 97 ( fe=ge' : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] ) | |
| 98 ( eqa : Equalizer A e f g ) | |
| 99 → Equalizer A (A [ e o h-1 ] ) f g | |
| 100 equalizer+iso {a} {b} {c} {c'} {f} {g} {e} h-1 h hh-1=1 h-1h=1 fe=ge' eqa = record { | |
| 222 | 101 fe=ge = fe=ge1 ; |
| 102 k = λ j eq → A [ h o k eqa j eq ] ; | |
| 230 | 103 ek=h = ek=h1 ; |
| 222 | 104 uniqueness = uniqueness1 |
| 105 } where | |
| 234 | 106 fe=ge1 : A [ A [ f o A [ e o h-1 ] ] ≈ A [ g o A [ e o h-1 ] ] ] |
| 107 fe=ge1 = fe=ge' | |
| 222 | 108 ek=h1 : {d : Obj A} {j : Hom A d a} {eq : A [ A [ f o j ] ≈ A [ g o j ] ]} → |
| 234 | 109 A [ A [ A [ e o h-1 ] o A [ h o k eqa j eq ] ] ≈ j ] |
| 222 | 110 ek=h1 {d} {j} {eq} = let open ≈-Reasoning (A) in |
| 111 begin | |
| 234 | 112 ( e o h-1 ) o ( h o k eqa j eq ) |
| 113 ≈↑⟨ assoc ⟩ | |
| 114 e o ( h-1 o ( h o k eqa j eq ) ) | |
| 115 ≈⟨ cdr assoc ⟩ | |
| 116 e o (( h-1 o h) o k eqa j eq ) | |
| 117 ≈⟨ cdr (car h-1h=1 ) ⟩ | |
| 118 e o (id1 A c o k eqa j eq ) | |
| 119 ≈⟨ cdr idL ⟩ | |
| 120 e o k eqa j eq | |
| 222 | 121 ≈⟨ ek=h eqa ⟩ |
| 122 j | |
| 123 ∎ | |
| 124 uniqueness1 : {d : Obj A} {h' : Hom A d a} {eq : A [ A [ f o h' ] ≈ A [ g o h' ] ]} {j : Hom A d c'} → | |
| 234 | 125 A [ A [ A [ e o h-1 ] o j ] ≈ h' ] → |
| 222 | 126 A [ A [ h o k eqa h' eq ] ≈ j ] |
| 127 uniqueness1 {d} {h'} {eq} {j} ej=h = let open ≈-Reasoning (A) in | |
| 128 begin | |
| 129 h o k eqa h' eq | |
| 234 | 130 ≈⟨ cdr (uniqueness eqa ( begin |
| 131 e o ( h-1 o j ) | |
| 132 ≈⟨ assoc ⟩ | |
| 133 (e o h-1 ) o j | |
| 134 ≈⟨ ej=h ⟩ | |
| 135 h' | |
| 136 ∎ )) ⟩ | |
| 137 h o ( h-1 o j ) | |
| 138 ≈⟨ assoc ⟩ | |
| 139 (h o h-1 ) o j | |
| 140 ≈⟨ car hh-1=1 ⟩ | |
| 141 id1 A c' o j | |
| 142 ≈⟨ idL ⟩ | |
| 222 | 143 j |
| 144 ∎ | |
| 145 | |
| 225 | 146 -- |
| 147 -- If we have two equalizers on c and c', there are isomorphic pair h, h' | |
| 148 -- | |
| 149 -- h : c → c' h' : c' → c | |
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150 -- e' = h o e |
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151 -- e = h' o e' |
| 225 | 152 |
| 153 | |
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154 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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155 ( eqa : Equalizer A e f g) → ( eqa' : Equalizer A e' f g ) |
| 234 | 156 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 157 → Hom A c c' -- should be e' = c-sio-l o e |
| 234 | 158 c-iso-l {c} {c'} eqa eqa' keqa = equalizer keqa |
| 226 | 159 |
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160 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 ) |
| 234 | 161 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) |
| 241 | 162 → Hom A c' c -- e = c-sio-r o e' |
| 230 | 163 c-iso-r {c} {c'} eqa eqa' keqa = k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) |
| 223 | 164 |
| 234 | 165 -- e' f |
| 230 | 166 -- c'----------> a ------->b f e j = g e j |
| 167 -- ^ g | |
| 168 -- |k h | |
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169 -- | h = e(eqaj) o k jhek = jh (uniqueness) |
| 230 | 170 -- | |
| 171 -- c j o (k (eqa ef ef) j ) = id c h = e(eqaj) | |
| 172 -- | |
| 173 -- h j e f = h j e g → h = 'j e f | |
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174 -- h = j e f -> j = j' |
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175 -- |
| 228 | 176 |
| 241 | 177 -- e = c-iso-l o e' is assumed by equalizer's degree of freedom |
| 234 | 178 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 ) |
| 179 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa)) (A [ f o e' ]) (A [ g o e' ]) ) | |
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180 → A [ A [ c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa ] ≈ id1 A c' ] |
| 234 | 181 c-iso→ {c} {c'} {a} {b} {f} {g} eqa eqa' keqa = let open ≈-Reasoning (A) in begin |
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182 c-iso-l eqa eqa' keqa o c-iso-r eqa eqa' keqa |
| 234 | 183 ≈⟨⟩ |
| 184 equalizer keqa o k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) | |
| 185 ≈⟨ ek=h keqa ⟩ | |
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186 id1 A c' |
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187 ∎ |
| 226 | 188 |
| 234 | 189 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 ) |
| 190 → ( keqa : Equalizer A (k eqa' e (fe=ge eqa )) (A [ f o e' ]) (A [ g o e' ]) ) | |
| 191 → ( keqa' : Equalizer A (k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') )) (A [ f o e ]) (A [ g o e ]) ) | |
| 241 | 192 -- e' = c-iso-r o e is assumed by equalizer's degree of freedom |
| 193 → { e'->e : A [ e' ≈ A [ e o equalizer keqa' ] ] } -- implicit assumption , which should be refl | |
| 234 | 194 → A [ A [ c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa ] ≈ id1 A c ] |
| 195 c-iso← {c} {c'} {a} {b} {f} {g} {e} {e'} eqa eqa' keqa keqa' {e'->e} = let open ≈-Reasoning (A) in begin | |
| 196 c-iso-r eqa eqa' keqa o c-iso-l eqa eqa' keqa | |
| 197 ≈⟨⟩ | |
| 198 k keqa (id1 A c') ( f1=g1 (fe=ge eqa') (id1 A c') ) o k eqa' e (fe=ge eqa ) | |
| 199 ≈⟨⟩ | |
| 200 equalizer keqa' o k eqa' e (fe=ge eqa ) | |
| 201 ≈⟨ cdr ( begin | |
| 202 k eqa' e (fe=ge eqa ) | |
| 203 ≈⟨ uniqueness eqa' ( begin | |
| 204 e' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
| 205 ≈⟨ car e'->e ⟩ | |
| 206 ( e o equalizer keqa' ) o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c)) | |
| 207 ≈↑⟨ assoc ⟩ | |
| 208 e o ( equalizer keqa' o k keqa' (id1 A c) (f1=g1 (fe=ge eqa) (id1 A c))) | |
| 209 ≈⟨ cdr ( ek=h keqa' ) ⟩ | |
| 210 e o id1 A c | |
| 211 ≈⟨ idR ⟩ | |
| 212 e | |
| 213 ∎ )⟩ | |
| 214 k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
| 215 ∎ )⟩ | |
| 216 equalizer keqa' o k keqa' (id1 A c) ( f1=g1 (fe=ge eqa) (id1 A c) ) | |
| 217 ≈⟨ ek=h keqa' ⟩ | |
| 218 id1 A c | |
| 219 ∎ | |
| 220 | |
| 221 | |
| 230 | 222 |
| 225 | 223 ---- |
| 224 -- | |
| 225 -- An equalizer satisfies Burroni equations | |
| 226 -- | |
| 241 | 227 -- congs are not yet done |
| 225 | 228 ---- |
| 229 | |
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230 lemma-equ1 : {a b c : Obj A} (f g : Hom A a b) → (e : Hom A c a ) → |
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231 ( eqa : {a b c : Obj A} → (f g : Hom A a b) → {e : Hom A c a } → Equalizer A e f g ) |
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232 → Burroni A {c} {a} {b} f g e |
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233 lemma-equ1 {a} {b} {c} f g e eqa = record { |
| 245 | 234 α = λ {a} {b} {c} f g e → equalizer (eqa {a} {b} {c} f g {e} ) ; -- Hom A c a |
| 242 | 235 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 ] ))) ] ) |
| 236 (lemma-equ4 {a} {b} {c} {d} f g h ) ; -- Hom A c d | |
| 245 | 237 δ = λ {a} {b} {c} e f → k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f); -- Hom A a c |
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238 cong-α = λ {a b c e f g g'} eq → cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq ; |
| 247 | 239 cong-γ = λ {a} {_} {c} {d} {f} {g} {h} {h'} eq → cong-γ1 {a} {c} {d} {f} {g} {h} {h'} eq ; |
| 245 | 240 cong-δ = λ {a b c e f f'} f=f' → cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' ; |
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241 b1 = fe=ge (eqa {a} {b} {c} f g {e}) ; |
| 226 | 242 b2 = lemma-b2 ; |
| 243 b3 = lemma-b3 ; | |
| 230 | 244 b4 = lemma-b4 |
| 211 | 245 } where |
| 216 | 246 -- |
| 247 -- e eqa f g f | |
| 248 -- c ----------> a ------->b | |
| 230 | 249 -- ^ g |
| 250 -- | | |
| 216 | 251 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) |
| 230 | 252 -- | |
| 216 | 253 -- d |
| 230 | 254 -- |
| 255 -- | |
| 216 | 256 -- e o id1 ≈ e → k e ≈ id |
| 257 | |
| 211 | 258 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 259 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 240 | 260 lemma-b3 : {a b d : Obj A} (f : Hom A a b ) { h : Hom A d a } → A [ A [ equalizer (eqa f f ) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 261 lemma-b3 {a} {b} {d} f {h} = let open ≈-Reasoning (A) in | |
| 230 | 262 begin |
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diff
changeset
|
263 equalizer (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) |
| 215 | 264 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 265 id1 A a |
| 266 ∎ | |
| 230 | 267 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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268 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 | 269 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 270 begin |
|
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diff
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|
271 f o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 272 ≈⟨ assoc ⟩ |
|
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273 (f o h) o equalizer (eqa (f o h) ( g o h )) |
| 221 | 274 ≈⟨ fe=ge (eqa (A [ f o h ]) (A [ g o h ])) ⟩ |
|
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275 (g o h) o equalizer (eqa (f o h) ( g o h )) |
| 212 | 276 ≈↑⟨ assoc ⟩ |
|
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277 g o ( h o equalizer (eqa (f o h) ( g o h ))) |
| 212 | 278 ∎ |
| 245 | 279 cong-α1 : {a b c : Obj A } → { e : Hom A c a } |
| 280 → {f g g' : Hom A a b } → A [ g ≈ g' ] → A [ equalizer (eqa {a} {b} {c} f g {e} )≈ equalizer (eqa {a} {b} {c} f g' {e} ) ] | |
| 281 cong-α1 {a} {b} {c} {e} {f} {g} {g'} eq = let open ≈-Reasoning (A) in refl-hom | |
| 247 | 282 cong-γ1 : {a c d : Obj A } → {f g : Hom A a b} {h h' : Hom A d a } → A [ h ≈ h' ] → { e : Hom A c a} → |
|
246
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diff
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|
283 A [ k (eqa f g {e} ) {d} ( A [ h o (equalizer ( eqa (A [ f o h ] ) (A [ g o h ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h ) |
|
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diff
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284 ≈ k (eqa f g {e} ) {d} ( A [ h' o (equalizer ( eqa (A [ f o h' ] ) (A [ g o h' ] ) {id1 A d} )) ] ) (lemma-equ4 {a} {b} {c} {d} f g h' ) ] |
| 247 | 285 cong-γ1 {a} {c} {d} {f} {g} {h} {h'} h=h' {e} = let open ≈-Reasoning (A) in begin |
| 245 | 286 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 ) |
| 287 ≈⟨ uniqueness (eqa f g) ( begin | |
|
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288 e o ( k (eqa f f {e}) (id1 A a) (f1=f1 f) o h) |
|
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|
289 ≈⟨ assoc ⟩ |
|
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diff
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290 (e o k (eqa f f {e}) (id1 A a) (f1=f1 f)) o h |
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diff
changeset
|
291 ≈⟨ car ( ek=h (eqa f f {e})) ⟩ |
|
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diff
changeset
|
292 id1 A a o h |
|
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diff
changeset
|
293 ≈⟨ idL ⟩ |
|
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diff
changeset
|
294 h |
|
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diff
changeset
|
295 ≈↑⟨ idR ⟩ |
|
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diff
changeset
|
296 h o (id1 A d ) |
|
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diff
changeset
|
297 ≈⟨⟩ |
| 245 | 298 h o equalizer (eqa ( f o h ) ( g o h )) |
| 299 ∎ )⟩ | |
|
246
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diff
changeset
|
300 k (eqa f f {e}) (id1 A a) (f1=f1 f) o h |
|
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diff
changeset
|
301 ≈⟨ cdr h=h' ⟩ |
|
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|
302 k (eqa f f {e}) (id1 A a) (f1=f1 f) o h' |
|
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diff
changeset
|
303 ≈↑⟨ uniqueness (eqa f g) ( begin |
|
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diff
changeset
|
304 e o ( k (eqa f f {e}) (id1 A a) (f1=f1 f) o h') |
|
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diff
changeset
|
305 ≈⟨ assoc ⟩ |
|
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changeset
|
306 (e o k (eqa f f {e}) (id1 A a) (f1=f1 f)) o h' |
|
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changeset
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307 ≈⟨ car ( ek=h (eqa f f {e})) ⟩ |
|
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diff
changeset
|
308 id1 A a o h' |
|
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diff
changeset
|
309 ≈⟨ idL ⟩ |
|
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changeset
|
310 h' |
|
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|
311 ≈↑⟨ idR ⟩ |
|
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changeset
|
312 h' o (id1 A d ) |
|
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diff
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|
313 ≈⟨⟩ |
|
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changeset
|
314 h' o equalizer (eqa ( f o h' ) ( g o h' )) |
|
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diff
changeset
|
315 ∎ )⟩ |
| 245 | 316 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' ) |
| 317 ∎ | |
| 318 cong-δ1 : {a b c : Obj A} {e : Hom A c a } {f f' : Hom A a b} → A [ f ≈ f' ] → A [ k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f) ≈ | |
| 319 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (lemma-equ2 f') ] | |
| 247 | 320 cong-δ1 {a} {b} {c} {e} {f} {f'} f=f' = let open ≈-Reasoning (A) in |
| 321 begin | |
| 322 k (eqa {a} {b} {c} f f {e} ) (id1 A a) (lemma-equ2 f) | |
| 323 ≈⟨ uniqueness (eqa f f) ( begin | |
| 324 e o k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (lemma-equ2 f') | |
| 325 ≈⟨ ek=h (eqa {a} {b} {c} f' f' {e} ) ⟩ | |
| 326 id1 A a | |
| 327 ∎ )⟩ | |
| 328 k (eqa {a} {b} {c} f' f' {e} ) (id1 A a) (lemma-equ2 f') | |
| 329 ∎ | |
| 330 | |
| 230 | 331 lemma-b2 : {d : Obj A} {h : Hom A d a} → A [ |
|
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diff
changeset
|
332 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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diff
changeset
|
333 ≈ A [ h o equalizer (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 226 | 334 lemma-b2 {d} {h} = let open ≈-Reasoning (A) in |
| 212 | 335 begin |
|
233
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diff
changeset
|
336 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 | 337 ≈⟨ ek=h (eqa f g) ⟩ |
|
233
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diff
changeset
|
338 h o equalizer (eqa (f o h ) ( g o h )) |
| 212 | 339 ∎ |
| 230 | 340 |
| 341 lemma-b4 : {d : Obj A} {j : Hom A d c} → A [ | |
|
236
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diff
changeset
|
342 A [ k (eqa f g) (A [ A [ equalizer (eqa f g) o j ] o |
|
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diff
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|
343 equalizer (eqa (A [ f o A [ equalizer (eqa f g {e}) o j ] ]) (A [ g o A [ equalizer (eqa f g {e} ) o j ] ])) ]) |
|
233
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diff
changeset
|
344 (lemma-equ4 {a} {b} {c} f g (A [ equalizer (eqa f g) o j ])) o |
|
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232
diff
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|
345 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 | 346 ≈ j ] |
| 230 | 347 lemma-b4 {d} {j} = let open ≈-Reasoning (A) in |
| 215 | 348 begin |
|
236
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parents:
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diff
changeset
|
349 ( k (eqa f g) (( ( equalizer (eqa f g) o j ) o equalizer (eqa (( f o ( equalizer (eqa f g {e}) o j ) )) (( g o ( equalizer (eqa f g {e}) o j ) ))) )) |
|
233
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diff
changeset
|
350 (lemma-equ4 {a} {b} {c} f g (( equalizer (eqa f g) o j ))) o |
|
236
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diff
changeset
|
351 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 ) ))) ) |
|
235
8835015a3e1a
passed let;s remove yellow
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parents:
234
diff
changeset
|
352 ≈⟨ car ((uniqueness (eqa f g) ( begin |
|
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234
diff
changeset
|
353 equalizer (eqa f g) o j |
|
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234
diff
changeset
|
354 ≈↑⟨ idR ⟩ |
|
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234
diff
changeset
|
355 (equalizer (eqa f g) o j ) o id1 A d |
| 241 | 356 ≈⟨⟩ -- here we decide e (fej) (gej)' is id1 A d |
|
236
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235
diff
changeset
|
357 ((equalizer (eqa f g) o j) o equalizer (eqa (f o equalizer (eqa f g {e}) o j) (g o equalizer (eqa f g {e}) o j))) |
|
235
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parents:
234
diff
changeset
|
358 ∎ ))) ⟩ |
|
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234
diff
changeset
|
359 j o 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 ) ))) |
|
8835015a3e1a
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234
diff
changeset
|
360 ≈⟨ cdr ((uniqueness (eqa (( f o ( equalizer (eqa f g) o j ) )) (( f o ( equalizer (eqa f g) o j ) ))) ( begin |
|
236
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diff
changeset
|
361 equalizer (eqa (f o equalizer (eqa f g {e} ) o j) (f o equalizer (eqa f g {e}) o j)) o id1 A d |
|
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diff
changeset
|
362 ≈⟨ idR ⟩ |
|
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diff
changeset
|
363 equalizer (eqa (f o equalizer (eqa f g {e}) o j) (f o equalizer (eqa f g {e}) o j)) |
| 241 | 364 ≈⟨⟩ -- here we decide e (fej) (fej)' is id1 A d |
|
235
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234
diff
changeset
|
365 id1 A d |
|
8835015a3e1a
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diff
changeset
|
366 ∎ ))) ⟩ |
|
8835015a3e1a
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234
diff
changeset
|
367 j o id1 A d |
|
8835015a3e1a
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parents:
234
diff
changeset
|
368 ≈⟨ idR ⟩ |
| 222 | 369 j |
|
235
8835015a3e1a
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parents:
234
diff
changeset
|
370 ∎ |
| 211 | 371 |
| 372 | |
|
238
c8db99cdf72a
Burrnoi to Equalizer problem written
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diff
changeset
|
373 lemma-equ2 : {a b c : Obj A} (f g : Hom A a b) (e : Hom A c a ) |
| 245 | 374 → ( bur : Burroni A {c} {a} {b} f g e ) → Equalizer A {c} {a} {b} (α bur f g e) f g |
|
238
c8db99cdf72a
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diff
changeset
|
375 lemma-equ2 {a} {b} {c} f g e bur = record { |
|
c8db99cdf72a
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diff
changeset
|
376 fe=ge = fe=ge1 ; |
|
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diff
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|
377 k = k1 ; |
|
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diff
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378 ek=h = λ {d} {h} {eq} → ek=h1 {d} {h} {eq} ; |
|
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diff
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379 uniqueness = λ {d} {h} {eq} {k'} ek=h → uniqueness1 {d} {h} {eq} {k'} ek=h |
|
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diff
changeset
|
380 } where |
|
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diff
changeset
|
381 k1 : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c |
|
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diff
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|
382 k1 {d} h fh=gh = β bur {d} {a} {b} f g h |
| 245 | 383 fe=ge1 : A [ A [ f o (α bur f g e) ] ≈ A [ g o (α bur f g e) ] ] |
|
238
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diff
changeset
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384 fe=ge1 = b1 bur |
| 245 | 385 ek=h1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ (α bur f g e) o k1 {d} h eq ] ≈ h ] |
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386 ek=h1 {d} {h} {eq} = let open ≈-Reasoning (A) in |
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387 begin |
| 245 | 388 α bur f g e o k1 h eq |
| 239 | 389 ≈⟨⟩ |
| 245 | 390 α bur f g e o ( γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
| 239 | 391 ≈⟨ assoc ⟩ |
| 245 | 392 ( α bur f g e o γ bur {a} {b} {c} f g h ) o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 239 | 393 ≈⟨ car (b2 bur) ⟩ |
| 245 | 394 ( h o ( α bur ( f o h ) ( g o h ) (id1 A d))) o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 239 | 395 ≈↑⟨ assoc ⟩ |
| 245 | 396 h o ((( α bur ( f o h ) ( g o h ) (id1 A d) )) o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
| 240 | 397 ≈↑⟨ cdr ( car ( cong-α bur eq)) ⟩ |
| 245 | 398 h o ((( α bur ( f o h ) ( f o h ) (id1 A d)))o δ bur {d} {b} {d} (id1 A d) (f o h) ) |
| 240 | 399 ≈⟨ cdr (b3 bur {d} {b} {d} (f o h) {id1 A d} ) ⟩ |
| 239 | 400 h o id1 A d |
| 240 | 401 ≈⟨ idR ⟩ |
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402 h |
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403 ∎ |
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404 uniqueness1 : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → |
| 245 | 405 A [ A [ (α bur f g e) o k' ] ≈ h ] → A [ k1 {d} h eq ≈ k' ] |
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406 uniqueness1 {d} {h} {eq} {k'} ek=h = let open ≈-Reasoning (A) in |
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407 begin |
| 240 | 408 k1 {d} h eq |
| 239 | 409 ≈⟨⟩ |
| 245 | 410 γ bur {a} {b} {c} f g h o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 240 | 411 ≈↑⟨ car (cong-γ bur {a} {b} {c} {d} ek=h ) ⟩ |
| 245 | 412 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id1 A d) (f o h) |
| 413 ≈↑⟨ cdr (cong-δ bur (resp ek=h refl-hom )) ⟩ | |
| 414 γ bur f g (A [ α bur f g e o k' ]) o δ bur {d} {b} {d} (id1 A d) ( f o ( α bur f g e o k') ) | |
| 240 | 415 ≈⟨ b4 bur ⟩ |
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416 k' |
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417 ∎ |
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418 |
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419 |
| 225 | 420 -- end |
| 212 | 421 |
| 422 | |
| 423 | |
| 215 | 424 |
| 425 |