Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 217:306f07bece85
add equalizer+h
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Wed, 04 Sep 2013 12:13:27 +0900 |
| parents | 0135419f375c |
| children | 749a1ecbc0b5 |
| 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 ] |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
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 ] |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
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 ]) ] | |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
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 | |
| 59 -- ---> d ---> | |
| 60 -- i h | |
| 61 | |
| 62 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 ) | |
| 63 → A [ A [ h o i ] ≈ e eqa ] | |
| 64 → Equalizer A {c} (A [ f o h ]) (A [ g o h ] ) | |
| 65 equalizer+h {a} {b} {c} {d} {f} {g} eqa i h eq = record { | |
| 66 e = i ; -- Hom A a d | |
| 67 ef=eg = ef=eg1 ; | |
| 68 k = λ j eq' → k eqa (A [ h o j ]) (fhj=ghj j eq' ) ; | |
| 69 ek=h = ek=h1 ; | |
| 70 uniqueness = uniqueness1 | |
| 71 } where | |
| 72 fhj=ghj : {d' : Obj A } → (j : Hom A d' d ) → | |
| 73 A [ A [ A [ f o h ] o j ] ≈ A [ A [ g o h ] o j ] ] → | |
| 74 A [ A [ f o A [ h o j ] ] ≈ A [ g o A [ h o j ] ] ] | |
| 75 fhj=ghj j eq' = let open ≈-Reasoning (A) in | |
| 76 begin | |
| 77 f o ( h o j ) | |
| 78 ≈⟨ assoc ⟩ | |
| 79 (f o h ) o j | |
| 80 ≈⟨ eq' ⟩ | |
| 81 (g o h ) o j | |
| 82 ≈↑⟨ assoc ⟩ | |
| 83 g o ( h o j ) | |
| 84 ∎ | |
| 85 ef=eg1 : A [ A [ A [ f o h ] o i ] ≈ A [ A [ g o h ] o i ] ] | |
| 86 ef=eg1 = let open ≈-Reasoning (A) in | |
| 87 begin | |
| 88 ( f o h ) o i | |
| 89 ≈↑⟨ assoc ⟩ | |
| 90 f o (h o i ) | |
| 91 ≈⟨ cdr eq ⟩ | |
| 92 f o (e eqa) | |
| 93 ≈⟨ ef=eg eqa ⟩ | |
| 94 g o (e eqa) | |
| 95 ≈↑⟨ cdr eq ⟩ | |
| 96 g o (h o i ) | |
| 97 ≈⟨ assoc ⟩ | |
| 98 ( g o h ) o i | |
| 99 ∎ | |
| 100 ek=h1 : {d' : Obj A} {h' : Hom A d' d} {eq' : A [ A [ A [ f o h ] o h' ] ≈ A [ A [ g o h ] o h' ] ]} → | |
| 101 A [ A [ i o k eqa (A [ h o h' ]) (fhj=ghj h' eq') ] ≈ h' ] | |
| 102 ek=h1 {d'} {h'} {eq'} = let open ≈-Reasoning (A) in | |
| 103 begin | |
| 104 i o k eqa (h o h' ) (fhj=ghj h' eq') | |
| 105 ≈⟨ {!!} ⟩ | |
| 106 h' | |
| 107 ∎ | |
| 108 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} → | |
| 109 A [ A [ i o k' ] ≈ h' ] → A [ k eqa (A [ h o h' ]) (fhj=ghj h' eq') ≈ k' ] | |
| 110 uniqueness1 {d'} {h'} {eq'} {k'} ik=h = let open ≈-Reasoning (A) in | |
| 111 begin | |
| 112 k eqa (A [ h o h' ]) (fhj=ghj h' eq') | |
| 113 ≈⟨ uniqueness eqa ( begin | |
| 114 e eqa o k' | |
| 115 ≈↑⟨ car eq ⟩ | |
| 116 (h o i ) o k' | |
| 117 ≈↑⟨ assoc ⟩ | |
| 118 h o (i o k') | |
| 119 ≈⟨ cdr ik=h ⟩ | |
| 120 h o h' | |
| 121 ∎ ) ⟩ | |
| 122 k' | |
| 123 ∎ | |
| 215 | 124 |
| 211 | 125 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
| 126 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
| 209 | 127 lemma-equ1 A {a} {b} {c} f g eqa = record { |
| 216 | 128 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 214 | 129 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 | 130 δ = λ {a} f → k (eqa f f) (id1 A a) (lemma-equ2 f); -- Hom A a c |
| 211 | 131 b1 = ef=eg (eqa f g) ; |
| 212 | 132 b2 = lemma-equ5 ; |
| 133 b3 = lemma-equ3 ; | |
| 215 | 134 b4 = lemma-equ6 |
| 211 | 135 } where |
| 216 | 136 -- |
| 137 -- e eqa f g f | |
| 138 -- c ----------> a ------->b | |
| 139 -- ^ g | |
| 140 -- | | |
| 141 -- |k₁ = e eqa (f o (e (eqa f g))) (g o (e (eqa f g)))) | |
| 142 -- | | |
| 143 -- d | |
| 144 -- | |
| 145 -- | |
| 146 -- e o id1 ≈ e → k e ≈ id | |
| 147 | |
| 211 | 148 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] |
| 149 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 213 | 150 lemma-equ3 : A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] |
| 151 lemma-equ3 = let open ≈-Reasoning (A) in | |
| 211 | 152 begin |
| 153 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 215 | 154 ≈⟨ ek=h (eqa f f ) ⟩ |
| 211 | 155 id1 A a |
| 156 ∎ | |
| 214 | 157 lemma-equ4 : {a b c d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 212 | 158 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 | 159 lemma-equ4 {a} {b} {c} {d} f g h = let open ≈-Reasoning (A) in |
| 212 | 160 begin |
| 161 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 162 ≈⟨ assoc ⟩ | |
| 163 (f o h) o e (eqa (f o h) ( g o h )) | |
| 164 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
| 165 (g o h) o e (eqa (f o h) ( g o h )) | |
| 166 ≈↑⟨ assoc ⟩ | |
| 167 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 168 ∎ | |
| 169 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 214 | 170 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 | 171 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] |
| 172 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 173 begin | |
| 215 | 174 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) |
| 175 ≈⟨ ek=h (eqa f g) ⟩ | |
| 212 | 176 h o e (eqa (f o h ) ( g o h )) |
| 177 ∎ | |
| 215 | 178 lemma-equ6 : {d : Obj A} {k₁ : Hom A d c} → A [ |
| 179 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 ] ])) ]) | |
| 180 (lemma-equ4 {a} {b} {c} f g (A [ e (eqa f g) o k₁ ])) o | |
| 181 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 ] ])) ] | |
| 182 ≈ k₁ ] | |
| 183 lemma-equ6 {d} {k₁} = let open ≈-Reasoning (A) in | |
| 184 begin | |
| 185 ( 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 ) ))) )) | |
| 186 (lemma-equ4 {a} {b} {c} f g (( e (eqa f g) o k₁ ))) o | |
| 187 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 ) ))) ) | |
| 188 ≈⟨ car ( uniqueness (eqa f g) ( begin | |
| 189 e (eqa f g) o k₁ | |
| 190 ≈⟨ {!!} ⟩ | |
| 191 (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)) | |
| 192 ∎ )) ⟩ | |
| 193 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 ) ))) | |
| 194 ≈⟨ cdr ( uniqueness (eqa (( f o ( e (eqa f g) o k₁ ) )) (( f o ( e (eqa f g) o k₁ ) ))) ( begin | |
| 195 e (eqa (f o e (eqa f g) o k₁) (f o e (eqa f g) o k₁)) o id1 A d | |
| 196 ≈⟨ {!!} ⟩ | |
| 197 id1 A d | |
| 198 ∎ )) ⟩ | |
| 199 k₁ o id1 A d | |
| 200 ≈⟨ idR ⟩ | |
| 201 k₁ | |
| 202 ∎ | |
| 211 | 203 |
| 204 | |
| 212 | 205 |
| 206 | |
| 207 | |
| 215 | 208 |
| 209 |