Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 212:8b3d3f69b725
b2
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Tue, 03 Sep 2013 01:11:59 +0900 |
| parents | 8c738327df19 |
| children | f2faee0897c7 |
| 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 |
| 26 ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] | |
| 27 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c | |
| 28 ke=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 } → A [ A [ e o k' ] ≈ h ] → | |
| 30 A [ k {d} h eq ≈ k' ] | |
| 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 |
| 37 γ : {a b d : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A c d | |
| 38 δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c | |
| 39 b1 : A [ A [ f o α f g ] ≈ A [ g o α f g ] ] | |
| 40 b2 : {d : Obj A } → {h : Hom A d a } → A [ A [ ( α f g) o (γ f g h) ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ] | |
| 211 | 41 b3 : {e : Obj A} → A [ A [ α f f o δ 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 ] |
| 212 | 43 b4 : {d : Obj A } {k : Hom A d c} → A [ A [ γ f g ( A [ α f g o k ] ) o δ (A [ f o A [ α f g o k ] ] ) ] ≈ ? ] |
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44 -- A [ α f g o β f g h ] ≈ h |
| 212 | 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 a d |
| 46 -- β {d} {e} {a} {b} f g h = A [ γ {a} {b} {d} f g h o δ (A [ f o h ]) ] | |
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47 |
| 209 | 48 open Equalizer |
| 49 open EqEqualizer | |
| 50 | |
| 211 | 51 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) → {a b c : Obj A} (f g : Hom A a b) → |
| 52 ( {a b c : Obj A} → (f g : Hom A a b) → Equalizer A {c} f g ) → EqEqualizer A {c} f g | |
| 209 | 53 lemma-equ1 A {a} {b} {c} f g eqa = record { |
| 211 | 54 α = λ f g → e (eqa f g ) ; -- Hom A c a |
| 212 | 55 γ = λ {a} {b} {d} f g h → ( k (eqa f g ) ( A [ h o (e ( eqa (A [ f o h ] ) (A [ g o h ] ))) ] ) (lemma-equ4 {a} {b} {d} f g h ) ) ; -- Hom A c d |
| 56 δ = λ f → k (eqa f f) (id1 A (Category.dom A f)) (lemma-equ2 f); -- Hom A a c | |
| 211 | 57 b1 = ef=eg (eqa f g) ; |
| 212 | 58 b2 = lemma-equ5 ; |
| 59 b3 = lemma-equ3 ; | |
| 60 b4 = {!!} | |
| 211 | 61 } where |
| 62 lemma-equ2 : {a b : Obj A} (f : Hom A a b) → A [ A [ f o id1 A a ] ≈ A [ f o id1 A a ] ] | |
| 63 lemma-equ2 f = let open ≈-Reasoning (A) in refl-hom | |
| 64 lemma-equ3 : {e' : Obj A} → A [ A [ e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) ] ≈ id1 A a ] | |
| 65 lemma-equ3 {e'} = let open ≈-Reasoning (A) in | |
| 66 begin | |
| 67 e (eqa f f) o k (eqa f f) (id1 A a) (lemma-equ2 f) | |
| 68 ≈⟨ ke=h (eqa f f ) (lemma-equ2 f) ⟩ | |
| 69 id1 A a | |
| 70 ∎ | |
| 212 | 71 lemma-equ4 : {a b d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → |
| 72 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 ])) ] ] ] | |
| 73 lemma-equ4 {a} {b} {d} f g h = let open ≈-Reasoning (A) in | |
| 74 begin | |
| 75 f o ( h o e (eqa (f o h) ( g o h ))) | |
| 76 ≈⟨ assoc ⟩ | |
| 77 (f o h) o e (eqa (f o h) ( g o h )) | |
| 78 ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩ | |
| 79 (g o h) o e (eqa (f o h) ( g o h )) | |
| 80 ≈↑⟨ assoc ⟩ | |
| 81 g o ( h o e (eqa (f o h) ( g o h ))) | |
| 82 ∎ | |
| 83 lemma-equ5 : {d : Obj A} {h : Hom A d a} → A [ | |
| 84 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 f g h) ] | |
| 85 ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ] | |
| 86 lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in | |
| 87 begin | |
| 88 e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 f g h) | |
| 89 ≈⟨ ke=h (eqa f g) (lemma-equ4 f g h) ⟩ | |
| 90 h o e (eqa (f o h ) ( g o h )) | |
| 91 ∎ | |
| 211 | 92 |
| 93 | |
| 212 | 94 |
| 95 | |
| 96 |