Mercurial > hg > Members > kono > Proof > category
annotate equalizer.agda @ 210:51c57efe89b9
α b1
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 02 Sep 2013 22:21:51 +0900 |
| parents | 4e138cc953f3 |
| children | 8c738327df19 |
| 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 open import Category.Sets | |
| 19 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where | |
| 20 | |
| 21 open import HomReasoning | |
| 22 open import cat-utility | |
| 23 | |
| 209 | 24 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 | 25 field |
| 209 | 26 e : Hom A c a |
| 27 ef=eg : A [ A [ f o e ] ≈ A [ g o e ] ] | |
| 28 k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c | |
| 29 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 ] | |
| 30 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 ] → | |
| 31 A [ k {d} h eq ≈ k' ] | |
| 32 equalizer : Hom A c a | |
| 33 equalizer = e | |
| 206 | 34 |
| 209 | 35 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 | 36 field |
| 210 | 37 α : (f : Hom A a b) → (g : Hom A a b ) → Hom A c a |
| 38 -- γ : {d e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → Hom A c e | |
| 39 -- δ : {a b : Obj A} → (f : Hom A a b) → Hom A a c | |
| 40 b1 : {e : Obj A } → A [ A [ f o α f g ] ≈ A [ g o α f g ] ] | |
| 41 -- b2 : {e d : Obj A } → {h : Hom A d a } → A [ A [ α {e} f g o γ f g h ] ≈ A [ h o α {c} (A [ f o h ]) (A [ g o h ]) ] ] | |
| 42 -- b3 : {e : Obj A} → A [ A [ α f f o δ f ] ≈ id1 A a ] | |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
43 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] |
| 210 | 44 -- 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 ] ] ) ] ≈ k ] |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
45 -- A [ α f g o β f g h ] ≈ h |
| 210 | 46 -- 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 e |
| 47 -- β {d} f g h = A [ γ f g h o δ {d} (A [ f o h ]) ] | |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
48 |
| 209 | 49 open Equalizer |
| 50 open EqEqualizer | |
| 51 | |
| 52 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b c : Obj A} (f g : Hom A a b) → Equalizer A {c} f g → EqEqualizer A {c} f g | |
| 53 lemma-equ1 A {a} {b} {c} f g eqa = record { | |
| 210 | 54 α = λ f g → e eqa ; -- Hom A c a |
| 55 -- γ = λ {d} {e} {a} {b} f g h → {!!} ; -- Hom A c e | |
| 56 -- δ = λ {a} {b} f → {!!} ; -- Hom A a c | |
| 57 b1 = ef=eg eqa -- ; | |
| 58 -- b2 = {!!} ; | |
| 59 -- b3 = {!!} ; | |
| 60 -- b4 = {!!} | |
|
207
22811f7a04e1
Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
206
diff
changeset
|
61 } |