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comparison equalizer.agda @ 207:22811f7a04e1

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Equalizer problems have written
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 02 Sep 2013 16:54:02 +0900
parents 3a5e2a22e053
children a1e5d2a3d3bd
comparison
equal deleted inserted replaced
206:3a5e2a22e053 207:22811f7a04e1
19 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where 19 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where
20 20
21 open import HomReasoning 21 open import HomReasoning
22 open import cat-utility 22 open import cat-utility
23 23
24 record Equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where 24 record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
25 field 25 field
26 equalizer : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) → Hom A c d 26 equalizer : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) → Hom A c d
27 equalize : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) → 27 equalize : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) →
28 A [ A [ f o f' ] ≈ A [ A [ g o g' ] o equalizer f' g' ] ] 28 A [ A [ f o f' ] ≈ A [ A [ g o g' ] o equalizer f' g' ] ]
29 uniqueness : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) ( e : Hom A c d ) → 29 uniqueness : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) ( e : Hom A c d ) →
30 A [ A [ f o f' ] ≈ A [ A [ g o g' ] o e ] ] → A [ e ≈ equalizer f' g' ] 30 A [ A [ f o f' ] ≈ A [ A [ g o g' ] o e ] ] → A [ e ≈ equalizer f' g' ]
31 31
32 record EqEqualizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where 32 record EqEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where
33 field 33 field
34 α : {d a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → Hom A d a 34 α : {e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → Hom A e a
35 γ : {d c : Obj A} → (f : Hom A c b) → (g : Hom A c b ) → (h : Hom A d c ) → Hom A d c 35 γ : {c 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
36 δ : {a b : Obj A} → (f : Hom A a b) → Hom A a a 36 δ : {e a b : Obj A} → (f : Hom A a b) → Hom A a e
37 β : {c a b d : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A c d ) → Hom A c a 37 b1 : {e : Obj A} → A [ A [ f o α {e} f g ] ≈ A [ g o α {e} f g ] ]
38 b1 : {c : Obj A} → A [ A [ f o α {c} f g ] ≈ A [ g o α {c} f g ] ] 38 b2 : {c d : Obj A } → {h : Hom A d a } → A [ A [ α {c} f g o γ {c} f g h ] ≈ A [ h o α (A [ f o h ]) (A [ g o h ]) ] ]
39 b2 : {c 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 ]) ] ] 39 b3 : {e : Obj A} → A [ A [ α {e} f f o δ {e} f ] ≈ id1 A a ]
40 b3 : A [ A [ α f f o δ f ] ≈ id1 A a ] 40 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ]
41 b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] 41 b4 : {c d : Obj A } {k : Hom A c a} → A [ A [ γ f g ( A [ α f g o k ] ) o δ {c} (A [ f o A [ α f g o k ] ] ) ] ≈ k ]
42 b5 : {c d : Obj A } → {h : Hom A d a } → A [ β f g h ≈ A [ γ f g h o δ (A [ f o h ]) ] ] 42 -- A [ α f g o β f g h ] ≈ h
43 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
44 β {d} f g h = A [ γ f g h o δ {d} (A [ f o h ]) ]
45
46 lemma-equ1 : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) → Equalizer A f g → EqEqualizer A f g
47 lemma-equ1 A {a} {b} f g eqa = record {
48 α = {!!} ;
49 γ = {!!} ;
50 δ = {!!} ;
51 b1 = {!!} ;
52 b2 = {!!} ;
53 b3 = {!!} ;
54 b4 = {!!}
55 }