Mercurial > hg > Members > kono > Proof > category

comparison equalizer.agda @ 210:51c57efe89b9

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α b1
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 02 Sep 2013 22:21:51 +0900
parents 4e138cc953f3
children 8c738327df19
comparison
equal deleted inserted replaced
209:4e138cc953f3 210:51c57efe89b9
32 equalizer : Hom A c a 32 equalizer : Hom A c a
33 equalizer = e 33 equalizer = e
34 34
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 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
36 field 36 field
37 α : {e a : Obj A } → (f : Hom A a b) → (g : Hom A a b ) → Hom A e a 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 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 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 {e} {a} f g ] ≈ A [ g o α {e} {a} f g ] ] 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 ]) ] ] 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 ] 42 -- b3 : {e : Obj A} → A [ A [ α f f o δ f ] ≈ id1 A a ]
43 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ] 43 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ]
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 ] 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 ]
45 -- A [ α f g o β f g h ] ≈ h 45 -- A [ α f g o β f g h ] ≈ h
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 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 ]) ] 47 -- β {d} f g h = A [ γ f g h o δ {d} (A [ f o h ]) ]
48 48
49 open Equalizer 49 open Equalizer
50 open EqEqualizer 50 open EqEqualizer
51 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 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 { 53 lemma-equ1 A {a} {b} {c} f g eqa = record {
54 α = λ {e'} {a} f g → ? ; -- e' -> c c -> a, Hom A e' a 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 55 -- γ = λ {d} {e} {a} {b} f g h → {!!} ; -- Hom A c e
56 δ = λ {a} {b} f → {!!} ; -- Hom A a c 56 -- δ = λ {a} {b} f → {!!} ; -- Hom A a c
57 b1 = {!!} ; 57 b1 = ef=eg eqa -- ;
58 b2 = {!!} ; 58 -- b2 = {!!} ;
59 b3 = {!!} ; 59 -- b3 = {!!} ;
60 b4 = {!!} 60 -- b4 = {!!}
61 } 61 }