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changeset 212:8b3d3f69b725

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b2
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Tue, 03 Sep 2013 01:11:59 +0900
parents 8c738327df19
children f2faee0897c7
files equalizer.agda
diffstat 1 files changed, 37 insertions(+), 13 deletions(-) [+]
line wrap: on
line diff
--- a/equalizer.agda	Mon Sep 02 23:18:40 2013 +0900
+++ b/equalizer.agda	Tue Sep 03 01:11:59 2013 +0900
@@ -33,17 +33,17 @@
 
 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
    field
-      α : (f : Hom A a b) → (g : Hom A a b ) →  Hom A c a
---      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 
-      δ : (f : Hom A a b) → Hom A a c 
-      b1 : {e : Obj A } →  A [ A [ f  o α  f g ] ≈ A [ g  o α f g ] ]
---      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 ]) ] ]
+      α : {a b c : Obj A } → (f : Hom A a b) → (g : Hom A a b ) →  Hom A c a
+      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
+      δ : {a b c : Obj A } → (f : Hom A a b) → Hom A a c 
+      b1 : A [ A [ f  o α  f g ] ≈ A [ g  o α f g ] ]
+      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 ]) ] ]
       b3 :  {e : Obj A} → A [ A [ α f f o δ f ] ≈ id1 A a ]
       -- b4 :  {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o  k ] ) ≈ k ]
---      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 ]
+      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 ] ] ) ] ≈ ? ]
    --  A [ α f g o β f g h ] ≈ h
---   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
---   β {d} f g h =  A [ γ f g h o δ {d} (A [ f o h ]) ] 
+--   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
+--   β {d} {e} {a} {b} f g h =  A [ γ {a} {b} {d} f g h o δ (A [ f o h ]) ] 
 
 open Equalizer
 open EqEqualizer
@@ -52,12 +52,12 @@
          ( {a b c : Obj A} → (f g : Hom A a b)  → Equalizer A {c} f g ) → EqEqualizer A {c} f g
 lemma-equ1  A {a} {b} {c} f g eqa = record {
       α = λ f g →  e (eqa f g ) ; -- Hom A c  a
---      γ = λ {d} {e} {a} {b} f g h → {!!} ;  -- Hom A c e
-      δ =  λ f → k (eqa f f) (id1 A a)  (lemma-equ2 f); -- Hom A a c
+      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
+      δ =  λ f → k (eqa f f) (id1 A (Category.dom A f))  (lemma-equ2 f); -- Hom A a c
       b1 = ef=eg (eqa f g) ;
---      b2 = {!!} ;
-      b3 = lemma-equ3 -- ;
---      b4 = {!!} 
+      b2 = lemma-equ5 ;
+      b3 = lemma-equ3 ;
+      b4 = {!!} 
  } where
      lemma-equ2 : {a b : Obj A} (f : Hom A a b)  → A [ A [ f o id1 A a ]  ≈ A [ f o id1 A a ] ]
      lemma-equ2 f =   let open ≈-Reasoning (A) in refl-hom
@@ -68,5 +68,29 @@
              ≈⟨ ke=h (eqa f f ) (lemma-equ2 f) ⟩
                   id1 A a
              ∎
+     lemma-equ4 :  {a b d : Obj A}  → (f : Hom A a b) → (g : Hom A a b ) → (h : Hom A d a ) → 
+                      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 ])) ] ] ]
+     lemma-equ4 {a} {b} {d} f g h  = let open ≈-Reasoning (A) in
+             begin
+                   f o ( h o e (eqa (f o h) ( g o h )))
+             ≈⟨ assoc ⟩
+                   (f o h) o e (eqa (f o h) ( g o h ))
+             ≈⟨ ef=eg (eqa (A [ f o h ]) (A [ g o h ])) ⟩
+                   (g o h) o e (eqa (f o h) ( g o h ))
+             ≈↑⟨ assoc ⟩
+                   g o ( h o e (eqa (f o h) ( g o h )))
+             ∎
+     lemma-equ5 :  {d : Obj A} {h : Hom A d a} → A [ 
+                      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) ]
+                    ≈ A [ h o e (eqa (A [ f o h ]) (A [ g o h ])) ] ]
+     lemma-equ5 {d} {h} = let open ≈-Reasoning (A) in
+             begin
+                    e (eqa f g) o k (eqa f g) (h o e (eqa (f o h) (g o h))) (lemma-equ4 f g h) 
+             ≈⟨ ke=h (eqa f g) (lemma-equ4 f g h) ⟩
+                    h o e (eqa (f o h ) ( g o h ))
+             ∎
 
 
+
+
+