Mercurial > hg > Members > kono > Proof > category

annotate equalizer.agda @ 207:22811f7a04e1

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
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
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
205
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
1 ---
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2 --
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 -- Equalizer
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4 --
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 -- f' f
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 -- c --------> a ----------> b
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 -- | . ---------->
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 -- | . g
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9 -- |h .
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 -- v . g'
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11 -- d
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12 --
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14 ----
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 open import Category -- https://github.com/konn/category-agda
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 open import Level
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18 open import Category.Sets
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
19 module equalizer { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ } where
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
20
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
21 open import HomReasoning
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 open import cat-utility
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23
207
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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
205
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 field
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26 equalizer : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) → Hom A c d
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
27 equalize : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) →
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 A [ A [ f o f' ] ≈ A [ A [ g o g' ] o equalizer f' g' ] ]
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29 uniqueness : {c d : Obj A} (f' : Hom A c a) (g' : Hom A d a) ( e : Hom A c d ) →
242adb6669da equalizer
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 A [ A [ f o f' ] ≈ A [ A [ g o g' ] o e ] ] → A [ e ≈ equalizer f' g' ]
206
3a5e2a22e053 on going
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 205
diff changeset
31
207
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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
206
3a5e2a22e053 on going
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 205
diff changeset
33 field
207
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
34 α : {e a b : Obj A} → (f : Hom A a b) → (g : Hom A a b ) → Hom A e a
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
36 δ : {e a b : Obj A} → (f : Hom A a b) → Hom A a e
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
37 b1 : {e : Obj A} → A [ A [ f o α {e} f g ] ≈ A [ g o α {e} f g ] ]
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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 ]) ] ]
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
39 b3 : {e : Obj A} → A [ A [ α {e} f f o δ {e} f ] ≈ id1 A a ]
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
40 -- b4 : {c d : Obj A } {k : Hom A c a} → A [ β f g ( A [ α f g o k ] ) ≈ k ]
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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 ]
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
42 -- A [ α f g o β f g h ] ≈ h
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
44 β {d} f g h = A [ γ f g h o δ {d} (A [ f o h ]) ]
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
45
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
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
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
47 lemma-equ1 A {a} {b} f g eqa = record {
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
48 α = {!!} ;
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
49 γ = {!!} ;
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
50 δ = {!!} ;
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
51 b1 = {!!} ;
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
52 b2 = {!!} ;
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
53 b3 = {!!} ;
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
54 b4 = {!!}
22811f7a04e1 Equalizer problems have written
Shinji KONO <kono@ie.u-ryukyu.ac.jp>
parents: 206
diff changeset
55 }