Mercurial > hg > Members > kono > Proof > category

comparison CCChom.agda @ 784:f27d966939f8

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add CCC hom
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Wed, 17 Apr 2019 14:47:39 +0900
parents CCC.agda@bded2347efa4
children a67959bcd44b
comparison
equal deleted inserted replaced
783:bded2347efa4 784:f27d966939f8
1 open import Level
2 open import Category
3 module CCChom where
4
5 open import HomReasoning
6 open import cat-utility
7 open import Data.Product renaming (_×_ to _∧_)
8 open import Category.Constructions.Product
9 open import Relation.Binary.PropositionalEquality
10
11 open Functor
12
13 -- ccc-1 : Hom A a 1 ≅ {*}
14 -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b )
15 -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c
16
17 data One {c : Level} : Set c where
18 OneObj : One -- () in Haskell ( or any one object set )
19
20 OneCat : Category Level.zero Level.zero Level.zero
21 OneCat = record {
22 Obj = One ;
23 Hom = λ a b → One ;
24 _o_ = λ{a} {b} {c} x y → OneObj ;
25 _≈_ = λ x y → x ≡ y ;
26 Id = λ{a} → OneObj ;
27 isCategory = record {
28 isEquivalence = record {refl = refl ; trans = trans ; sym = sym } ;
29 identityL = λ{a b f} → lemma {a} {b} {f} ;
30 identityR = λ{a b f} → lemma {a} {b} {f} ;
31 o-resp-≈ = λ{a b c f g h i} _ _ → refl ;
32 associative = λ{a b c d f g h } → refl
33 }
34 } where
35 lemma : {a b : One {Level.zero}} → { f : One {Level.zero}} → OneObj ≡ f
36 lemma {a} {b} {f} with f
37 ... | OneObj = refl
38
39 record IsoS {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') (a b : Obj A) ( a' b' : Obj B )
40 : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' ) where
41 field
42 ≅→ : Hom A a b → Hom B a' b'
43 ≅← : Hom B a' b' → Hom A a b
44 iso→ : {f : Hom B a' b' } → B [ ≅→ ( ≅← f) ≈ f ]
45 iso← : {f : Hom A a b } → A [ ≅← ( ≅→ f) ≈ f ]
46
47
48 record IsCCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (１ : Obj A)
49 ( _*_ : Obj A → Obj A → Obj A ) ( _^_ : Obj A → Obj A → Obj A ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
50 field
51 ccc-1 : {a : Obj A} → -- Hom A a １ ≅ {*}
52 IsoS A OneCat a １ OneObj OneObj
53 ccc-2 : {a b c : Obj A} → -- Hom A c ( a * b ) ≅ ( Hom A c a ) * ( Hom A c b )
54 IsoS A (A × A) c (a * b) (c , c ) (a , b )
55 ccc-3 : {a b c : Obj A} → -- Hom A a ( c ^ b ) ≅ Hom A ( a * b ) c
56 IsoS A A a (c ^ b) (a * b) c
57
58
59 record CCChom {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
60 field
61 one : Obj A
62 _*_ : Obj A → Obj A → Obj A
63 _^_ : Obj A → Obj A → Obj A
64 isCCChom : IsCCChom A one _*_ _^_
65
66 open import HomReasoning
67
68 open import CCC
69
70 CCC→hom : {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( c : CCC A ) → CCChom A
71 CCC→hom A c = record {
72 one = CCC.１ c
73 ; _*_ = CCC._∧_ c
74 ; _^_ = CCC._<=_ c
75 ; isCCChom = record {
76 ccc-1 = {!!}
77 ; ccc-2 = {!!}
78 ; ccc-3 = {!!}
79 }
80 }