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comparison CCC.agda @ 780:b44c1c6ce646

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CCC in Hom form
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Mon, 08 Oct 2018 16:48:27 +0900
parents 6b4bd02efd80
children 340708e8d54f
comparison
equal deleted inserted replaced
779:6b4bd02efd80 780:b44c1c6ce646
2 open import Category 2 open import Category
3 module CCC where 3 module CCC where
4 4
5 open import HomReasoning 5 open import HomReasoning
6 open import cat-utility 6 open import cat-utility
7 open import Data.Product renaming (_×_ to _*_) 7 open import Data.Product renaming (_×_ to _∧_)
8 open import Category.Constructions.Product
9 open import Relation.Binary.PropositionalEquality
8 10
9 open Functor 11 open Functor
10 12
11 -- ccc-1 : Hom A a 1 ≅ {*} 13 -- ccc-1 : Hom A a 1 ≅ {*}
12 -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b ) 14 -- ccc-2 : Hom A c (a × b) ≅ (Hom A c a ) × ( Hom A c b )
13 -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c 15 -- ccc-3 : Hom A a (c ^ b) ≅ Hom A (a × b) c
14 16
15 record _≅_ {c₁ c₂ ℓ ℓ' : Level} {A : Category c₁ c₂ ℓ} {a b : Obj A} (f : Hom A a b) (S : Set ℓ') : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ ℓ' ) where 17 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 )
18 : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' ) where
16 field 19 field
17 ≅→ : {!!} 20 ≅→ : Hom A a b → Hom B a' b'
18 ≅← : {!!} 21 ≅← : Hom B a' b' → Hom A a b
19 iso→ : {!!} 22 iso→ : {f : Hom B a' b' } → B [ ≅→ ( ≅← f) ≈ f ]
20 iso← : {!!} 23 iso← : {f : Hom A a b } → A [ ≅← ( ≅→ f) ≈ f ]
21 24
22 data One {c : Level} : Set c where 25 data One {c : Level} : Set c where
23 OneObj : One -- () in Haskell ( or any one object set ) 26 OneObj : One -- () in Haskell ( or any one object set )
24 27
28 OneCat : Category Level.zero Level.zero Level.zero
29 OneCat = record {
30 Obj = One ;
31 Hom = λ a b → One ;
32 _o_ = λ{a} {b} {c} x y → OneObj ;
33 _≈_ = λ x y → x ≡ y ;
34 Id = λ{a} → OneObj ;
35 isCategory = record {
36 isEquivalence = record {refl = refl ; trans = trans ; sym = sym } ;
37 identityL = λ{a b f} → lemma {a} {b} {f} ;
38 identityR = λ{a b f} → lemma {a} {b} {f} ;
39 o-resp-≈ = λ{a b c f g h i} _ _ → refl ;
40 associative = λ{a b c d f g h } → refl
41 }
42 } where
43 lemma : {a b : One {Level.zero}} → { f : One {Level.zero}} → OneObj ≡ f
44 lemma {a} {b} {f} with f
45 ... | OneObj = refl
46
25 47
26 record isCCC {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (one : Obj A) 48 record isCCC {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (one : Obj A)
27 ( _×_ : Obj A → Obj A → Obj A ) ( _^_ : Obj A → Obj A → Obj A ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where 49 ( _*_ : Obj A → Obj A → Obj A ) ( _^_ : Obj A → Obj A → Obj A ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ) where
28 field 50 field
29 ccc-1 : {a : Obj A} → Hom A a one ≅ One {ℓ} 51 ccc-1 : {a : Obj A} → -- Hom A a one ≅ {*}
30 ccc-2 : {a b c : Obj A} → Hom A c ( a × b ) ≅ ( Hom A c a ) * ( Hom A c b ) 52 IsoS A OneCat a one OneObj OneObj
31 ccc-3 : {a b c : Obj A} → Hom A a ( c ^ b ) ≅ Hom A ( a × b ) c 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
32 57
33 58
34 59