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