Use dotted expression in Cat
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@ -17,7 +17,6 @@ open import Cat.Category.NaturalTransformation
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open import Cat.Equality
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open Equality.Data.Product
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open Functor using (fmap ; omap)
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open Category using (Object ; 𝟙)
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-- The category of categories
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@ -169,14 +168,19 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
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-- Basically proves that `Cat ℓ ℓ` is cartesian closed.
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module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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private
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open Data.Product
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open import Cat.Categories.Fun
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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Categoryℓ = Category ℓ ℓ
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open Fun ℂ 𝔻 renaming (identity to idN)
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private
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:omap: : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
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:omap: (F , A) = omap F A
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:omap: (F , A) = F.omap A
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where
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module F = Functor F
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prodObj : Categoryℓ
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prodObj = Fun
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@ -193,28 +197,31 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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B : Object ℂ
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B = proj₂ cod
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module F = Functor F
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module G = Functor G
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:fmap: : (pobj : NaturalTransformation F G × ℂ [ A , B ])
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→ 𝔻 [ omap F A , omap G B ]
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→ 𝔻 [ F.omap A , G.omap B ]
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:fmap: ((θ , θNat) , f) = result
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where
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θA : 𝔻 [ omap F A , omap G A ]
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θA : 𝔻 [ F.omap A , G.omap A ]
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θA = θ A
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θB : 𝔻 [ omap F B , omap G B ]
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θB : 𝔻 [ F.omap B , G.omap B ]
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θB = θ B
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F→f : 𝔻 [ omap F A , omap F B ]
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F→f = fmap F f
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G→f : 𝔻 [ omap G A , omap G B ]
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G→f = fmap G f
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l : 𝔻 [ omap F A , omap G B ]
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l = 𝔻 [ θB ∘ F→f ]
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r : 𝔻 [ omap F A , omap G B ]
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r = 𝔻 [ G→f ∘ θA ]
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F→f : 𝔻 [ F.omap A , F.omap B ]
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F→f = F.fmap f
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G→f : 𝔻 [ G.omap A , G.omap B ]
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G→f = G.fmap f
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l : 𝔻 [ F.omap A , G.omap B ]
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l = 𝔻 [ θB ∘ F.fmap f ]
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r : 𝔻 [ F.omap A , G.omap B ]
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r = 𝔻 [ G.fmap f ∘ θA ]
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-- There are two choices at this point,
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-- but I suppose the whole point is that
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-- by `θNat f` we have `l ≡ r`
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-- lem : 𝔻 [ θ B ∘ F .fmap f ] ≡ 𝔻 [ G .fmap f ∘ θ A ]
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-- lem = θNat f
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result : 𝔻 [ omap F A , omap G B ]
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result : 𝔻 [ F.omap A , G.omap B ]
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result = l
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open CatProduct renaming (obj to _×p_) using ()
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@ -237,23 +244,27 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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:ident: = begin
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:fmap: {c} {c} (𝟙 (prodObj ×p ℂ) {c}) ≡⟨⟩
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:fmap: {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
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𝔻 [ identityTrans F C ∘ fmap F (𝟙 ℂ)] ≡⟨⟩
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𝔻 [ 𝟙 𝔻 ∘ fmap F (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
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fmap F (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
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𝔻 [ identityTrans F C ∘ F.fmap (𝟙 ℂ)] ≡⟨⟩
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𝔻 [ 𝟙 𝔻 ∘ F.fmap (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
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F.fmap (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
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𝟙 𝔻 ∎
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where
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open module 𝔻 = Category 𝔻
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open module F = Functor F
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module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
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private
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F = F×A .proj₁
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A = F×A .proj₂
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G = G×B .proj₁
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B = G×B .proj₂
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H = H×C .proj₁
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C = H×C .proj₂
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module F = Functor F
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module G = Functor G
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module H = Functor H
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-- Not entirely clear what this is at this point:
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_P⊕_ = Category._∘_ (prodObj ×p ℂ) {F×A} {G×B} {H×C}
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module _
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-- NaturalTransformation F G × ℂ .Arrow A B
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{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
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@ -279,31 +290,28 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
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ηθNat = proj₂ ηθNT
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:isDistributive: :
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ fmap F ( ℂ [ g ∘ f ] ) ]
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≡ 𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ 𝔻 [ θ B ∘ fmap F f ] ]
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𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F.fmap ( ℂ [ g ∘ f ] ) ]
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≡ 𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ θ B ∘ F.fmap f ] ]
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:isDistributive: = begin
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𝔻 [ (ηθ C) ∘ fmap F (ℂ [ g ∘ f ]) ]
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𝔻 [ (ηθ C) ∘ F.fmap (ℂ [ g ∘ f ]) ]
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≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
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𝔻 [ fmap H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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𝔻 [ H.fmap (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.isDistributive) ⟩
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𝔻 [ 𝔻 [ fmap H g ∘ fmap H f ] ∘ (ηθ A) ]
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≡⟨ sym isAssociative ⟩
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𝔻 [ fmap H g ∘ 𝔻 [ fmap H f ∘ ηθ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ fmap H g ∘ φ ]) isAssociative ⟩
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𝔻 [ fmap H g ∘ 𝔻 [ 𝔻 [ fmap H f ∘ η A ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ fmap H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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𝔻 [ fmap H g ∘ 𝔻 [ 𝔻 [ η B ∘ fmap G f ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ fmap H g ∘ φ ]) (sym isAssociative) ⟩
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𝔻 [ fmap H g ∘ 𝔻 [ η B ∘ 𝔻 [ fmap G f ∘ θ A ] ] ]
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≡⟨ isAssociative ⟩
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𝔻 [ 𝔻 [ fmap H g ∘ η B ] ∘ 𝔻 [ fmap G f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ fmap G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ 𝔻 [ fmap G f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ φ ]) (sym (θNat f)) ⟩
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𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ 𝔻 [ θ B ∘ fmap F f ] ] ∎
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where
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open Category 𝔻
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module H = Functor H
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𝔻 [ 𝔻 [ H.fmap g ∘ H.fmap f ] ∘ (ηθ A) ]
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≡⟨ sym 𝔻.isAssociative ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ H.fmap f ∘ ηθ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) 𝔻.isAssociative ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ 𝔻 [ H.fmap f ∘ η A ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ 𝔻 [ η B ∘ G.fmap f ] ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) (sym 𝔻.isAssociative) ⟩
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𝔻 [ H.fmap g ∘ 𝔻 [ η B ∘ 𝔻 [ G.fmap f ∘ θ A ] ] ]
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≡⟨ 𝔻.isAssociative ⟩
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𝔻 [ 𝔻 [ H.fmap g ∘ η B ] ∘ 𝔻 [ G.fmap f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ G.fmap f ∘ θ A ] ]) (sym (ηNat g)) ⟩
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𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ G.fmap f ∘ θ A ] ]
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≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ φ ]) (sym (θNat f)) ⟩
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𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ θ B ∘ F.fmap f ] ] ∎
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eval : Functor (CatProduct.obj prodObj ℂ) 𝔻
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-- :eval: : Functor (prodObj ×p ℂ) 𝔻
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