Use dotted expression in Cat
This commit is contained in:
parent
5ad506a09f
commit
48672b01bd
|
|
@ -17,7 +17,6 @@ open import Cat.Category.NaturalTransformation
|
|||
open import Cat.Equality
|
||||
open Equality.Data.Product
|
||||
|
||||
open Functor using (fmap ; omap)
|
||||
open Category using (Object ; 𝟙)
|
||||
|
||||
-- The category of categories
|
||||
|
|
@ -169,14 +168,19 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
|
|||
|
||||
-- Basically proves that `Cat ℓ ℓ` is cartesian closed.
|
||||
module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
|
||||
open Data.Product
|
||||
open import Cat.Categories.Fun
|
||||
private
|
||||
open Data.Product
|
||||
open import Cat.Categories.Fun
|
||||
module ℂ = Category ℂ
|
||||
module 𝔻 = Category 𝔻
|
||||
|
||||
Categoryℓ = Category ℓ ℓ
|
||||
open Fun ℂ 𝔻 renaming (identity to idN)
|
||||
private
|
||||
:omap: : Functor ℂ 𝔻 × Object ℂ → Object 𝔻
|
||||
:omap: (F , A) = omap F A
|
||||
:omap: (F , A) = F.omap A
|
||||
where
|
||||
module F = Functor F
|
||||
|
||||
prodObj : Categoryℓ
|
||||
prodObj = Fun
|
||||
|
|
@ -193,28 +197,31 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
|
|||
B : Object ℂ
|
||||
B = proj₂ cod
|
||||
|
||||
module F = Functor F
|
||||
module G = Functor G
|
||||
|
||||
:fmap: : (pobj : NaturalTransformation F G × ℂ [ A , B ])
|
||||
→ 𝔻 [ omap F A , omap G B ]
|
||||
→ 𝔻 [ F.omap A , G.omap B ]
|
||||
:fmap: ((θ , θNat) , f) = result
|
||||
where
|
||||
θA : 𝔻 [ omap F A , omap G A ]
|
||||
θA : 𝔻 [ F.omap A , G.omap A ]
|
||||
θA = θ A
|
||||
θB : 𝔻 [ omap F B , omap G B ]
|
||||
θB : 𝔻 [ F.omap B , G.omap B ]
|
||||
θB = θ B
|
||||
F→f : 𝔻 [ omap F A , omap F B ]
|
||||
F→f = fmap F f
|
||||
G→f : 𝔻 [ omap G A , omap G B ]
|
||||
G→f = fmap G f
|
||||
l : 𝔻 [ omap F A , omap G B ]
|
||||
l = 𝔻 [ θB ∘ F→f ]
|
||||
r : 𝔻 [ omap F A , omap G B ]
|
||||
r = 𝔻 [ G→f ∘ θA ]
|
||||
F→f : 𝔻 [ F.omap A , F.omap B ]
|
||||
F→f = F.fmap f
|
||||
G→f : 𝔻 [ G.omap A , G.omap B ]
|
||||
G→f = G.fmap f
|
||||
l : 𝔻 [ F.omap A , G.omap B ]
|
||||
l = 𝔻 [ θB ∘ F.fmap f ]
|
||||
r : 𝔻 [ F.omap A , G.omap B ]
|
||||
r = 𝔻 [ G.fmap f ∘ θA ]
|
||||
-- There are two choices at this point,
|
||||
-- but I suppose the whole point is that
|
||||
-- by `θNat f` we have `l ≡ r`
|
||||
-- lem : 𝔻 [ θ B ∘ F .fmap f ] ≡ 𝔻 [ G .fmap f ∘ θ A ]
|
||||
-- lem = θNat f
|
||||
result : 𝔻 [ omap F A , omap G B ]
|
||||
result : 𝔻 [ F.omap A , G.omap B ]
|
||||
result = l
|
||||
|
||||
open CatProduct renaming (obj to _×p_) using ()
|
||||
|
|
@ -237,23 +244,27 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
|
|||
:ident: = begin
|
||||
:fmap: {c} {c} (𝟙 (prodObj ×p ℂ) {c}) ≡⟨⟩
|
||||
:fmap: {c} {c} (idN F , 𝟙 ℂ) ≡⟨⟩
|
||||
𝔻 [ identityTrans F C ∘ fmap F (𝟙 ℂ)] ≡⟨⟩
|
||||
𝔻 [ 𝟙 𝔻 ∘ fmap F (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
|
||||
fmap F (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
|
||||
𝔻 [ identityTrans F C ∘ F.fmap (𝟙 ℂ)] ≡⟨⟩
|
||||
𝔻 [ 𝟙 𝔻 ∘ F.fmap (𝟙 ℂ)] ≡⟨ proj₂ 𝔻.isIdentity ⟩
|
||||
F.fmap (𝟙 ℂ) ≡⟨ F.isIdentity ⟩
|
||||
𝟙 𝔻 ∎
|
||||
where
|
||||
open module 𝔻 = Category 𝔻
|
||||
open module F = Functor F
|
||||
|
||||
module _ {F×A G×B H×C : Functor ℂ 𝔻 × Object ℂ} where
|
||||
F = F×A .proj₁
|
||||
A = F×A .proj₂
|
||||
G = G×B .proj₁
|
||||
B = G×B .proj₂
|
||||
H = H×C .proj₁
|
||||
C = H×C .proj₂
|
||||
-- Not entirely clear what this is at this point:
|
||||
_P⊕_ = Category._∘_ (prodObj ×p ℂ) {F×A} {G×B} {H×C}
|
||||
private
|
||||
F = F×A .proj₁
|
||||
A = F×A .proj₂
|
||||
G = G×B .proj₁
|
||||
B = G×B .proj₂
|
||||
H = H×C .proj₁
|
||||
C = H×C .proj₂
|
||||
module F = Functor F
|
||||
module G = Functor G
|
||||
module H = Functor H
|
||||
-- Not entirely clear what this is at this point:
|
||||
_P⊕_ = Category._∘_ (prodObj ×p ℂ) {F×A} {G×B} {H×C}
|
||||
|
||||
module _
|
||||
-- NaturalTransformation F G × ℂ .Arrow A B
|
||||
{θ×f : NaturalTransformation F G × ℂ [ A , B ]}
|
||||
|
|
@ -279,31 +290,28 @@ module CatExponential {ℓ : Level} (ℂ 𝔻 : Category ℓ ℓ) where
|
|||
ηθNat = proj₂ ηθNT
|
||||
|
||||
:isDistributive: :
|
||||
𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ fmap F ( ℂ [ g ∘ f ] ) ]
|
||||
≡ 𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ 𝔻 [ θ B ∘ fmap F f ] ]
|
||||
𝔻 [ 𝔻 [ η C ∘ θ C ] ∘ F.fmap ( ℂ [ g ∘ f ] ) ]
|
||||
≡ 𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ θ B ∘ F.fmap f ] ]
|
||||
:isDistributive: = begin
|
||||
𝔻 [ (ηθ C) ∘ fmap F (ℂ [ g ∘ f ]) ]
|
||||
𝔻 [ (ηθ C) ∘ F.fmap (ℂ [ g ∘ f ]) ]
|
||||
≡⟨ ηθNat (ℂ [ g ∘ f ]) ⟩
|
||||
𝔻 [ fmap H (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
|
||||
𝔻 [ H.fmap (ℂ [ g ∘ f ]) ∘ (ηθ A) ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ φ ∘ ηθ A ]) (H.isDistributive) ⟩
|
||||
𝔻 [ 𝔻 [ fmap H g ∘ fmap H f ] ∘ (ηθ A) ]
|
||||
≡⟨ sym isAssociative ⟩
|
||||
𝔻 [ fmap H g ∘ 𝔻 [ fmap H f ∘ ηθ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ fmap H g ∘ φ ]) isAssociative ⟩
|
||||
𝔻 [ fmap H g ∘ 𝔻 [ 𝔻 [ fmap H f ∘ η A ] ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ fmap H g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
|
||||
𝔻 [ fmap H g ∘ 𝔻 [ 𝔻 [ η B ∘ fmap G f ] ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ fmap H g ∘ φ ]) (sym isAssociative) ⟩
|
||||
𝔻 [ fmap H g ∘ 𝔻 [ η B ∘ 𝔻 [ fmap G f ∘ θ A ] ] ]
|
||||
≡⟨ isAssociative ⟩
|
||||
𝔻 [ 𝔻 [ fmap H g ∘ η B ] ∘ 𝔻 [ fmap G f ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ fmap G f ∘ θ A ] ]) (sym (ηNat g)) ⟩
|
||||
𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ 𝔻 [ fmap G f ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ φ ]) (sym (θNat f)) ⟩
|
||||
𝔻 [ 𝔻 [ η C ∘ fmap G g ] ∘ 𝔻 [ θ B ∘ fmap F f ] ] ∎
|
||||
where
|
||||
open Category 𝔻
|
||||
module H = Functor H
|
||||
𝔻 [ 𝔻 [ H.fmap g ∘ H.fmap f ] ∘ (ηθ A) ]
|
||||
≡⟨ sym 𝔻.isAssociative ⟩
|
||||
𝔻 [ H.fmap g ∘ 𝔻 [ H.fmap f ∘ ηθ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) 𝔻.isAssociative ⟩
|
||||
𝔻 [ H.fmap g ∘ 𝔻 [ 𝔻 [ H.fmap f ∘ η A ] ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) (cong (λ φ → 𝔻 [ φ ∘ θ A ]) (sym (ηNat f))) ⟩
|
||||
𝔻 [ H.fmap g ∘ 𝔻 [ 𝔻 [ η B ∘ G.fmap f ] ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ H.fmap g ∘ φ ]) (sym 𝔻.isAssociative) ⟩
|
||||
𝔻 [ H.fmap g ∘ 𝔻 [ η B ∘ 𝔻 [ G.fmap f ∘ θ A ] ] ]
|
||||
≡⟨ 𝔻.isAssociative ⟩
|
||||
𝔻 [ 𝔻 [ H.fmap g ∘ η B ] ∘ 𝔻 [ G.fmap f ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ φ ∘ 𝔻 [ G.fmap f ∘ θ A ] ]) (sym (ηNat g)) ⟩
|
||||
𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ G.fmap f ∘ θ A ] ]
|
||||
≡⟨ cong (λ φ → 𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ φ ]) (sym (θNat f)) ⟩
|
||||
𝔻 [ 𝔻 [ η C ∘ G.fmap g ] ∘ 𝔻 [ θ B ∘ F.fmap f ] ] ∎
|
||||
|
||||
eval : Functor (CatProduct.obj prodObj ℂ) 𝔻
|
||||
-- :eval: : Functor (prodObj ×p ℂ) 𝔻
|
||||
|
|
|
|||
Loading…
Reference in New Issue