cat/src/Cat/Category/NaturalTransformation.agda

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{-# OPTIONS --allow-unsolved-metas --cubical #-}
module Cat.Category.NaturalTransformation where
open import Agda.Primitive
open import Data.Product
open import Cubical
open import Cat.Category
open import Cat.Category.Functor hiding (identity)
module NaturalTransformation {c c' d d' : Level}
( : Category c c') (𝔻 : Category d d') where
open Category using (Object ; 𝟙)
module _ (F G : Functor 𝔻) where
private
module F = Functor F
module G = Functor G
-- What do you call a non-natural tranformation?
Transformation : Set (c d')
Transformation = (C : Object ) 𝔻 [ F.func* C , G.func* C ]
Natural : Transformation Set (c (c' d'))
Natural θ
= {A B : Object }
(f : [ A , B ])
𝔻 [ θ B F.func→ f ] 𝔻 [ G.func→ f θ A ]
NaturalTransformation : Set (c c' d')
NaturalTransformation = Σ Transformation Natural
NaturalTransformation≡ : {α β : NaturalTransformation}
(eq₁ : α .proj₁ β .proj₁)
(eq₂ : PathP
(λ i {A B : Object } (f : [ A , B ])
𝔻 [ eq₁ i B F.func→ f ]
𝔻 [ G.func→ f eq₁ i A ])
(α .proj₂) (β .proj₂))
α β
NaturalTransformation≡ eq₁ eq₂ i = eq₁ i , eq₂ i
identityTrans : (F : Functor 𝔻) Transformation F F
identityTrans F C = 𝟙 𝔻
identityNatural : (F : Functor 𝔻) Natural F F (identityTrans F)
identityNatural F {A = A} {B = B} f = begin
𝔻 [ identityTrans F B F→ f ] ≡⟨⟩
𝔻 [ 𝟙 𝔻 F→ f ] ≡⟨ proj₂ 𝔻.isIdentity
F→ f ≡⟨ sym (proj₁ 𝔻.isIdentity)
𝔻 [ F→ f 𝟙 𝔻 ] ≡⟨⟩
𝔻 [ F→ f identityTrans F A ]
where
module F = Functor F
F→ = F.func→
module 𝔻 = Category 𝔻
identity : (F : Functor 𝔻) NaturalTransformation F F
identity F = identityTrans F , identityNatural F
module _ {F G H : Functor 𝔻} where
private
module F = Functor F
module G = Functor G
module H = Functor H
_∘nt_ : Transformation G H Transformation F G Transformation F H
(θ ∘nt η) C = 𝔻 [ θ C η C ]
NatComp _:⊕:_ : NaturalTransformation G H NaturalTransformation F G NaturalTransformation F H
proj₁ ((θ , _) :⊕: (η , _)) = θ ∘nt η
proj₂ ((θ , θNat) :⊕: (η , ηNat)) {A} {B} f = begin
𝔻 [ (θ ∘nt η) B F.func→ f ] ≡⟨⟩
𝔻 [ 𝔻 [ θ B η B ] F.func→ f ] ≡⟨ sym isAssociative
𝔻 [ θ B 𝔻 [ η B F.func→ f ] ] ≡⟨ cong (λ φ 𝔻 [ θ B φ ]) (ηNat f)
𝔻 [ θ B 𝔻 [ G.func→ f η A ] ] ≡⟨ isAssociative
𝔻 [ 𝔻 [ θ B G.func→ f ] η A ] ≡⟨ cong (λ φ 𝔻 [ φ η A ]) (θNat f)
𝔻 [ 𝔻 [ H.func→ f θ A ] η A ] ≡⟨ sym isAssociative
𝔻 [ H.func→ f 𝔻 [ θ A η A ] ] ≡⟨⟩
𝔻 [ H.func→ f (θ ∘nt η) A ]
where
open Category 𝔻
NatComp = _:⊕:_