148 lines
5.5 KiB
Agda
148 lines
5.5 KiB
Agda
-- This module Essentially just provides the data for natural transformations
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--
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-- This includes:
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--
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-- The types:
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--
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-- * Transformation - a family of functors
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-- * Natural - naturality condition for transformations
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-- * NaturalTransformation - both of the above
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--
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-- Elements of the above:
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--
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-- * identityTrans - the identity transformation
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-- * identityNatural - naturality for the above
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-- * identity - both of the above
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--
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-- Functions for manipulating the above:
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--
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-- * A composition operator.
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{-# OPTIONS --cubical #-}
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open import Cat.Prelude
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open import Data.Nat using (_≤′_ ; ≤′-refl ; ≤′-step)
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module Nat = Data.Nat
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open import Cat.Category
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open import Cat.Category.Functor
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module Cat.Category.NaturalTransformation
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{ℓc ℓc' ℓd ℓd' : Level}
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(ℂ : Category ℓc ℓc') (𝔻 : Category ℓd ℓd') where
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open Category using (Object)
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private
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module ℂ = Category ℂ
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module 𝔻 = Category 𝔻
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module _ (F G : Functor ℂ 𝔻) where
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private
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module F = Functor F
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module G = Functor G
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-- What do you call a non-natural tranformation?
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Transformation : Set (ℓc ⊔ ℓd')
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Transformation = (C : Object ℂ) → 𝔻 [ F.omap C , G.omap C ]
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Natural : Transformation → Set (ℓc ⊔ (ℓc' ⊔ ℓd'))
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Natural θ
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= {A B : Object ℂ}
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→ (f : ℂ [ A , B ])
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→ 𝔻 [ θ B ∘ F.fmap f ] ≡ 𝔻 [ G.fmap f ∘ θ A ]
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NaturalTransformation : Set (ℓc ⊔ ℓc' ⊔ ℓd')
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NaturalTransformation = Σ Transformation Natural
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-- Think I need propPi and that arrows are sets
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propIsNatural : (θ : _) → isProp (Natural θ)
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propIsNatural θ x y i {A} {B} f = 𝔻.arrowsAreSets _ _ (x f) (y f) i
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NaturalTransformation≡ : {α β : NaturalTransformation}
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→ (eq₁ : α .fst ≡ β .fst)
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→ α ≡ β
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NaturalTransformation≡ eq = lemSig propIsNatural _ _ eq
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identityTrans : (F : Functor ℂ 𝔻) → Transformation F F
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identityTrans F C = 𝔻.identity
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identityNatural : (F : Functor ℂ 𝔻) → Natural F F (identityTrans F)
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identityNatural F {A = A} {B = B} f = begin
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𝔻 [ identityTrans F B ∘ F→ f ] ≡⟨⟩
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𝔻 [ 𝔻.identity ∘ F→ f ] ≡⟨ 𝔻.leftIdentity ⟩
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F→ f ≡⟨ sym 𝔻.rightIdentity ⟩
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𝔻 [ F→ f ∘ 𝔻.identity ] ≡⟨⟩
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𝔻 [ F→ f ∘ identityTrans F A ] ∎
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where
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module F = Functor F
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F→ = F.fmap
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identity : (F : Functor ℂ 𝔻) → NaturalTransformation F F
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identity F = identityTrans F , identityNatural F
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module _ {F G H : Functor ℂ 𝔻} where
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private
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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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T[_∘_] : Transformation G H → Transformation F G → Transformation F H
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T[ θ ∘ η ] C = 𝔻 [ θ C ∘ η C ]
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NT[_∘_] : NaturalTransformation G H → NaturalTransformation F G → NaturalTransformation F H
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fst NT[ (θ , _) ∘ (η , _) ] = T[ θ ∘ η ]
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snd NT[ (θ , θNat) ∘ (η , ηNat) ] {A} {B} f = begin
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𝔻 [ T[ θ ∘ η ] B ∘ F.fmap f ] ≡⟨⟩
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𝔻 [ 𝔻 [ θ B ∘ η B ] ∘ F.fmap f ] ≡⟨ sym 𝔻.isAssociative ⟩
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𝔻 [ θ B ∘ 𝔻 [ η B ∘ F.fmap f ] ] ≡⟨ cong (λ φ → 𝔻 [ θ B ∘ φ ]) (ηNat f) ⟩
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𝔻 [ θ B ∘ 𝔻 [ G.fmap f ∘ η A ] ] ≡⟨ 𝔻.isAssociative ⟩
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𝔻 [ 𝔻 [ θ B ∘ G.fmap f ] ∘ η A ] ≡⟨ cong (λ φ → 𝔻 [ φ ∘ η A ]) (θNat f) ⟩
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𝔻 [ 𝔻 [ H.fmap f ∘ θ A ] ∘ η A ] ≡⟨ sym 𝔻.isAssociative ⟩
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𝔻 [ H.fmap f ∘ 𝔻 [ θ A ∘ η A ] ] ≡⟨⟩
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𝔻 [ H.fmap f ∘ T[ θ ∘ η ] A ] ∎
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module Properties where
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module _ {F G : Functor ℂ 𝔻} where
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transformationIsSet : isSet (Transformation F G)
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transformationIsSet _ _ p q i j C = 𝔻.arrowsAreSets _ _ (λ l → p l C) (λ l → q l C) i j
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naturalIsProp : (θ : Transformation F G) → isProp (Natural F G θ)
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naturalIsProp θ θNat θNat' = lem
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where
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lem : (λ _ → Natural F G θ) [ (λ f → θNat f) ≡ (λ f → θNat' f) ]
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lem = λ i f → 𝔻.arrowsAreSets _ _ (θNat f) (θNat' f) i
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naturalIsSet : (θ : Transformation F G) → isSet (Natural F G θ)
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naturalIsSet θ =
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ntypeCumulative {n = 1}
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(Data.Nat.≤′-step Data.Nat.≤′-refl)
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(naturalIsProp θ)
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naturalTransformationIsSet : isSet (NaturalTransformation F G)
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naturalTransformationIsSet = sigPresSet transformationIsSet naturalIsSet
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module _
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{F G H I : Functor ℂ 𝔻}
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{θ : NaturalTransformation F G}
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{η : NaturalTransformation G H}
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{ζ : NaturalTransformation H I} where
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-- isAssociative : NT[ ζ ∘ NT[ η ∘ θ ] ] ≡ NT[ NT[ ζ ∘ η ] ∘ θ ]
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isAssociative
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: NT[_∘_] {F} {H} {I} ζ (NT[_∘_] {F} {G} {H} η θ)
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≡ NT[_∘_] {F} {G} {I} (NT[_∘_] {G} {H} {I} ζ η) θ
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isAssociative
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= lemSig (naturalIsProp {F = F} {I}) _ _
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(funExt (λ _ → 𝔻.isAssociative))
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module _ {F G : Functor ℂ 𝔻} {θNT : NaturalTransformation F G} where
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private
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propNat = naturalIsProp {F = F} {G}
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rightIdentity : (NT[_∘_] {F} {F} {G} θNT (identity F)) ≡ θNT
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rightIdentity = lemSig propNat _ _ (funExt (λ _ → 𝔻.rightIdentity))
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leftIdentity : (NT[_∘_] {F} {G} {G} (identity G) θNT) ≡ θNT
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leftIdentity = lemSig propNat _ _ (funExt (λ _ → 𝔻.leftIdentity))
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isIdentity
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: (NT[_∘_] {F} {G} {G} (identity G) θNT) ≡ θNT
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× (NT[_∘_] {F} {F} {G} θNT (identity F)) ≡ θNT
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isIdentity = leftIdentity , rightIdentity
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