148 lines
4.4 KiB
Agda
148 lines
4.4 KiB
Agda
{-# OPTIONS --cubical #-}
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module Cat.Category.Functor where
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open import Agda.Primitive
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open import Cubical
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open import Function
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open import Cat.Category
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open Category hiding (_∘_ ; raw)
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module _ {ℓc ℓc' ℓd ℓd'}
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(ℂ : Category ℓc ℓc')
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(𝔻 : Category ℓd ℓd')
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where
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private
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ℓ = ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd'
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𝓤 = Set ℓ
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record RawFunctor : 𝓤 where
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field
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func* : Object ℂ → Object 𝔻
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func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
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record IsFunctor (F : RawFunctor) : 𝓤 where
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open RawFunctor F
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field
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ident : {c : Object ℂ} → func→ (𝟙 ℂ {c}) ≡ 𝟙 𝔻 {func* c}
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distrib : {A B C : Object ℂ} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
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→ func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ]
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record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
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field
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raw : RawFunctor
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{{isFunctor}} : IsFunctor raw
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private
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module R = RawFunctor raw
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func* : Object ℂ → Object 𝔻
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func* = R.func*
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func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
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func→ = R.func→
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open IsFunctor
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open Functor
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-- TODO: Is `IsFunctor` a proposition?
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module _
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{ℓa ℓb : Level}
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{ℂ 𝔻 : Category ℓa ℓb}
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{F : RawFunctor ℂ 𝔻}
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where
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private
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module 𝔻 = IsCategory (isCategory 𝔻)
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-- isProp : Set ℓ
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-- isProp = (x y : A) → x ≡ y
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IsFunctorIsProp : isProp (IsFunctor _ _ F)
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IsFunctorIsProp isF0 isF1 i = record
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{ ident = 𝔻.arrowIsSet _ _ isF0.ident isF1.ident i
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; distrib = 𝔻.arrowIsSet _ _ isF0.distrib isF1.distrib i
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}
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where
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module isF0 = IsFunctor isF0
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module isF1 = IsFunctor isF1
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-- Alternate version of above where `F` is indexed by an interval
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module _
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{ℓa ℓb : Level}
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{ℂ 𝔻 : Category ℓa ℓb}
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{F : I → RawFunctor ℂ 𝔻}
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where
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private
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module 𝔻 = IsCategory (isCategory 𝔻)
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IsProp' : {ℓ : Level} (A : I → Set ℓ) → Set ℓ
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IsProp' A = (a0 : A i0) (a1 : A i1) → A [ a0 ≡ a1 ]
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IsFunctorIsProp' : IsProp' λ i → IsFunctor _ _ (F i)
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IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
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(\ F → IsFunctorIsProp {F = F}) (\ i → F i)
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where
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open import Cubical.NType.Properties using (lemPropF)
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module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
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Functor≡ : {F G : Functor ℂ 𝔻}
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→ (eq* : func* F ≡ func* G)
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→ (eq→ : (λ i → ∀ {x y} → ℂ [ x , y ] → 𝔻 [ eq* i x , eq* i y ])
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[ func→ F ≡ func→ G ])
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→ F ≡ G
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Functor≡ {F} {G} eq* eq→ i = record
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{ raw = eqR i
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; isFunctor = eqIsF i
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}
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where
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eqR : raw F ≡ raw G
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eqR i = record { func* = eq* i ; func→ = eq→ i }
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eqIsF : (λ i → IsFunctor ℂ 𝔻 (eqR i)) [ isFunctor F ≡ isFunctor G ]
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eqIsF = IsFunctorIsProp' (isFunctor F) (isFunctor G)
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module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
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private
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F* = func* F
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F→ = func→ F
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G* = func* G
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G→ = func→ G
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module _ {a0 a1 a2 : Object A} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where
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dist : (F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡ C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ]
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dist = begin
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(F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡⟨ refl ⟩
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F→ (G→ (A [ α1 ∘ α0 ])) ≡⟨ cong F→ (G .isFunctor .distrib)⟩
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F→ (B [ G→ α1 ∘ G→ α0 ]) ≡⟨ F .isFunctor .distrib ⟩
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C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ] ∎
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_∘fr_ : RawFunctor A C
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RawFunctor.func* _∘fr_ = F* ∘ G*
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RawFunctor.func→ _∘fr_ = F→ ∘ G→
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instance
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isFunctor' : IsFunctor A C _∘fr_
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isFunctor' = record
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{ ident = begin
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(F→ ∘ G→) (𝟙 A) ≡⟨ refl ⟩
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F→ (G→ (𝟙 A)) ≡⟨ cong F→ (G .isFunctor .ident)⟩
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F→ (𝟙 B) ≡⟨ F .isFunctor .ident ⟩
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𝟙 C ∎
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; distrib = dist
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}
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_∘f_ : Functor A C
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raw _∘f_ = _∘fr_
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-- The identity functor
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identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
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identity = record
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{ raw = record
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{ func* = λ x → x
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; func→ = λ x → x
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}
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; isFunctor = record
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{ ident = refl
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; distrib = refl
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}
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}
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