cat/src/Cat/Category/Functor.agda

{-# OPTIONS --cubical #-}
module Cat.Category.Functor where

open import Agda.Primitive
open import Cubical
open import Function

open import Cat.Category

open Category hiding (_∘_ ; raw)

module _ {ℓc ℓc' ℓd ℓd'}
    (ℂ : Category ℓc ℓc')
    (𝔻 : Category ℓd ℓd')
    where

  private
    ℓ = ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd'
    𝓤 = Set ℓ

  record RawFunctor : 𝓤 where
    field
      func* : Object ℂ → Object 𝔻
      func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]

  record IsFunctor (F : RawFunctor) : 𝓤 where
    open RawFunctor F
    field
      ident   : {c : Object ℂ} → func→ (𝟙 ℂ {c}) ≡ 𝟙 𝔻 {func* c}
      distrib : {A B C : Object ℂ} {f : ℂ [ A , B ]} {g : ℂ [ B , C ]}
        → func→ (ℂ [ g ∘ f ]) ≡ 𝔻 [ func→ g ∘ func→ f ]

  record Functor : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
    field
      raw : RawFunctor
      {{isFunctor}} : IsFunctor raw

    private
      module R = RawFunctor raw

    func* : Object ℂ → Object 𝔻
    func* = R.func*

    func→ : ∀ {A B} → ℂ [ A , B ] → 𝔻 [ func* A , func* B ]
    func→ = R.func→

open IsFunctor
open Functor

-- TODO: Is `IsFunctor` a proposition?
module _
    {ℓa ℓb : Level}
    {ℂ 𝔻 : Category ℓa ℓb}
    {F : RawFunctor ℂ 𝔻}
    where
  private
    module 𝔻 = IsCategory (isCategory 𝔻)

  -- isProp  : Set ℓ
  -- isProp  = (x y : A) → x ≡ y

  IsFunctorIsProp : isProp (IsFunctor _ _ F)
  IsFunctorIsProp isF0 isF1 i = record
    { ident = 𝔻.arrowIsSet isF0.ident isF1.ident i
    ; distrib = 𝔻.arrowIsSet isF0.distrib isF1.distrib i
    }
    where
      module isF0 = IsFunctor isF0
      module isF1 = IsFunctor isF1

-- Alternate version of above where `F` is a path in 
module _
    {ℓa ℓb : Level}
    {ℂ 𝔻 : Category ℓa ℓb}
    {F : I → RawFunctor ℂ 𝔻}
    where
  private
    module 𝔻 = IsCategory (isCategory 𝔻)
  IsProp' : {ℓ : Level} (A : I → Set ℓ) → Set ℓ
  IsProp' A = (a0 : A i0) (a1 : A i1) → A [ a0 ≡ a1 ]

  IsFunctorIsProp' : IsProp' λ i → IsFunctor _ _ (F i)
  IsFunctorIsProp' isF0 isF1 = lemPropF {B = IsFunctor ℂ 𝔻}
    (\ F → IsFunctorIsProp {F = F}) (\ i → F i)
    where
      open import Cubical.GradLemma using (lemPropF)

module _ {ℓ ℓ' : Level} {ℂ 𝔻 : Category ℓ ℓ'} where
  Functor≡ : {F G : Functor ℂ 𝔻}
    → (eq* : func* F ≡ func* G)
    → (eq→ : (λ i → ∀ {x y} → ℂ [ x , y ] → 𝔻 [ eq* i x , eq* i y ])
        [ func→ F ≡ func→ G ])
    → F ≡ G
  Functor≡ {F} {G} eq* eq→ i = record
    { raw = eqR i
    ; isFunctor = eqIsF i
    }
    where
      eqR : raw F ≡ raw G
      eqR i = record { func* = eq* i ; func→ = eq→ i }
      eqIsF : (λ i →  IsFunctor ℂ 𝔻 (eqR i)) [ isFunctor F ≡ isFunctor G ]
      eqIsF = IsFunctorIsProp' (isFunctor F) (isFunctor G)

module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
  private
    F* = func* F
    F→ = func→ F
    G* = func* G
    G→ = func→ G
    module _ {a0 a1 a2 : Object A} {α0 : A [ a0 , a1 ]} {α1 : A [ a1 , a2 ]} where

      dist : (F→ ∘ G→) (A [ α1 ∘ α0 ]) ≡ C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ]
      dist = begin
        (F→ ∘ G→) (A [ α1 ∘ α0 ])         ≡⟨ refl ⟩
        F→ (G→ (A [ α1 ∘ α0 ]))           ≡⟨ cong F→ (G .isFunctor .distrib)⟩
        F→ (B [ G→ α1 ∘ G→ α0 ])          ≡⟨ F .isFunctor .distrib ⟩
        C [ (F→ ∘ G→) α1 ∘ (F→ ∘ G→) α0 ] ∎

    _∘fr_ : RawFunctor A C
    RawFunctor.func* _∘fr_ = F* ∘ G*
    RawFunctor.func→ _∘fr_ = F→ ∘ G→
    instance
      isFunctor' : IsFunctor A C _∘fr_
      isFunctor' = record
        { ident = begin
          (F→ ∘ G→) (𝟙 A) ≡⟨ refl ⟩
          F→ (G→ (𝟙 A))   ≡⟨ cong F→ (G .isFunctor .ident)⟩
          F→ (𝟙 B)        ≡⟨ F .isFunctor .ident ⟩
          𝟙 C             ∎
        ; distrib = dist
        }

  _∘f_ : Functor A C
  raw _∘f_ = _∘fr_

-- The identity functor
identity : ∀ {ℓ ℓ'} → {C : Category ℓ ℓ'} → Functor C C
identity = record
  { raw = record
    { func* = λ x → x
    ; func→ = λ x → x
    }
  ; isFunctor = record
    { ident = refl
    ; distrib = refl
    }
  }