cat/src/Cat/Functor.agda

module Cat.Functor where

open import Agda.Primitive
open import Cubical
open import Function

open import Cat.Category

record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category ℓd ℓd')
  : Set (ℓc ⊔ ℓc' ⊔ ℓd ⊔ ℓd') where
  private
    open module C = Category C
    open module D = Category D
  field
    func* : C.Object → D.Object
    func→ : {dom cod : C.Object} → C.Arrow dom cod → D.Arrow (func* dom) (func* cod)
    ident   : { c : C.Object } → func→ (C.𝟙 {c}) ≡ D.𝟙 {func* c}
    -- TODO: Avoid use of ugly explicit arguments somehow.
    -- This guy managed to do it:
    --    https://github.com/copumpkin/categories/blob/master/Categories/Functor/Core.agda
    distrib : { c c' c'' : C.Object} {a : C.Arrow c c'} {a' : C.Arrow c' c''}
      → func→ (a' C.⊕ a) ≡ func→ a' D.⊕ func→ a

module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
  private
    open module F = Functor F
    open module G = Functor G
    open module A = Category A
    open module B = Category B
    open module C = Category C

    F* = F.func*
    F→ = F.func→
    G* = G.func*
    G→ = G.func→
    module _ {a0 a1 a2 : A.Object} {α0 : A.Arrow a0 a1} {α1 : A.Arrow a1 a2} where

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

  functor-comp : Functor A C
  functor-comp =
    record
      { func* = F* ∘ G*
      ; func→ = F→ ∘ G→
      ; ident = begin
        (F→ ∘ G→) (A.𝟙) ≡⟨ refl ⟩
        F→ (G→ (A.𝟙))   ≡⟨ cong F→ G.ident ⟩
        F→ (B.𝟙)        ≡⟨ F.ident ⟩
        C.𝟙             ∎
      ; distrib = dist
      }

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