63 lines
2.3 KiB
Agda
63 lines
2.3 KiB
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
|
|||
|
|
constructor functor
|
|||
|
|
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' : Level} → {C : Category {ℓ} {ℓ'}} → Functor C C
|
|||
|
|
-- Identity = record { F* = λ x → x ; F→ = λ x → x ; ident = refl ; distrib = refl }
|
|||
|
|
identity = functor (λ x → x) (λ x → x) (refl) (refl)
|