cat/src/Cat/Functor.agda

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2018-01-08 21:54:53 +00:00
module Cat.Functor where
open import Agda.Primitive
open import Cubical
open import Function
open import Cat.Category
record Functor {c c' d d'} (C : Category {c} {c'}) (D : Category {d} {d'})
: Set (c c' d d') 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 _ { ' : Level} {A B C : Category {} {'}} (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 : { ' : Level} {C : Category {} {'}} 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
}