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Frederik Hanghøj Iversen 2018-01-21 15:19:15 +01:00
parent ea3e14af96
commit b158b1d420
1 changed files with 19 additions and 20 deletions
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39
src/Cat/Functor.agda
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@ -22,25 +22,24 @@ record Functor {ℓc ℓc' ℓd ℓd'} (C : Category ℓc ℓc') (D : Category
→ func→ (a' C.⊕ a) ≡ func→ a' D.⊕ func→ a → 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 module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : Functor A B) where
open Functor
open Category
private private
open module F = Functor F F* = F .func*
open module G = Functor G F→ = F .func→
open module A = Category A G* = G .func*
open module B = Category B G→ = G .func→
open module C = Category C _A⊕_ = A ._⊕_
_B⊕_ = B ._⊕_
_C⊕_ = C ._⊕_
module _ {a0 a1 a2 : A .Object} {α0 : A .Arrow a0 a1} {α1 : A .Arrow a1 a2} where
F* = F.func* dist : (F→ ∘ G→) (α1 A⊕ α0) ≡ (F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0
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 dist = begin
(F→ ∘ G→) (α1 A.⊕ α0) ≡⟨ refl ⟩ (F→ ∘ G→) (α1 A⊕ α0) ≡⟨ refl ⟩
F→ (G→ (α1 A.⊕ α0)) ≡⟨ cong F→ G.distrib ⟩ F→ (G→ (α1 A⊕ α0)) ≡⟨ cong F→ (G .distrib)⟩
F→ ((G→ α1) B.⊕ (G→ α0)) ≡⟨ F.distrib ⟩ F→ ((G→ α1) B⊕ (G→ α0)) ≡⟨ F .distrib ⟩
(F→ ∘ G→) α1 C.⊕ (F→ ∘ G→) α0 ∎ (F→ ∘ G→) α1 C⊕ (F→ ∘ G→) α0 ∎
functor-comp : Functor A C functor-comp : Functor A C
functor-comp = functor-comp =
@ -48,10 +47,10 @@ module _ {ℓ ℓ' : Level} {A B C : Category ℓ ℓ'} (F : Functor B C) (G : F
{ func* = F* ∘ G* { func* = F* ∘ G*
; func→ = F→ ∘ G→ ; func→ = F→ ∘ G→
; ident = begin ; ident = begin
(F→ ∘ G→) (A.𝟙) ≡⟨ refl ⟩ (F→ ∘ G→) (A .𝟙) ≡⟨ refl ⟩
F→ (G→ (A.𝟙)) ≡⟨ cong F→ G.ident ⟩ F→ (G→ (A .𝟙)) ≡⟨ cong F→ (G .ident)⟩
F→ (B.𝟙) ≡⟨ F.ident ⟩ F→ (B .𝟙) ≡⟨ F .ident ⟩
C.𝟙 ∎ C .𝟙 ∎
; distrib = dist ; distrib = dist
} }