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Type-synonyms for Representable functors and Presheafs

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Frederik Hanghøj Iversen 2018-01-17 12:16:07 +01:00
parent 902b953ad0
commit acacfac31c
1 changed files with 9 additions and 3 deletions
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12
src/Cat/Categories/Sets.agda
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@ -25,7 +25,10 @@ Sets {ℓ} = record
; ident = funExt (λ x → refl) , funExt (λ x → refl) ; ident = funExt (λ x → refl) , funExt (λ x → refl)
} }
representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Functor ℂ (Sets {ℓ'}) Representable : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
Representable {ℓ' = ℓ'} ℂ = Functor ℂ (Sets {ℓ'})
representable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object ℂ → Representable ℂ
representable {ℂ = ℂ} A = record representable {ℂ = ℂ} A = record
{ func* = λ B → ℂ.Arrow A B { func* = λ B → ℂ.Arrow A B
; func→ = λ f g → f ℂ.⊕ g ; func→ = λ f g → f ℂ.⊕ g
@ -35,8 +38,11 @@ representable {ℂ = ℂ} A = record
where where
open module ℂ = Category ℂ open module ℂ = Category ℂ
coRepresentable : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Functor (Opposite ℂ) (Sets {ℓ'}) Presheaf : {ℓ ℓ' : Level} → (ℂ : Category {ℓ} {ℓ'}) → Set (ℓ ⊔ lsuc ℓ')
coRepresentable {ℂ = ℂ} B = record Presheaf {ℓ' = ℓ'} ℂ = Functor (Opposite ℂ) (Sets {ℓ'})
presheaf : {ℓ ℓ' : Level} → {ℂ : Category {ℓ} {ℓ'}} → Category.Object (Opposite ℂ) → Presheaf ℂ
presheaf {ℂ = ℂ} B = record
{ func* = λ A → ℂ.Arrow A B { func* = λ A → ℂ.Arrow A B
; func→ = λ f g → g ℂ.⊕ f ; func→ = λ f g → g ℂ.⊕ f
; ident = funExt λ x → fst ℂ.ident ; ident = funExt λ x → fst ℂ.ident