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Implement category of presheaves

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This commit is contained in:
Frederik Hanghøj Iversen 2018-03-05 14:50:53 +01:00
parent ce4dd83969
commit bb379fa196
1 changed files with 19 additions and 13 deletions
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32
src/Cat/Categories/Fun.agda
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@ -62,9 +62,7 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
_h⊕g_ = NT[_∘_] {B} {C} {D} _h⊕g_ = NT[_∘_] {B} {C} {D}
:isAssociative: : L ≡ R :isAssociative: : L ≡ R
:isAssociative: = lemSig (naturalIsProp {F = A} {D}) :isAssociative: = lemSig (naturalIsProp {F = A} {D})
L R (funExt (λ x → isAssociative)) L R (funExt (λ x → 𝔻.isAssociative))
where
open Category 𝔻
private private
module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where module _ {A B : Functor ℂ 𝔻} {f : NaturalTransformation A B} where
@ -107,14 +105,22 @@ module Fun {ℓc ℓc' ℓd ℓd' : Level} (ℂ : Category ℓc ℓc') (𝔻 : C
Category.raw Fun = RawFun Category.raw Fun = RawFun
module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where module _ {ℓ ℓ' : Level} (ℂ : Category ℓ ℓ') where
open import Cat.Categories.Sets private
open NaturalTransformation (opposite ℂ) (𝓢𝓮𝓽 ℓ') open import Cat.Categories.Sets
open NaturalTransformation (opposite ℂ) (𝓢𝓮𝓽 ℓ')
-- Restrict the functors to Presheafs. -- Restrict the functors to Presheafs.
RawPresh : RawCategory (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ') rawPresh : RawCategory (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
RawPresh = record rawPresh = record
{ Object = Presheaf ℂ { Object = Presheaf ℂ
; Arrow = NaturalTransformation ; Arrow = NaturalTransformation
; 𝟙 = λ {F} → identity F ; 𝟙 = λ {F} → identity F
; _∘_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H} ; _∘_ = λ {F G H} → NT[_∘_] {F = F} {G = G} {H = H}
} }
instance
isCategory : IsCategory rawPresh
isCategory = Fun.:isCategory: _ _
Presh : Category (ℓ ⊔ lsuc ℓ') (ℓ ⊔ ℓ')
Category.raw Presh = rawPresh
Category.isCategory Presh = isCategory