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Clean-up yoneda embedding

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Frederik Hanghøj Iversen 2018-03-05 14:04:04 +01:00
parent 1bf565b87a
commit 7fbca1aeeb
1 changed files with 24 additions and 31 deletions
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49
src/Cat/Category/Yoneda.agda
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@ -5,23 +5,18 @@ module Cat.Category.Yoneda where
open import Agda.Primitive open import Agda.Primitive
open import Data.Product open import Data.Product
open import Cubical open import Cubical
open import Cubical.NType.Properties
open import Cat.Category open import Cat.Category
open import Cat.Category.Functor open import Cat.Category.Functor
open import Cat.Equality open import Cat.Equality
open Equality.Data.Product
-- TODO: We want to avoid defining the yoneda embedding going through the open import Cat.Categories.Fun
-- category of categories (since it doesn't exist). open import Cat.Categories.Sets
open import Cat.Categories.Cat using (RawCat) open import Cat.Categories.Cat
module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
private private
open import Cat.Categories.Fun
open import Cat.Categories.Sets
module Cat = Cat.Categories.Cat
open import Cat.Category.Exponential
open Functor
𝓢 = Sets ℓ 𝓢 = Sets ℓ
open Fun (opposite ℂ) 𝓢 open Fun (opposite ℂ) 𝓢
prshf = presheaf ℂ prshf = presheaf ℂ
@ -34,33 +29,31 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
-- --
-- In stead we'll use an ad-hoc definition -- which is definitionally -- In stead we'll use an ad-hoc definition -- which is definitionally
-- equivalent to that other one. -- equivalent to that other one.
_⇑_ = Cat.CatExponential.prodObj _⇑_ = CatExponential.prodObj
module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where module _ {A B : ℂ.Object} (f : ℂ [ A , B ]) where
:func→: : NaturalTransformation (prshf A) (prshf B) :func→: : NaturalTransformation (prshf A) (prshf B)
:func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.isAssociative :func→: = (λ C x → ℂ [ f ∘ x ]) , λ f₁ → funExt λ _ → ℂ.isAssociative
module _ {c : Category.Object ℂ} where rawYoneda : RawFunctor ℂ Fun
eqTrans : (λ _ → Transformation (prshf c) (prshf c)) RawFunctor.func* rawYoneda = prshf
[ (λ _ x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c) ] RawFunctor.func→ rawYoneda = :func→:
eqTrans = funExt λ x → funExt λ x → ℂ.isIdentity .proj₂ open RawFunctor rawYoneda
open import Cubical.NType.Properties isIdentity : IsIdentity
open import Cat.Categories.Fun isIdentity {c} = lemSig (naturalIsProp {F = prshf c} {prshf c}) _ _ eq
:ident: : :func→: (ℂ.𝟙 {c}) ≡ Category.𝟙 Fun {A = prshf c}
:ident: = lemSig (naturalIsProp {F = prshf c} {prshf c}) _ _ eq
where where
eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c) eq : (λ C x → ℂ [ ℂ.𝟙 ∘ x ]) ≡ identityTrans (prshf c)
eq = funExt λ A → funExt λ B → proj₂ ℂ.isIdentity eq = funExt λ A → funExt λ B → proj₂ ℂ.isIdentity
isDistributive : IsDistributive
isDistributive = {!!}
instance
isFunctor : IsFunctor ℂ Fun rawYoneda
IsFunctor.isIdentity isFunctor = isIdentity
IsFunctor.isDistributive isFunctor = isDistributive
yoneda : Functor ℂ Fun yoneda : Functor ℂ Fun
yoneda = record Functor.raw yoneda = rawYoneda
{ raw = record Functor.isFunctor yoneda = isFunctor
{ func* = prshf
; func→ = :func→:
}
; isFunctor = record
{ isIdentity = :ident:
; isDistributive = {!!}
}
}