One step closer to yoneda
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@ -58,35 +58,35 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where
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open Exponential
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private
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Catℓ = Cat ℓ ℓ
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prshf = presheaf {ℂ = ℂ}
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-- Exp : Set (lsuc (lsuc ℓ))
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-- Exp = Exponential (Cat (lsuc ℓ) ℓ)
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-- Sets (Opposite ℂ)
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-- Exp : Set (lsuc (lsuc ℓ))
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-- Exp = Exponential (Cat (lsuc ℓ) ℓ)
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-- Sets (Opposite ℂ)
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_⇑_ : (A B : Catℓ .Object) → Catℓ .Object
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A ⇑ B = (exponent A B) .obj
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where
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open HasExponentials (Cat.hasExponentials ℓ)
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_⇑_ : (A B : Catℓ .Object) → Catℓ .Object
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A ⇑ B = (exponent A B) .obj
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where
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open HasExponentials (Cat.hasExponentials ℓ)
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-- private
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-- -- I need `Sets` to be a `Category ℓ ℓ` but it simlpy isn't.
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-- Setz : Category ℓ ℓ
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-- Setz = {!Sets!}
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-- :func*: : ℂ .Object → (Setz ⇑ Opposite ℂ) .Object
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-- :func*: A = {!!}
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module _ {A B : ℂ .Object} (f : ℂ .Arrow A B) where
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:func→: : NaturalTransformation (prshf A) (prshf B)
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:func→: = (λ C x → (ℂ ._⊕_ f x)) , λ f₁ → funExt λ x → lem
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where
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lem = (ℂ .isCategory) .IsCategory.assoc
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module _ {c : ℂ .Object} where
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eqTrans : (:func→: (ℂ .𝟙 {c})) .proj₁ ≡ (Fun .𝟙 {o = prshf c}) .proj₁
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eqTrans = funExt λ x → funExt λ x → ℂ .isCategory .IsCategory.ident .proj₂
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eqNat : (i : I) → Natural (prshf c) (prshf c) (eqTrans i)
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eqNat i f = {!!}
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-- prsh = presheaf {ℂ = ℂ}
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-- k = prsh {!!}
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-- :func*:' : ℂ .Object → Presheaf ℂ
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-- :func*:' = prsh
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-- module _ {A B : ℂ .Object} (f : ℂ .Arrow A B) where
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-- open import Cat.Categories.Fun
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-- :func→:' : NaturalTransformation (prsh A) (prsh B)
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:ident: : (:func→: (ℂ .𝟙 {c})) ≡ (Fun .𝟙 {o = prshf c})
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:ident: i = eqTrans i , eqNat i
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yoneda : Functor ℂ (Fun {ℂ = Opposite ℂ} {𝔻 = Sets {ℓ}})
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yoneda = record
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{ func* = presheaf {ℂ = ℂ}
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; func→ = {!!}
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; ident = {!!}
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{ func* = prshf
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; func→ = :func→:
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; ident = :ident:
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; distrib = {!!}
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}
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