diff --git a/src/Cat/Category/Properties.agda b/src/Cat/Category/Properties.agda index 6edaca1..7056504 100644 --- a/src/Cat/Category/Properties.agda +++ b/src/Cat/Category/Properties.agda @@ -58,35 +58,35 @@ module _ {ℓ : Level} {ℂ : Category ℓ ℓ} where open Exponential private Catℓ = Cat ℓ ℓ + prshf = presheaf {ℂ = ℂ} - -- Exp : Set (lsuc (lsuc ℓ)) - -- Exp = Exponential (Cat (lsuc ℓ) ℓ) - -- Sets (Opposite ℂ) + -- Exp : Set (lsuc (lsuc ℓ)) + -- Exp = Exponential (Cat (lsuc ℓ) ℓ) + -- Sets (Opposite ℂ) - _⇑_ : (A B : Catℓ .Object) → Catℓ .Object - A ⇑ B = (exponent A B) .obj - where - open HasExponentials (Cat.hasExponentials ℓ) + _⇑_ : (A B : Catℓ .Object) → Catℓ .Object + A ⇑ B = (exponent A B) .obj + where + open HasExponentials (Cat.hasExponentials ℓ) - -- private - -- -- I need `Sets` to be a `Category ℓ ℓ` but it simlpy isn't. - -- Setz : Category ℓ ℓ - -- Setz = {!Sets!} - -- :func*: : ℂ .Object → (Setz ⇑ Opposite ℂ) .Object - -- :func*: A = {!!} + module _ {A B : ℂ .Object} (f : ℂ .Arrow A B) where + :func→: : NaturalTransformation (prshf A) (prshf B) + :func→: = (λ C x → (ℂ ._⊕_ f x)) , λ f₁ → funExt λ x → lem + where + lem = (ℂ .isCategory) .IsCategory.assoc + module _ {c : ℂ .Object} where + eqTrans : (:func→: (ℂ .𝟙 {c})) .proj₁ ≡ (Fun .𝟙 {o = prshf c}) .proj₁ + eqTrans = funExt λ x → funExt λ x → ℂ .isCategory .IsCategory.ident .proj₂ + eqNat : (i : I) → Natural (prshf c) (prshf c) (eqTrans i) + eqNat i f = {!!} - -- prsh = presheaf {ℂ = ℂ} - -- k = prsh {!!} - -- :func*:' : ℂ .Object → Presheaf ℂ - -- :func*:' = prsh - -- module _ {A B : ℂ .Object} (f : ℂ .Arrow A B) where - -- open import Cat.Categories.Fun - -- :func→:' : NaturalTransformation (prsh A) (prsh B) + :ident: : (:func→: (ℂ .𝟙 {c})) ≡ (Fun .𝟙 {o = prshf c}) + :ident: i = eqTrans i , eqNat i yoneda : Functor ℂ (Fun {ℂ = Opposite ℂ} {𝔻 = Sets {ℓ}}) yoneda = record - { func* = presheaf {ℂ = ℂ} - ; func→ = {!!} - ; ident = {!!} + { func* = prshf + ; func→ = :func→: + ; ident = :ident: ; distrib = {!!} }