diff --git a/src/Cat/Categories/Fun.agda b/src/Cat/Categories/Fun.agda index 4ae4251..4d32e47 100644 --- a/src/Cat/Categories/Fun.agda +++ b/src/Cat/Categories/Fun.agda @@ -6,6 +6,7 @@ open import Cubical open import Function open import Data.Product import Cubical.GradLemma +module UIP = Cubical.GradLemma open import Cat.Category open import Cat.Category.Functor @@ -117,11 +118,29 @@ module _ {ℓc ℓc' ℓd ℓd' : Level} {ℂ : Category ℓc ℓc'} {𝔻 : Cat lem : (λ _ → Natural F G θ) [ (λ f → θNat f) ≡ (λ f → θNat' f) ] lem = λ i f → 𝔻.arrowIsSet _ _ (θNat f) (θNat' f) i - naturalTransformationIsSets : isSet (NaturalTransformation F G) + naturalTransformationIsSets f : isSet (NaturalTransformation F G) + f a b p q i = res + where + k : (θ : Transformation F G) → (xx yy : Natural F G θ) → xx ≡ yy + k θ x y = let kk = naturalIsProp θ x y in {!!} + res : a ≡ b + res j = {!!} , {!!} + -- naturalTransformationIsSets σa σb p q + -- where + -- -- In Andrea's proof `lemSig` he proves something very similiar to + -- -- what I'm doing here, just for `Cubical.FromPathPrelude.Σ` rather + -- -- than `Σ`. In that proof, he just needs *one* proof that the first + -- -- components are equal - hence the arbitrary usage of `p` here. + -- secretSauce : proj₁ σa ≡ proj₁ σb + -- secretSauce i = proj₁ (p i) + -- lemSig : σa ≡ σb + -- lemSig i = (secretSauce i) , (UIP.lemPropF naturalIsProp secretSauce) {proj₂ σa} {proj₂ σb} i + -- res : p ≡ q + -- res = {!!} naturalTransformationIsSets (θ , θNat) (η , ηNat) p q i j = θ-η -- `i or `j - `p'` or `q'`? - , refl {x = t} i + , {!!} -- UIP.lemPropF {B = Natural F G} (λ x → {!!}) {(θ , θNat)} {(η , ηNat)} {!!} i -- naturalIsSet i (λ i → {!!} i) {!!} {!!} i j -- naturalIsSet {!p''!} {!p''!} {!!} i j -- λ f k → 𝔻.arrowIsSet (λ l → proj₂ (p l) f k) (λ l → proj₂ (p l) f k) {!!} {!!}