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@ -137,53 +137,27 @@ module _ {ℓa ℓb : Level} {C : RawCategory ℓa ℓb} {ℂ : IsCategory C} wh
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propUnivalent a b i = propPi (λ iso → propHasLevel ⟨-2⟩) a b i
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propUnivalent a b i = propPi (λ iso → propHasLevel ⟨-2⟩) a b i
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module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
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module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
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-- TODO, provable by using arrow-is-set and that isProp (isEquiv _ _ _)
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-- This lemma will be useful to prove the equality of two categories.
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IsCategory-is-prop : isProp (IsCategory ℂ)
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IsCategory-is-prop x y i = record
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-- Why choose `x`'s `propIsAssociative`?
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-- Well, probably it could be pulled out of the record.
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{ assoc = x.propIsAssociative x.assoc y.assoc i
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; ident = ident' i
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; arrowIsSet = x.propArrowIsSet x.arrowIsSet y.arrowIsSet i
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; univalent = eqUni i
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}
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where
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module x = IsCategory x
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module y = IsCategory y
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ident' = x.propIsIdentity x.ident y.ident
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ident'' = ident' i
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xuni : x.Univalent
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xuni = x.univalent
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yuni : y.Univalent
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yuni = y.univalent
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open RawCategory ℂ
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open RawCategory ℂ
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Pp : (x.ident ≡ y.ident) → I → Set (ℓa ⊔ ℓb)
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private
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Pp eqIdent i = {A B : Object} →
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module _ (x y : IsCategory ℂ) where
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isEquiv (A ≡ B) (A ≅ B)
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module IC = IsCategory
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(λ A≡B →
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module X = IsCategory x
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transp
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module Y = IsCategory y
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(λ j →
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ident = X.propIsIdentity X.ident Y.ident
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Σ-syntax (Arrow A (A≡B j))
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done : x ≡ y
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(λ f → Σ-syntax (Arrow (A≡B j) A) (λ g → g ∘ f ≡ 𝟙 × f ∘ g ≡ 𝟙)))
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( 𝟙
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, 𝟙
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, ident' i
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)
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)
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T : I → Set (ℓa ⊔ ℓb)
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T : I → Set (ℓa ⊔ ℓb)
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T = Pp {!ident'!}
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T i = {A B : Object} →
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open Cubical.NType.Properties
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isEquiv (A ≡ B) (A ≅ B)
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test : (λ _ → x.Univalent) [ xuni ≡ xuni ]
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(λ eq → transp (λ i₁ → A ≅ eq i₁) (𝟙 , 𝟙 , ident i))
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test = refl
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eqUni : T [ X.univalent ≡ Y.univalent ]
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t = {!!}
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P : (uni : x.Univalent) → xuni ≡ uni → Set (ℓa ⊔ ℓb)
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P = {!!}
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-- T i0 ≡ x.Univalent
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-- T i1 ≡ y.Univalent
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eqUni : T [ xuni ≡ yuni ]
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eqUni = {!!}
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eqUni = {!!}
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IC.assoc (done i) = X.propIsAssociative X.assoc Y.assoc i
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IC.ident (done i) = ident i
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IC.arrowIsSet (done i) = X.propArrowIsSet X.arrowIsSet Y.arrowIsSet i
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IC.univalent (done i) = eqUni i
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propIsCategory : isProp (IsCategory ℂ)
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propIsCategory = done
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record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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record Category (ℓa ℓb : Level) : Set (lsuc (ℓa ⊔ ℓb)) where
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field
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field
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