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Factor out a useful type-family

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Frederik Hanghøj Iversen 2018-02-20 17:33:02 +01:00
parent 860c91f913
commit ff496aae09
1 changed files with 12 additions and 5 deletions
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17
src/Cat/Category.agda
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@ -143,13 +143,20 @@ module _ {ℓa} {ℓb} {ℂ : RawCategory ℓa ℓb} where
module IC = IsCategory module IC = IsCategory
module X = IsCategory x module X = IsCategory x
module Y = IsCategory y module Y = IsCategory y
-- ident : X.ident {?} ≡ Y.ident
ident : (λ _ → IsIdentity 𝟙) [ X.ident ≡ Y.ident ]
ident = X.propIsIdentity X.ident Y.ident ident = X.propIsIdentity X.ident Y.ident
done : x ≡ y -- A version of univalence indexed by the identity proof.
T : I → Set (ℓa ⊔ ℓb) -- Not of course that since it's defined where `RawCategory ℂ` has been opened
T i = {A B : Object} → -- this is specialized to that category.
Univ : IsIdentity 𝟙 → Set _
Univ idnt = {A B : Y.Raw.Object} →
isEquiv (A ≡ B) (A ≅ B) isEquiv (A ≡ B) (A ≅ B)
(λ eq → transp (λ i₁ → A ≅ eq i₁) (𝟙 , 𝟙 , ident i)) (λ eq → transp (λ j → A ≅ eq j) (𝟙 , 𝟙 , idnt))
eqUni : T [ X.univalent ≡ Y.univalent ] done : x ≡ y
U : ∀ {a : IsIdentity 𝟙} → (λ _ → IsIdentity 𝟙) [ X.ident ≡ a ] → (b : Univ a) → Set _
U eqwal bbb = (λ i → Univ (eqwal i)) [ X.univalent ≡ bbb ]
eqUni : U ident Y.univalent
eqUni = {!!} eqUni = {!!}
IC.assoc (done i) = X.propIsAssociative X.assoc Y.assoc i IC.assoc (done i) = X.propIsAssociative X.assoc Y.assoc i
IC.ident (done i) = ident i IC.ident (done i) = ident i