diff --git a/formalizations/guarded-cubical/Cubical/HigherCategories/ThinDoubleCategory/Constructions/Constructions.agda b/formalizations/guarded-cubical/Cubical/HigherCategories/ThinDoubleCategory/Constructions/Constructions.agda index 37a27cab355d2cef1d278e693c900525d253255a..6144152f02532c65e665d8994e05fe5b7700d983 100644 --- a/formalizations/guarded-cubical/Cubical/HigherCategories/ThinDoubleCategory/Constructions/Constructions.agda +++ b/formalizations/guarded-cubical/Cubical/HigherCategories/ThinDoubleCategory/Constructions/Constructions.agda @@ -7,7 +7,8 @@ module Cubical.HigherCategories.ThinDoubleCategory.Constructions.Constructions w open import Cubical.Foundations.Prelude open import Cubical.Data.Sigma - +open import Cubical.Data.Unit.Base +open import Cubical.Data.Unit.Properties open import Cubical.Categories.Displayed.Base open import Cubical.Categories.Displayed.Limits.BinProduct @@ -31,6 +32,7 @@ open import Cubical.Categories.Instances.Sets open import Cubical.Foundations.HLevels open import Cubical.Categories.Functor.Base open import Cubical.Foundations.Equiv +open import Cubical.Foundations.Isomorphism private variable @@ -119,9 +121,24 @@ module _ (C : ThinDoubleCat ℓ ℓ' ℓ'' ℓ''') where Term : (t : 1Terminal (VCat C)) -> Type (ℓ-max (ℓ-max (ℓ-max ℓ'' ℓ''') (ℓ-max ℓ ℓ)) (ℓ-max ℓ' ℓ')) Term t = Terminalᴰ (Preorderᴰ→Catᴰ (Squares C)) - (transport {!!} (terminalToUniversalElement t ×UE terminalToUniversalElement t)) - - + (transport + (cong (λ p -> UniversalElement (VCat C ×C VCat C) p) + (sym (cong (λ u -> Constant ( ( VCat C ×C VCat C ) ^op) (SET ℓ-zero) u) (Σ≡Prop (λ _ -> isPropIsSet) (isoToPath (iso to inv sec retr))) + ∙ Const-product {C = (VCat C) ^op} {D = (VCat C) ^op} {x = (Unit , isSetUnit)}))) + (terminalToUniversalElement t ×UE terminalToUniversalElement t)) + where + to : Unit -> Σ Unit (λ _ → Unit) + to x = x , x + + inv : Σ Unit (λ _ → Unit) -> Unit + inv x = x .fst + + sec : section to inv + sec x = refl + + retr : retract to inv + retr x = refl + -- "manual" way -- (Preorderᴰ→Catᴰ (Squares C)) (terminalToUniversalElement -- ((fst t , fst t) ,