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If C is a univalent category, then the type of universal elements of a presheaf on C should be a property. We now have an application of this in the cartesian product solver.
This should follow from proving that the category of elements of a presheaf on a univalent category is univalent, because we have already shown universal elements are equivalent to terminal objects on the category of elements, and terminal objects are already shown to be a prop.
Proving this is going to take some work though because there isn't much in cubical for working with univalent categories and because you run into transport hell pretty much immediately. The following properties look useful:
Transport along isoToPath in SET is equal to applying the function (the equivalent of uaβ in Foundations.Univalence).
If F is a functor whose domain and codomain are univalent, then isoToPath of F-iso F iso is equal to cong (F .F-ob) of isoToPath
And possibly more...
The text was updated successfully, but these errors were encountered:
If C is a univalent category, then the type of universal elements of a presheaf on C should be a property. We now have an application of this in the cartesian product solver.
This should follow from proving that the category of elements of a presheaf on a univalent category is univalent, because we have already shown universal elements are equivalent to terminal objects on the category of elements, and terminal objects are already shown to be a prop.
Proving this is going to take some work though because there isn't much in cubical for working with univalent categories and because you run into transport hell pretty much immediately. The following properties look useful:
SET
is equal to applying the function (the equivalent ofuaβ
inFoundations.Univalence
).F
is a functor whose domain and codomain are univalent, then isoToPath ofF-iso F iso
is equal tocong (F .F-ob)
ofisoToPath
And possibly more...
The text was updated successfully, but these errors were encountered: