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Applications of Univalence #1

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maxsnew opened this issue Mar 18, 2023 · 2 comments
Closed

Applications of Univalence #1

maxsnew opened this issue Mar 18, 2023 · 2 comments
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@maxsnew
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maxsnew commented Mar 18, 2023

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:

  1. Transport along isoToPath in SET is equal to applying the function (the equivalent of uaβ in Foundations.Univalence).
  2. 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...

@maxsnew maxsnew added the enhancement New feature or request label Mar 18, 2023
@maxsnew maxsnew self-assigned this Mar 25, 2023
stschaef referenced this issue in stschaef/cubical-categorical-logic May 18, 2023
Move code from experiment back into profunctor branch
stschaef added a commit that referenced this issue Jun 8, 2023
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maxsnew commented Jul 12, 2023

Completed in branch universal-element-prop. I'll merge in after the cubical changes PR is merged

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maxsnew commented Jul 24, 2023

Implemented in #40

@maxsnew maxsnew closed this as completed Jul 24, 2023
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