cat/BACKLOG.md

Backlog

Prove postulates in Cat.Wishlist:

• ntypeCommulative : n ≤ m → HasLevel ⟨ n ⟩₋₂ A → HasLevel ⟨ m ⟩₋₂ A

Prove univalence for the category of

• the opposite category
• sets This does not follow trivially from Cubical.Univalence.univalence because objects are not Set but hSet
• functors and natural transformations

Prove:

• isProp (Product ...)

• isProp (HasProducts ...)

• Functor ✓

• Applicative Functor ✗

  • Lax monoidal functor ✗
    • Monoidal functor ✗
  • Tensorial strength ✗

• Category ✓

  • Monoidal category ✗

• Monad

  • Monoidal monad ✓
  • Kleisli monad ✓
  • Kleisli ≃ Monoidal ✓
  • Problem 2.3 in [voe]
    • 1st contruction ~ monoidal ✓
    • 2nd contruction ~ klesli ✓
      • 1st ≃ 2nd ✗ I've managed to set this up so I should be able to reuse the proof that Kleisli ≃ Monoidal, but I don't know why it doesn't typecheck.