cat/BACKLOG.md

Backlog

Prove postulates in Cat.Wishlist:

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

Prove that these two formulations of univalence are equivalent:

∀ A B → isEquiv (A ≡ B) (A ≅ B) (idToIso A B)
∀ A   → isContr (Σ[ X ∈ Object ] A ≅ X)

Prove univalence for the category of

• the opposite category
• functors and natural transformations

Prove:

• isProp (Product ...)
• isProp (HasProducts ...)

Ideas for future work

It would be nice if my formulation of monads is not so "stand-alone" as it is at the moment.

We can built up the notion of monads and related concept in multiple ways as demonstrated in the two equivalent formulations of monads (kleisli/monoidal): There seems to be a category-theoretic approach and an approach more in the style of functional programming as e.g. the related typeclasses in the standard library of Haskell.

It would be nice to build up this hierarchy in two ways: The "category-theoretic" way and the "functional programming" way.

Here is an overview of some of the concepts that need to be developed to acheive this:

• 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 ✓