cat/src/Cat/Wishlist.agda

{-# OPTIONS --allow-unsolved-metas #-}
module Cat.Wishlist where

open import Level hiding (suc)
open import Cubical
open import Cubical.NType
open import Data.Nat using (_≤_ ; z≤n ; s≤s ; zero ; suc)
open import Agda.Builtin.Sigma

open import Cubical.NType.Properties

step : ∀ {ℓ} {A : Set ℓ} → isContr A → (x y : A) → isContr (x ≡ y)
step (a , contr) x y = {!p , c!}
  -- where
  -- p : x ≡ y
  -- p = begin
  --   x ≡⟨ sym (contr x) ⟩
  --   a ≡⟨ contr y ⟩
  --   y ∎
  -- c : (q : x ≡ y) → p ≡ q
  -- c q i j = contr (p {!!}) {!!}

-- Contractible types have any given homotopy level.
contrInitial : {ℓ : Level} {A : Set ℓ} → ∀ n → isContr A → HasLevel n A
contrInitial ⟨-2⟩ contr = contr
-- lem' (S ⟨-2⟩) (a , contr) = {!step!}
contrInitial (S ⟨-2⟩) (a , contr) x y = begin
  x ≡⟨ sym (contr x) ⟩
  a ≡⟨ contr y ⟩
  y ∎
contrInitial (S (S n)) contr x y = {!lvl!} -- Why is this not well-founded?
  where
  c : isContr (x ≡ y)
  c = step contr x y
  lvl : HasLevel (S n) (x ≡ y)
  lvl = contrInitial {A = x ≡ y} (S n) c

module _ {ℓ : Level} {A : Set ℓ} where
  ntypeCommulative : ∀ {n m} → n ≤ m → HasLevel ⟨ n ⟩₋₂ A → HasLevel ⟨ m ⟩₋₂ A
  ntypeCommulative {n = zero} {m} z≤n lvl = {!contrInitial ⟨ m ⟩₋₂ lvl!}
  ntypeCommulative {n = .(suc _)} {.(suc _)} (s≤s x) lvl = {!!}