42 lines
1.4 KiB
Agda
42 lines
1.4 KiB
Agda
{-# OPTIONS --allow-unsolved-metas #-}
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module Cat.Wishlist where
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open import Level hiding (suc)
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open import Cubical
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open import Cubical.NType
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open import Data.Nat using (_≤_ ; z≤n ; s≤s ; zero ; suc)
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open import Agda.Builtin.Sigma
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open import Cubical.NType.Properties
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step : ∀ {ℓ} {A : Set ℓ} → isContr A → (x y : A) → isContr (x ≡ y)
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step (a , contr) x y = {!p , c!}
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-- where
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-- p : x ≡ y
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-- p = begin
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-- x ≡⟨ sym (contr x) ⟩
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-- a ≡⟨ contr y ⟩
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-- y ∎
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-- c : (q : x ≡ y) → p ≡ q
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-- c q i j = contr (p {!!}) {!!}
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-- Contractible types have any given homotopy level.
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contrInitial : {ℓ : Level} {A : Set ℓ} → ∀ n → isContr A → HasLevel n A
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contrInitial ⟨-2⟩ contr = contr
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-- lem' (S ⟨-2⟩) (a , contr) = {!step!}
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contrInitial (S ⟨-2⟩) (a , contr) x y = begin
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x ≡⟨ sym (contr x) ⟩
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a ≡⟨ contr y ⟩
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y ∎
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contrInitial (S (S n)) contr x y = {!lvl!} -- Why is this not well-founded?
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where
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c : isContr (x ≡ y)
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c = step contr x y
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lvl : HasLevel (S n) (x ≡ y)
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lvl = contrInitial {A = x ≡ y} (S n) c
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module _ {ℓ : Level} {A : Set ℓ} where
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ntypeCommulative : ∀ {n m} → n ≤ m → HasLevel ⟨ n ⟩₋₂ A → HasLevel ⟨ m ⟩₋₂ A
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ntypeCommulative {n = zero} {m} z≤n lvl = {!contrInitial ⟨ m ⟩₋₂ lvl!}
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ntypeCommulative {n = .(suc _)} {.(suc _)} (s≤s x) lvl = {!!}
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