* **[HoTT] Defining n-sphere via HIT in cubical Agda**
**@ 2019-01-27 20:11 Yuhao Huang**
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From: Yuhao Huang @ 2019-01-27 20:11 UTC (permalink / raw)
To: Homotopy Type Theory
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I was experimenting cubical Agda with {-# OPTIONS --cubical --rewriting #-}
I tried to define S^2, S^3 separately and found HIT worked fine.
However when I was trying to write down Sphere n for general natural number
n it simply won't allow me to do so.
And the error message is "The target of a constructor must be the datatype
applied to its parameters".
So how do I define a dependent HIT?
PtType : Set₁
PtType = Σ Set (λ A → A)
Ω1 : PtType → PtType
Ω1 A = ((A .snd) ≡ (A .snd)) , refl
Ω : (n : ℕ) → PtType → PtType
Ω 0 A = A
Ω (suc n) A = Ω1 (Ω n A)
data S² : Set where
base : S²
surf : ((Ω 2 (S² , base)) .fst)
data S³ : Set where
base : S³
cell : ((Ω 3 (S³ , base)) .fst)
data Sphere : ℕ → Set where
base : (n : ℕ) → Sphere n
--cell : (n : ℕ) → ((Ω n ((Sphere n) , (base n))) .fst) -- Error!
222222222222
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