------------------------------------------------------------------------ -- H-levels ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} -- Partly based on Voevodsky's work on so-called univalent -- foundations. open import Equality module H-level {reflexive} (eq : ∀ {a p} → Equality-with-J a p reflexive) where open Derived-definitions-and-properties eq open import Logical-equivalence hiding (id; _∘_) open import Nat eq open import Prelude open import Surjection eq hiding (id; _∘_) private variable a ℓ : Level m n : ℕ A B : Type a ------------------------------------------------------------------------ -- H-levels -- H-levels ("homotopy levels"). H-level : ℕ → Type ℓ → Type ℓ H-level zero A = Contractible A H-level (suc zero) A = Is-proposition A H-level (suc (suc n)) A = {x y : A} → H-level (suc n) (x ≡ y) private -- Note that H-level 2 is a synonym for Is-set. H-level-2≡Is-set : H-level 2 A ≡ Is-set A H-level-2≡Is-set = refl _ -- For-iterated-equality n P A means that P holds for (equalities -- over)^n A. For-iterated-equality : ℕ → (Type ℓ → Type ℓ) → (Type ℓ → Type ℓ) For-iterated-equality zero P A = P A For-iterated-equality (suc n) P A = (x y : A) → For-iterated-equality n P (x ≡ y) -- An alternative definition of h-levels. -- -- In some cases this definition, with only two cases, is easier to -- use. In other cases the definition above, which is less complicated -- for positive h-levels, is easier to use. H-level′ : ℕ → Type ℓ → Type ℓ H-level′ = flip For-iterated-equality Contractible -- Propositions are propositional types. Proposition : (ℓ : Level) → Type (lsuc ℓ) Proposition _ = ∃ Is-proposition -- Types that are sets. Set : (ℓ : Level) → Type (lsuc ℓ) Set _ = ∃ Is-set -- The underlying type. ⌞_⌟ : Set ℓ → Type ℓ ⌞ A ⌟ = proj₁ A ------------------------------------------------------------------------ -- General properties -- H-level′ is upwards closed in its first argument. mono₁′ : ∀ n → H-level′ n A → H-level′ (1 + n) A mono₁′ (suc n) h x y = mono₁′ n (h x y) mono₁′ {A = A} zero h x y = trivial x y , irr where trivial : (x y : A) → x ≡ y trivial x y = x ≡⟨ sym $ proj₂ h x ⟩ proj₁ h ≡⟨ proj₂ h y ⟩∎ y ∎ irr : (x≡y : x ≡ y) → trivial x y ≡ x≡y irr = elim (λ {x y} x≡y → trivial x y ≡ x≡y) (λ x → trans-symˡ (proj₂ h x)) -- H-level and H-level′ are pointwise logically equivalent. H-level⇔H-level′ : H-level n A ⇔ H-level′ n A H-level⇔H-level′ = record { to = to _; from = from _ } where to : ∀ n → H-level n A → H-level′ n A to zero h = h to (suc zero) h = λ x → mono₁′ 0 (x , h x) x to (suc (suc n)) h = λ x y → to (suc n) h from : ∀ n → H-level′ n A → H-level n A from zero h = h from (suc zero) h x y = proj₁ (h x y) from (suc (suc n)) h {x = x} {y = y} = from (suc n) (h x y) -- If A has h-level 1 + n, then the types of equality proofs between -- elements of type A have h-level n. +⇒≡ : {x y : A} → H-level (suc n) A → H-level n (x ≡ y) +⇒≡ h = _⇔_.from H-level⇔H-level′ $ _⇔_.to H-level⇔H-level′ h _ _ -- H-level is upwards closed in its first argument. mono₁ : ∀ n → H-level n A → H-level (1 + n) A mono₁ n = _⇔_.from H-level⇔H-level′ ∘ mono₁′ n ∘ _⇔_.to H-level⇔H-level′ abstract mono : m ≤ n → H-level m A → H-level n A mono (≤-refl′ eq) = subst (λ n → H-level n _) eq mono (≤-step′ m≤n eq) = subst (λ n → H-level n _) eq ∘ mono₁ _ ∘ mono m≤n -- If A has h-level n, then the types of equality proofs between -- elements of type A also have h-level n. ⇒≡ : {x y : A} → ∀ n → H-level n A → H-level n (x ≡ y) ⇒≡ _ = +⇒≡ ∘ mono₁ _ -- If something is contractible given the assumption that it is -- inhabited, then it is propositional. [inhabited⇒contractible]⇒propositional : (A → Contractible A) → Is-proposition A [inhabited⇒contractible]⇒propositional h x = mono₁ 0 (h x) x -- If something has h-level (1 + n) given the assumption that it is -- inhabited, then it has h-level (1 + n). [inhabited⇒+]⇒+ : ∀ n → (A → H-level (1 + n) A) → H-level (1 + n) A [inhabited⇒+]⇒+ n h = _⇔_.from H-level⇔H-level′ λ x → _⇔_.to H-level⇔H-level′ (h x) x -- An alternative characterisation of sets and higher h-levels. -- -- This is Theorem 7.2.7 from the HoTT book. 2+⇔∀1+≡ : ∀ n → H-level (2 + n) A ⇔ ((x : A) → H-level (1 + n) (x ≡ x)) 2+⇔∀1+≡ n = record { to = λ h _ → h ; from = λ h → [inhabited⇒+]⇒+ _ (elim (λ {x y} _ → H-level (1 + n) (x ≡ y)) h) } -- If a propositional type is inhabited, then it is contractible. propositional⇒inhabited⇒contractible : {@0 A : Type a} → Is-proposition A → A → Contractible A propositional⇒inhabited⇒contractible p x = (x , p x) -- H-level′ n respects (split) surjections. respects-surjection′ : A ↠ B → ∀ n → H-level′ n A → H-level′ n B respects-surjection′ A↠B zero (x , irr) = (to x , irr′) where open _↠_ A↠B irr′ : ∀ y → to x ≡ y irr′ = λ y → to x ≡⟨ cong to (irr (from y)) ⟩ to (from y) ≡⟨ right-inverse-of y ⟩∎ y ∎ respects-surjection′ A↠B (suc n) h = λ x y → respects-surjection′ (↠-≡ A↠B) n (h (from x) (from y)) where open _↠_ A↠B -- H-level n respects (split) surjections. respects-surjection : A ↠ B → ∀ n → H-level n A → H-level n B respects-surjection A↠B n = _⇔_.from H-level⇔H-level′ ∘ respects-surjection′ A↠B n ∘ _⇔_.to H-level⇔H-level′