------------------------------------------------------------------------
-- H-levels
------------------------------------------------------------------------

{-# OPTIONS --without-K --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  : Level
    m n : 
    A B : Set a

------------------------------------------------------------------------
-- H-levels

-- H-levels ("homotopy levels").

H-level :   Set   Set 
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 _

-- An alternative definition.
--
-- 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′ :   Set   Set 
H-level′ zero    A = Contractible A
H-level′ (suc n) A = (x y : A)  H-level′ n (x  y)

-- Propositions are propositional types.

Proposition : ( : Level)  Set (lsuc )
Proposition _ =  Is-proposition

-- Types that are sets.

SET : ( : Level)  Set (lsuc )
SET _ =  Is-set

-- The underlying type.

Type : SET   Set 
Type 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  {x y} x≡y  trivial x y  x≡y)
              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  n  H-level n _) eq
  mono (≤-step′ m≤n eq) =
    subst  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  {x y} _  H-level (1 + n) (x  y)) h)
    }

-- If a propositional type is inhabited, then it is contractible.

propositional⇒inhabited⇒contractible :
  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′