Theory about equivalences

Definitions are in Core/Glue.agda but re-exported by this module

- isEquiv is a proposition ([isPropIsEquiv])
- Any isomorphism is an equivalence ([isoToEquiv])

There are more statements about equivalences in Equiv/Properties.agda:

- if f is an equivalence then (cong f) is an equivalence
- if f is an equivalence then precomposition with f is an equivalence
- if f is an equivalence then postcomposition with f is an equivalence

{-# OPTIONS --safe #-}
module Cubical.Foundations.Equiv where

open import Cubical.Foundations.Function
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.GroupoidLaws

open import Cubical.Foundations.Equiv.Base public
open import Cubical.Data.Sigma.Base

     ℓ' ℓ''  : Level
    A B C D : Type 

equivIsEquiv : (e : A  B)  isEquiv (equivFun e)
equivIsEquiv e = snd e

equivCtr : (e : A  B) (y : B)  fiber (equivFun e) y
equivCtr e y = e .snd .equiv-proof y .fst

equivCtrPath : (e : A  B) (y : B) 
  (v : fiber (equivFun e) y)  Path _ (equivCtr e y) v
equivCtrPath e y = e .snd .equiv-proof y .snd

-- Proof using isPropIsContr. This is slow and the direct proof below is better

isPropIsEquiv' : (f : A  B)  isProp (isEquiv f)
equiv-proof (isPropIsEquiv' f u0 u1 i) y =
  isPropIsContr (u0 .equiv-proof y) (u1 .equiv-proof y) i

-- Direct proof that computes quite ok (can be optimized further if
-- necessary, see:
-- https://github.com/mortberg/cubicaltt/blob/pi4s3_dimclosures/examples/brunerie2.ctt#L562

isPropIsEquiv : (f : A  B)  isProp (isEquiv f)
equiv-proof (isPropIsEquiv f p q i) y =
  let p2 = p .equiv-proof y .snd
      q2 = q .equiv-proof y .snd
  in p2 (q .equiv-proof y .fst) i
   , λ w j  hcomp  k  λ { (i = i0)  p2 w j
                            ; (i = i1)  q2 w (j  ~ k)
                            ; (j = i0)  p2 (q2 w (~ k)) i
                            ; (j = i1)  w })
                   (p2 w (i  j))

equivEq : {e f : A  B}  (h : e .fst  f .fst)  e  f
equivEq {e = e} {f = f} h = λ i  (h i) , isProp→PathP  i  isPropIsEquiv (h i)) (e .snd) (f .snd) i

module _ {f : A  B} (equivF : isEquiv f) where
  funIsEq : A  B
  funIsEq = f

  invIsEq : B  A
  invIsEq y = equivF .equiv-proof y .fst .fst

  secIsEq : section f invIsEq
  secIsEq y = equivF .equiv-proof y .fst .snd

  retIsEq : retract f invIsEq
  retIsEq a i = equivF .equiv-proof (f a) .snd (a , refl) i .fst

  commSqIsEq :  a  Square (secIsEq (f a)) refl (cong f (retIsEq a)) refl
  commSqIsEq a i = equivF .equiv-proof (f a) .snd (a , refl) i .snd

  commPathIsEq :  a  secIsEq (f a)  cong f (retIsEq a)
  commPathIsEq a i j =
       k  λ
        { (i = i0)  secIsEq (f a) j
        ; (i = i1)  f (retIsEq a (j  ~ k))
        ; (j = i0)  f (retIsEq a (i  ~ k))
        ; (j = i1)  f a
      (commSqIsEq a i j)

module _ (w : A  B) where
  invEq : B  A
  invEq = invIsEq (snd w)

  retEq : retract (w .fst) invEq
  retEq = retIsEq (snd w)

  secEq : section (w .fst) invEq
  secEq = secIsEq (snd w)

open Iso

equivToIso :  { ℓ'} {A : Type } {B : Type ℓ'}  A  B  Iso A B
fun (equivToIso e) = e .fst
inv (equivToIso e) = invEq e
rightInv (equivToIso e) = secEq e
leftInv (equivToIso e)  = retEq e

-- TODO: there should be a direct proof of this that doesn't use equivToIso
invEquiv : A  B  B  A
invEquiv e = isoToEquiv (invIso (equivToIso e))

invEquivIdEquiv : (A : Type )  invEquiv (idEquiv A)  idEquiv A
invEquivIdEquiv _ = equivEq refl

compEquiv : A  B  B  C  A  C
compEquiv f g .fst = g .fst  f .fst
compEquiv {A = A} {C = C} f g .snd .equiv-proof c = contr
  contractG = g .snd .equiv-proof c .snd

  secFiller : (a : A) (p : g .fst (f .fst a)  c)  _ {- square in A -}
  secFiller a p = compPath-filler (cong (invEq f  fst) (contractG (_ , p))) (retEq f a)

  contr : isContr (fiber (g .fst  f .fst) c)
  contr .fst .fst = invEq f (invEq g c)
  contr .fst .snd = cong (g .fst) (secEq f (invEq g c))  secEq g c
  contr .snd (a , p) i .fst = secFiller a p i1 i
  contr .snd (a , p) i .snd j =
       k  λ
        { (i = i1)  fSquare k
        ; (j = i0)  g .fst (f .fst (secFiller a p k i))
        ; (j = i1)  contractG (_  , p) i .snd k
      (g .fst (secEq f (contractG (_ , p) i .fst) j))
    fSquare : I  C
    fSquare k =
         l  λ
          { (j = i0)  g .fst (f .fst (retEq f a k))
          ; (j = i1)  p (k  l)
          ; (k = i0)  g .fst (secEq f (f .fst a) j)
          ; (k = i1)  p (j  l)
        (g .fst (f .snd .equiv-proof (f .fst a) .snd (a , refl) k .snd j))

compEquivIdEquiv : (e : A  B)  compEquiv (idEquiv A) e  e
compEquivIdEquiv e = equivEq refl

compEquivEquivId : (e : A  B)  compEquiv e (idEquiv B)  e
compEquivEquivId e = equivEq refl

invEquiv-is-rinv : (e : A  B)  compEquiv e (invEquiv e)  idEquiv A
invEquiv-is-rinv e = equivEq (funExt (retEq e))

invEquiv-is-linv : (e : A  B)  compEquiv (invEquiv e) e  idEquiv B
invEquiv-is-linv e = equivEq (funExt (secEq e))

compEquiv-assoc : (f : A  B) (g : B  C) (h : C  D)
                 compEquiv f (compEquiv g h)  compEquiv (compEquiv f g) h
compEquiv-assoc f g h = equivEq refl

LiftEquiv : A  Lift {i = } {j = ℓ'} A
LiftEquiv .fst a .lower = a
LiftEquiv .snd .equiv-proof = strictContrFibers lower

Lift≃Lift : (e : A  B)  Lift {j = ℓ'} A  Lift {j = ℓ''} B
Lift≃Lift e .fst a .lower = e .fst (a .lower)
Lift≃Lift e .snd .equiv-proof b .fst .fst .lower = invEq e (b .lower)
Lift≃Lift e .snd .equiv-proof b .fst .snd i .lower =
  e .snd .equiv-proof (b .lower) .fst .snd i
Lift≃Lift e .snd .equiv-proof b .snd (a , p) i .fst .lower =
  e .snd .equiv-proof (b .lower) .snd (a .lower , cong lower p) i .fst
Lift≃Lift e .snd .equiv-proof b .snd (a , p) i .snd j .lower =
  e .snd .equiv-proof (b .lower) .snd (a .lower , cong lower p) i .snd j

isContr→Equiv : isContr A  isContr B  A  B
isContr→Equiv Actr Bctr = isoToEquiv (isContr→Iso Actr Bctr)

propBiimpl→Equiv : (Aprop : isProp A) (Bprop : isProp B) (f : A  B) (g : B  A)  A  B
propBiimpl→Equiv Aprop Bprop f g = f , hf
  hf : isEquiv f
  hf .equiv-proof y .fst          = (g y , Bprop (f (g y)) y)
  hf .equiv-proof y .snd h i .fst = Aprop (g y) (h .fst) i
  hf .equiv-proof y .snd h i .snd = isProp→isSet' Bprop (Bprop (f (g y)) y) (h .snd)
                                                  (cong f (Aprop (g y) (h .fst))) refl i

isEquivPropBiimpl→Equiv : isProp A  isProp B
                         ((A  B) × (B  A))  (A  B)
isEquivPropBiimpl→Equiv {A = A} {B = B} Aprop Bprop = isoToEquiv isom where
  isom : Iso (Σ (A  B)  _  B  A)) (A  B)
  isom .fun (f , g) = propBiimpl→Equiv Aprop Bprop f g
  isom .inv e = equivFun e , invEq e
  isom .rightInv e = equivEq refl
  isom .leftInv _ = refl

equivΠCod :  {F : A  Type } {G : A  Type ℓ'}
         ((x : A)  F x  G x)  ((x : A)  F x)  ((x : A)  G x)
equivΠCod k .fst f x = k x .fst (f x)
equivΠCod k .snd .equiv-proof f .fst .fst x   = equivCtr (k x) (f x) .fst
equivΠCod k .snd .equiv-proof f .fst .snd i x = equivCtr (k x) (f x) .snd i
equivΠCod k .snd .equiv-proof f .snd (g , p) i .fst x =
  equivCtrPath (k x) (f x) (g x , λ j  p j x) i .fst
equivΠCod k .snd .equiv-proof f .snd (g , p) i .snd j x =
  equivCtrPath (k x) (f x) (g x , λ k  p k x) i .snd j

equivImplicitΠCod :  {F : A  Type } {G : A  Type ℓ'}
         ({x : A}  F x  G x)  ({x : A}  F x)  ({x : A}  G x)
equivImplicitΠCod k .fst f {x} = k {x} .fst (f {x})
equivImplicitΠCod k .snd .equiv-proof f .fst .fst {x}   = equivCtr (k {x}) (f {x}) .fst
equivImplicitΠCod k .snd .equiv-proof f .fst .snd i {x} = equivCtr (k {x}) (f {x}) .snd i
equivImplicitΠCod k .snd .equiv-proof f .snd (g , p) i .fst {x} =
  equivCtrPath (k {x}) (f {x}) (g {x} , λ j  p j {x}) i .fst
equivImplicitΠCod k .snd .equiv-proof f .snd (g , p) i .snd j {x} =
  equivCtrPath (k {x}) (f {x}) (g {x} , λ k  p k {x}) i .snd j

equiv→Iso : (A  B)  (C  D)  Iso (A  C) (B  D)
equiv→Iso h k .Iso.fun f b = equivFun k (f (invEq h b))
equiv→Iso h k .Iso.inv g a = invEq k (g (equivFun h a))
equiv→Iso h k .Iso.rightInv g = funExt λ b  secEq k _  cong g (secEq h b)
equiv→Iso h k .Iso.leftInv f = funExt λ a  retEq k _  cong f (retEq h a)

equiv→ : (A  B)  (C  D)  (A  C)  (B  D)
equiv→ h k = isoToEquiv (equiv→Iso h k)

equivΠ :  {ℓA ℓA' ℓB ℓB'} {A : Type ℓA} {A' : Type ℓA'}
  {B : A  Type ℓB} {B' : A'  Type ℓB'}
  (eA : A  A')
  (eB : (a : A)  B a  B' (eA .fst a))
   ((a : A)  B a)  ((a' : A')  B' a')
equivΠ {B' = B'} eA eB = isoToEquiv isom
  open Iso

  isom : Iso _ _
  isom .fun f a' =
    subst B' (secEq eA a') (eB _ .fst (f (invEq eA a')))
  isom .inv f' a =
    invEq (eB _) (f' (eA .fst a))
  isom .rightInv f' =
    funExt λ a' 
    cong (subst B' (secEq eA a')) (secEq (eB _) _)
     fromPathP (cong f' (secEq eA a'))
  isom .leftInv f =
    funExt λ a 
    invEq (eB a) (subst B' (secEq eA _) (eB _ .fst (f (invEq eA (eA .fst a)))))
      ≡⟨ cong  t  invEq (eB a) (subst B' t (eB _ .fst (f (invEq eA (eA .fst a))))))
           (commPathIsEq (snd eA) a) 
    invEq (eB a) (subst B' (cong (eA .fst) (retEq eA a)) (eB _ .fst (f (invEq eA (eA .fst a)))))
      ≡⟨ cong (invEq (eB a)) (fromPathP  i  eB _ .fst (f (retEq eA a i)))) 
    invEq (eB a) (eB a .fst (f a))
      ≡⟨ retEq (eB _) (f a) 
    f a

equivCompIso : (A  B)  (C  D)  Iso (A  C) (B  D)
equivCompIso h k .Iso.fun f = compEquiv (compEquiv (invEquiv h) f) k
equivCompIso h k .Iso.inv g = compEquiv (compEquiv h g) (invEquiv k)
equivCompIso h k .Iso.rightInv g = equivEq (equiv→Iso h k .Iso.rightInv (equivFun g))
equivCompIso h k .Iso.leftInv f = equivEq (equiv→Iso h k .Iso.leftInv (equivFun f))

equivComp : (A  B)  (C  D)  (A  C)  (B  D)
equivComp h k = isoToEquiv (equivCompIso h k)

-- Some helpful notation:
_≃⟨_⟩_ : (X : Type )  (X  B)  (B  C)  (X  C)
_ ≃⟨ f  g = compEquiv f g

_■ : (X : Type )  (X  X)
_■ = idEquiv

infixr  0 _≃⟨_⟩_
infix   1 _■

composesToId→Equiv : (f : A  B) (g : B  A)  f  g  idfun B  isEquiv f  isEquiv g
composesToId→Equiv f g id iseqf =
    (iso g f
          b   i  equiv-proof iseqf (f b) .snd (g (f b) , cong  h  h (f b)) id) (~ i) .fst)
                ∙∙ cong  x  equiv-proof iseqf (f b) .fst .fst) id
                ∙∙ λ i  equiv-proof iseqf (f b) .snd (b , refl) i .fst)
         λ a i  id i a)