{-
- Definitions of [section] and [retract]
- Definition of isomorphisms ([Iso])
- Any isomorphism is an equivalence ([isoToEquiv])
-}
{-# OPTIONS --safe #-}
module Cubical.Foundations.Isomorphism where
open import Cubical.Core.Everything
open import Cubical.Foundations.Prelude
open import Cubical.Foundations.GroupoidLaws
open import Cubical.Foundations.Function
open import Cubical.Foundations.Equiv.Base
private
variable
ℓ ℓ' : Level
A B C : Type ℓ
-- Section and retract
module _ {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} where
section : (f : A → B) → (g : B → A) → Type ℓ'
section f g = ∀ b → f (g b) ≡ b
-- NB: g is the retraction!
retract : (f : A → B) → (g : B → A) → Type ℓ
retract f g = ∀ a → g (f a) ≡ a
record Iso {ℓ ℓ'} (A : Type ℓ) (B : Type ℓ') : Type (ℓ-max ℓ ℓ') where
no-eta-equality
constructor iso
field
fun : A → B
inv : B → A
rightInv : section fun inv
leftInv  : retract fun inv
isIso : (A → B) → Type _
isIso {A = A} {B = B} f = Σ[ g ∈ (B → A) ] Σ[ _ ∈ section f g ] retract f g
isoFunInjective : (f : Iso A B) → (x y : A) → Iso.fun f x ≡ Iso.fun f y → x ≡ y
isoFunInjective f x y h = sym (Iso.leftInv f x) ∙∙ cong (Iso.inv f) h ∙∙ Iso.leftInv f y
isoInvInjective : (f : Iso A B) → (x y : B) → Iso.inv f x ≡ Iso.inv f y → x ≡ y
isoInvInjective f x y h = sym (Iso.rightInv f x) ∙∙ cong (Iso.fun f) h ∙∙ Iso.rightInv f y
-- Any iso is an equivalence
module _ (i : Iso A B) where
open Iso i renaming ( fun to f
; inv to g
; rightInv to s
; leftInv to t)
private
module _ (y : B) (x0 x1 : A) (p0 : f x0 ≡ y) (p1 : f x1 ≡ y) where
fill0 : I → I → A
fill0 i = hfill (λ k → λ { (i = i1) → t x0 k
; (i = i0) → g y })
(inS (g (p0 (~ i))))
fill1 : I → I → A
fill1 i = hfill (λ k → λ { (i = i1) → t x1 k
; (i = i0) → g y })
(inS (g (p1 (~ i))))
fill2 : I → I → A
fill2 i = hfill (λ k → λ { (i = i1) → fill1 k i1
; (i = i0) → fill0 k i1 })
(inS (g y))
p : x0 ≡ x1
p i = fill2 i i1
sq : I → I → A
sq i j = hcomp (λ k → λ { (i = i1) → fill1 j (~ k)
; (i = i0) → fill0 j (~ k)
; (j = i1) → t (fill2 i i1) (~ k)
; (j = i0) → g y })
(fill2 i j)
sq1 : I → I → B
sq1 i j = hcomp (λ k → λ { (i = i1) → s (p1 (~ j)) k
; (i = i0) → s (p0 (~ j)) k
; (j = i1) → s (f (p i)) k
; (j = i0) → s y k })
(f (sq i j))
lemIso : (x0 , p0) ≡ (x1 , p1)
lemIso i .fst = p i
lemIso i .snd = λ j → sq1 i (~ j)
isoToIsEquiv : isEquiv f
isoToIsEquiv .equiv-proof y .fst .fst = g y
isoToIsEquiv .equiv-proof y .fst .snd = s y
isoToIsEquiv .equiv-proof y .snd z = lemIso y (g y) (fst z) (s y) (snd z)

isoToEquiv : Iso A B → A ≃ B
isoToEquiv i .fst = i .Iso.fun
isoToEquiv i .snd = isoToIsEquiv i
isoToPath : Iso A B → A ≡ B
isoToPath {A = A} {B = B} f i =
Glue B (λ { (i = i0) → (A , isoToEquiv f)
; (i = i1) → (B , idEquiv B) })
open Iso
invIso : Iso A B → Iso B A
fun (invIso f) = inv f
inv (invIso f) = fun f
rightInv (invIso f) = leftInv f
leftInv (invIso f)  = rightInv f
compIso : Iso A B → Iso B C → Iso A C
fun (compIso i j)       = fun j ∘ fun i
inv (compIso i j) = inv i ∘ inv j
rightInv (compIso i j) b = cong (fun j) (rightInv i (inv j b)) ∙ rightInv j b
leftInv (compIso i j) a = cong (inv i) (leftInv j (fun i a)) ∙ leftInv i a
composesToId→Iso : (G : Iso A B) (g : B → A) → G .fun ∘ g ≡ idfun B → Iso B A
fun (composesToId→Iso _ g _)             = g
inv (composesToId→Iso j _ _) = fun j
rightInv (composesToId→Iso i g path) b =
sym (leftInv i (g (fun i b))) ∙∙ cong (λ g → inv i (g (fun i b))) path ∙∙ leftInv i b
leftInv (composesToId→Iso _ _ path) b i = path i b
idIso : Iso A A
fun idIso = idfun _
inv idIso = idfun _
rightInv idIso _ = refl
leftInv idIso _  = refl
compIsoIdL : (isom : Iso A B) → compIso idIso isom ≡ isom
fun (compIsoIdL isom i) = fun isom
inv (compIsoIdL isom i) = inv isom
rightInv (compIsoIdL isom i) b = lUnit (isom .rightInv b) (~ i)
leftInv (compIsoIdL isom i) a = rUnit (isom .leftInv a) (~ i)
compIsoIdR : (isom : Iso A B) → compIso isom idIso ≡ isom
fun (compIsoIdR isom i) = fun isom
inv (compIsoIdR isom i) = inv isom
rightInv (compIsoIdR isom i) b = rUnit (isom .rightInv b) (~ i)
leftInv (compIsoIdR isom i) a = lUnit (isom .leftInv a) (~ i)
LiftIso : Iso A (Lift {i = ℓ} {j = ℓ'} A)
fun LiftIso = lift
inv LiftIso = lower
rightInv LiftIso _ = refl
leftInv LiftIso _  = refl
isContr→Iso : isContr A → isContr B → Iso A B
fun (isContr→Iso _ Bctr) _ = Bctr .fst
inv (isContr→Iso Actr _) _ = Actr .fst
rightInv (isContr→Iso _ Bctr) = Bctr .snd
leftInv (isContr→Iso Actr _)  = Actr .snd
isContr→Iso' : isContr A → isContr B → (A → B) → Iso A B
fun (isContr→Iso' _ Bctr f) = f
inv (isContr→Iso' Actr _ _) _ = Actr .fst
rightInv (isContr→Iso' _ Bctr f) = isContr→isProp Bctr _
leftInv (isContr→Iso' Actr _ _)  = Actr .snd
isProp→Iso :  (Aprop : isProp A) (Bprop : isProp B) (f : A → B) (g : B → A) → Iso A B
fun (isProp→Iso _ _ f _) = f
inv (isProp→Iso _ _ _ g) = g
rightInv (isProp→Iso _ Bprop f g) b = Bprop (f (g b)) b
leftInv (isProp→Iso Aprop _ f g) a  = Aprop (g (f a)) a
domIso : ∀ {ℓ} {C : Type ℓ} → Iso A B → Iso (A → C) (B → C)
fun (domIso e) f b = f (inv e b)
inv (domIso e) f a = f (fun e a)
rightInv (domIso e) f i x = f (rightInv e x i)
leftInv (domIso e) f i x = f (leftInv e x i)
_Iso⟨_⟩_ : ∀ {ℓ ℓ' ℓ''} {B : Type ℓ'} {C : Type ℓ''} (X : Type ℓ) → Iso X B → Iso B C → Iso X C
_ Iso⟨ f ⟩ g = compIso f g
_∎Iso : ∀ {ℓ} (A : Type ℓ) → Iso A A
A ∎Iso = idIso {A = A}
infixr  0 _Iso⟨_⟩_
infix   1 _∎Iso
codomainIsoDep : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : A → Type ℓ'} {C : A → Type ℓ''}
→ ((a : A) → Iso (B a) (C a))
→ Iso ((a : A) → B a) ((a : A) → C a)
fun (codomainIsoDep is) f a = fun (is a) (f a)
inv (codomainIsoDep is) f a = inv (is a) (f a)
rightInv (codomainIsoDep is) f = funExt λ a → rightInv (is a) (f a)
leftInv (codomainIsoDep is) f = funExt λ a → leftInv (is a) (f a)
codomainIso : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''}
→ Iso B C
→ Iso (A → B) (A → C)
codomainIso z = codomainIsoDep λ _ → z

Iso≡Set : isSet A → isSet B → (f g : Iso A B)
→ ((x : A) → f .fun x ≡ g .fun x)
→ ((x : B) → f .inv x ≡ g .inv x)
→ f ≡ g
fun (Iso≡Set hA hB f g hfun hinv i) x = hfun x i
inv (Iso≡Set hA hB f g hfun hinv i) x = hinv x i
rightInv (Iso≡Set hA hB f g hfun hinv i) x j =
isSet→isSet' hB (rightInv f x) (rightInv g x) (λ i → hfun (hinv x i) i) refl i j
leftInv (Iso≡Set hA hB f g hfun hinv i) x j =
isSet→isSet' hA (leftInv f x) (leftInv g x) (λ i → hinv (hfun x i) i) refl i j