Some theory about Bi-Invertible Equivalences

- BiInvEquiv to Iso
- BiInvEquiv to Equiv
- BiInvEquiv to HAEquiv
- Iso to BiInvEquiv

{-# OPTIONS --cubical --safe #-}
module Cubical.Foundations.BiInvEquiv where

open import Cubical.Foundations.Prelude
open import Cubical.Foundations.Function
open import Cubical.Foundations.Isomorphism
open import Cubical.Foundations.Equiv
open import Cubical.Foundations.HAEquiv

record BiInvEquiv { ℓ'} (A : Type ) (B : Type ℓ') : Type (ℓ-max  ℓ') where
  constructor biInvEquiv
    fun : A  B
    invr : B  A
    invr-rightInv : section fun invr
    invl : B  A
    invl-leftInv : retract fun invl

  invr≡invl :  b  invr b  invl b
  invr≡invl b =            invr b   ≡⟨ sym (invl-leftInv (invr b)) 
                invl (fun (invr b)) ≡⟨ cong invl (invr-rightInv b) 
                invl b              

  invr-leftInv : retract fun invr
  invr-leftInv a = invr≡invl (fun a)  (invl-leftInv a)

  invr≡invl-leftInv :  a  PathP  j  invr≡invl (fun a) j  a) (invr-leftInv a) (invl-leftInv a)
  invr≡invl-leftInv a j i = compPath'-filler (invr≡invl (fun a)) (invl-leftInv a) (~ j) i

  invl-rightInv : section fun invl
  invl-rightInv a = sym (cong fun (invr≡invl a))  (invr-rightInv a)

  invr≡invl-rightInv :  a  PathP  j  fun (invr≡invl a j)  a) (invr-rightInv a) (invl-rightInv a)
  invr≡invl-rightInv a j i = compPath'-filler (sym (cong fun (invr≡invl a))) (invr-rightInv a) j i

module _ {} {A B : Type } (e : BiInvEquiv A B) where
  open BiInvEquiv e

  biInvEquiv→Iso-right : Iso A B
  Iso.fun biInvEquiv→Iso-right      = fun
  Iso.inv biInvEquiv→Iso-right      = invr
  Iso.rightInv biInvEquiv→Iso-right = invr-rightInv
  Iso.leftInv biInvEquiv→Iso-right  = invr-leftInv

  biInvEquiv→Iso-left : Iso A B
  Iso.fun biInvEquiv→Iso-left      = fun
  Iso.inv biInvEquiv→Iso-left      = invl
  Iso.rightInv biInvEquiv→Iso-left = invl-rightInv
  Iso.leftInv biInvEquiv→Iso-left  = invl-leftInv

  biInvEquiv→Equiv-right biInvEquiv→Equiv-left : A  B
  biInvEquiv→Equiv-right = fun , isoToIsEquiv biInvEquiv→Iso-right
  biInvEquiv→Equiv-left  = fun , isoToIsEquiv biInvEquiv→Iso-left

  -- since Iso.rightInv ends up getting modified during iso→HAEquiv, in some sense biInvEquiv→Iso-left
  --  is the most natural choice for forming a HAEquiv from a BiInvEquiv
  biInvEquiv→HAEquiv : HAEquiv A B
  biInvEquiv→HAEquiv = iso→HAEquiv biInvEquiv→Iso-left

module _ {} {A B : Type } (i : Iso A B) where
  open Iso i

  iso→BiInvEquiv : BiInvEquiv A B
  BiInvEquiv.fun iso→BiInvEquiv           = fun
  BiInvEquiv.invr iso→BiInvEquiv          = inv
  BiInvEquiv.invr-rightInv iso→BiInvEquiv = rightInv
  BiInvEquiv.invl iso→BiInvEquiv          = inv
  BiInvEquiv.invl-leftInv iso→BiInvEquiv  = leftInv