Library HoTT.Equiv.BiInv

(* -*- mode: coq; mode: visual-line -*- *)

Require Import HoTT.Basics HoTT.Types.

Local Open Scope path_scope.
Generalizable Variables A B f.

Bi-invertible maps

A map is "bi-invertible" if it has both a section and a retraction, not necessarily the same. This definition of equivalence was proposed by Andre Joyal.

Definition BiInv `(f : A → B) : Type
:= {g : B → A & g o f == idmap } × {h : B → A & f o h == idmap }.

It seems that the easiest way to show that bi-invertibility is equivalent to being an equivalence is also to show that both are h-props and that they are logically equivalent.

Definition isequiv_biinv `(f : A → B)
  : BiInv f → IsEquiv f.
Proof.
  intros [[g s] [h r]].
  exact (isequiv_adjointify f g
    (fun x ⇒ ap f (ap g (r x )^ @ s (h x)) @ r x)
    s).
Defined.

Definition biinv_isequiv `(f : A → B)
  : IsEquiv f → BiInv f.
Proof.
  intros [g s r adj].
  exact ((g; r), (g; s)).
Defined.

Definition iff_biinv_isequiv `(f : A → B)
  : BiInv f ↔ IsEquiv f.
Proof.
  split.
  - apply isequiv_biinv.
  - apply biinv_isequiv.
Defined.

Global Instance ishprop_biinv `{Funext} `(f : A → B) : IsHProp (BiInv f) | 0.
Proof.
  apply hprop_inhabited_contr.
  intros bif; pose (fe := isequiv_biinv f bif).
  apply @contr_prod.
  (* For this, we've done all the work already. *)
  - by apply contr_retr_equiv.
  - by apply contr_sect_equiv.
Defined.

Definition equiv_biinv_isequiv `{Funext} `(f : A → B)
  : BiInv f <~> IsEquiv f.
Proof.
  apply equiv_iff_hprop_uncurried, iff_biinv_isequiv.
Defined.