Library UniMath.MoreFoundations.NullHomotopies

Null homotopies, an aid for proving things about propositional truncation

Require Export UniMath.Foundations.PartD.
Require Import UniMath.Foundations.UnivalenceAxiom.
Require Import UniMath.MoreFoundations.Tactics.
Require Import UniMath.MoreFoundations.Notations.

Local Open Scope transport.

Definition nullHomotopyTo {X Y} (f:X ->Y) (y:Y) := ∏ x:X , f x = y.
Definition NullHomotopyTo {X Y} (f:X ->Y) := total2 (nullHomotopyTo f).
Definition NullHomotopyTo_center {X Y} (f:X ->Y) : NullHomotopyTo f -> Y := pr1.
Definition NullHomotopyTo_path {X Y} {f:X ->Y} (r:NullHomotopyTo f) := pr2 r.

Definition nullHomotopyFrom {X Y} (f:X ->Y) (y:Y) := ∏ x:X , y = f x.
Definition NullHomotopyFrom {X Y} (f:X ->Y) := total2 (nullHomotopyFrom f).
Definition NullHomotopyFrom_center {X Y} (f:X ->Y) : NullHomotopyFrom f -> Y := pr1.
Definition NullHomotopyFrom_path {X Y} {f:X ->Y} (r:NullHomotopyFrom f) := pr2 r.

Definition nullHomotopyTo_transport {X Y} {f:X ->Y} {y:Y} (h : nullHomotopyTo f y)
{y':Y} (p:y = y') (x:X) : (p # h ) x = h x @ p.
Show proof.

intros. induction p. apply pathsinv0. apply pathscomp0rid.

Lemma isaset_NullHomotopyTo {X Y} (i:isaset Y) (f:X ->Y) : isaset (NullHomotopyTo f).
Show proof.

  intros. apply (isofhleveltotal2 2).
  { apply i. }
  intros y. apply impred; intros x. apply isasetaprop. apply i.

Lemma isaprop_nullHomotopyTo {X Y} (is:isaset Y) (f:X ->Y) (y:Y) :
isaprop (nullHomotopyTo f y).
Show proof.

apply impred; intros x. apply is.

Lemma isaprop_NullHomotopyTo {X} {Y} (is:isaset Y) (f:X ->Y) :
∥ X ∥ -> isaprop (NullHomotopyTo f).
Show proof.

  apply factor_through_squash.
  apply isapropisaprop.
  intros x. apply invproofirrelevance. intros [r i] [s j].
  apply subtypePairEquality.
  - intros n. apply (isaprop_nullHomotopyTo is).
  - exact (!i x @ j x).

Definition cone_squash_map {X Y} (f:X ->Y) (y:Y) :
nullHomotopyTo f y -> ∥ X ∥ -> Y.
Show proof.

intros e h. exact (point_from (h (paths_to_prop y) (λ x , f x ,,e x))).

Goal ∏ X Y (y :Y ) (f :X ->Y ) (e :∏ m :X , f m = y ),
f = cone_squash_map f y e ∘ squash_element.
Show proof.

reflexivity.