lean2/hott/homotopy/connectedness.hlean

/-
Copyright (c) 2015 Ulrik Buchholtz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ulrik Buchholtz
-/
import types.trunc types.arrow_2 types.fiber

open eq is_trunc is_equiv nat equiv trunc function

namespace homotopy

  definition is_conn [reducible] (n : trunc_index) (A : Type) : Type :=
  is_contr (trunc n A)

  definition is_conn_map (n : trunc_index) {A B : Type} (f : A → B) : Type :=
  Πb : B, is_conn n (fiber f b)

  definition is_conn_of_map_to_unit (n : trunc_index) (A : Type)
    : is_conn_map n (λx : A, unit.star) → is_conn n A :=
  begin
    intro H, unfold is_conn_map at H,
    rewrite [-(ua (fiber.fiber_star_equiv A))],
    exact (H unit.star)
  end

  -- Lemma 7.5.2
  definition minus_one_conn_of_surjective {A B : Type} (f : A → B)
    : is_surjective f → is_conn_map -1 f :=
  begin
    intro H, intro b,
    exact @is_contr_of_inhabited_hprop (∥fiber f b∥) (is_trunc_trunc -1 (fiber f b)) (H b),
  end

  definition is_surjection_of_minus_one_conn {A B : Type} (f : A → B)
    : is_conn_map -1 f → is_surjective f :=
  begin
    intro H, intro b,
    exact @center (∥fiber f b∥) (H b),
  end

  definition merely_of_minus_one_conn {A : Type} : is_conn -1 A → ∥A∥ :=
  λH, @center (∥A∥) H

  definition minus_one_conn_of_merely {A : Type} : ∥A∥ → is_conn -1 A :=
  @is_contr_of_inhabited_hprop (∥A∥) (is_trunc_trunc -1 A)

  section
    open arrow

    variables {f g : arrow}

    -- Lemma 7.5.4
    definition retract_of_conn_is_conn [instance] (r : arrow_hom f g) [H : is_retraction r]
      (n : trunc_index) [K : is_conn_map n f] : is_conn_map n g :=
    begin
      intro b, unfold is_conn,
      apply is_contr_retract (trunc_functor n (retraction_on_fiber r b)),
      exact K (on_cod (arrow.is_retraction.sect r) b)
    end

  end

  -- Corollary 7.5.5
  definition is_conn_homotopy (n : trunc_index) {A B : Type} {f g : A → B}
    (p : f ~ g) (H : is_conn_map n f) : is_conn_map n g :=
  @retract_of_conn_is_conn _ _ (arrow.arrow_hom_of_homotopy p) (arrow.is_retraction_arrow_hom_of_homotopy p) n H 

end homotopy