lean2/hott/homotopy/susp.hlean

/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Ulrik Buchholtz

Declaration of suspension
-/

import hit.pushout types.pointed2 cubical.square .connectedness

open pushout unit eq equiv pointed is_equiv

definition susp' (A : Type) : Type := pushout (λ(a : A), star) (λ(a : A), star)

namespace susp

  definition north' {A : Type} : susp' A :=
  inl star

  definition pointed_susp [instance] [constructor] (X : Type)
    : pointed (susp' X) :=
  pointed.mk north'

end susp open susp

definition susp [constructor] (X : Type) : Type* :=
pointed.MK (susp' X) north'

notation `⅀` := susp

namespace susp
  variable {A : Type}

  definition north {A : Type} : susp A :=
  north'

  definition south {A : Type} : susp A :=
  inr star

  definition merid (a : A) : @north A = @south A :=
  glue a

  protected definition rec {P : susp A → Type} (PN : P north) (PS : P south)
    (Pm : Π(a : A), PN =[merid a] PS) (x : susp' A) : P x :=
  begin
    induction x with u u,
    { cases u, exact PN},
    { cases u, exact PS},
    { apply Pm},
  end

  protected definition rec_on [reducible] {P : susp A → Type} (y : susp' A)
    (PN : P north) (PS : P south) (Pm : Π(a : A), PN =[merid a] PS) : P y :=
  susp.rec PN PS Pm y

  theorem rec_merid {P : susp A → Type} (PN : P north) (PS : P south)
    (Pm : Π(a : A), PN =[merid a] PS) (a : A)
      : apd (susp.rec PN PS Pm) (merid a) = Pm a :=
  !rec_glue

  protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
    (x : susp' A) : P :=
  susp.rec PN PS (λa, pathover_of_eq _ (Pm a)) x

  protected definition elim_on [reducible] {P : Type} (x : susp' A)
    (PN : P) (PS : P)  (Pm : A → PN = PS) : P :=
  susp.elim PN PS Pm x

  theorem elim_merid {P : Type} {PN PS : P} (Pm : A → PN = PS) (a : A)
    : ap (susp.elim PN PS Pm) (merid a) = Pm a :=
  begin
    apply inj_inv !(pathover_constant (merid a)),
    rewrite [▸*,-apd_eq_pathover_of_eq_ap,↑susp.elim,rec_merid],
  end

  protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
    (x : susp' A) : Type :=
  pushout.elim_type (λx, PN) (λx, PS) Pm x

  protected definition elim_type_on [reducible] (x : susp' A)
    (PN : Type) (PS : Type)  (Pm : A → PN ≃ PS) : Type :=
  susp.elim_type PN PS Pm x

  theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
    (a : A) : transport (susp.elim_type PN PS Pm) (merid a) = Pm a :=
  !elim_type_glue

  theorem elim_type_merid_inv {A : Type} (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
    (a : A) : transport (susp.elim_type PN PS Pm) (merid a)⁻¹ = to_inv (Pm a) :=
  !elim_type_glue_inv

  protected definition merid_square {a a' : A} (p : a = a')
    : square (merid a) (merid a') idp idp :=
  by cases p; apply vrefl

end susp

attribute susp.north' susp.north susp.south [constructor]
attribute susp.rec susp.elim [unfold 6] [recursor 6]
attribute susp.elim_type [unfold 5]
attribute susp.rec_on susp.elim_on [unfold 3]
attribute susp.elim_type_on [unfold 2]

namespace susp

  open is_trunc is_conn trunc

  -- Theorem 8.2.1
  definition is_conn_susp [instance] (n : trunc_index) (A : Type)
    [H : is_conn n A] : is_conn (n .+1) (susp A) :=
  is_contr.mk (tr north)
  begin
    intro x, induction x with x, induction x,
    { reflexivity },
    { exact (trunc.rec (λa, ap tr (merid a)) (center (trunc n A))) },
    { generalize (center (trunc n A)),
      intro x, induction x with a',
      apply pathover_of_tr_eq,
      rewrite [eq_transport_Fr,idp_con],
      revert H, induction n with n IH: intro H,
      { apply is_prop.elim },
      { change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a'),
        generalize a',
        apply is_conn_fun.elim n
              (is_conn_fun_from_unit n A a)
              (λx : A, trunctype.mk' n (ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid x))),
        intros,
        change ap (@tr n .+2 (susp A)) (merid a) = ap tr (merid a),
        reflexivity }
    }
  end

  /- Flattening lemma -/

  open prod prod.ops
  section
    universe variable u
    parameters (A : Type) (PN PS : Type.{u}) (Pm : A → PN ≃ PS)
    include Pm

    local abbreviation P [unfold 5] := susp.elim_type PN PS Pm

    local abbreviation F : A × PN → PN := λz, z.2

    local abbreviation G : A × PN → PS := λz, Pm z.1 z.2

    protected definition flattening : sigma P ≃ pushout F G :=
    begin
      apply equiv.trans !pushout.flattening,
      fapply pushout.equiv,
      { exact sigma.equiv_prod A PN },
      { apply sigma.sigma_unit_left },
      { apply sigma.sigma_unit_left },
      { reflexivity },
      { reflexivity }
    end
  end

end susp

/- Functoriality and equivalence -/
namespace susp
  variables {A B : Type} (f : A → B)
  include f

  definition susp_functor' [unfold 4] : susp A → susp B :=
  begin
    intro x, induction x with a,
    { exact north },
    { exact south },
    { exact merid (f a) }
  end

  variable [Hf : is_equiv f]
  include Hf

  open is_equiv
  protected definition is_equiv_functor [instance] [constructor] : is_equiv (susp_functor' f) :=
  adjointify (susp_functor' f) (susp_functor' f⁻¹)
  abstract begin
    intro sb, induction sb with b, do 2 reflexivity,
    apply eq_pathover,
    rewrite [ap_id,-ap_compose' (susp_functor' f) (susp_functor' f⁻¹)],
    krewrite [susp.elim_merid,susp.elim_merid], apply transpose,
    apply susp.merid_square (right_inv f b)
  end end
  abstract begin
    intro sa, induction sa with a, do 2 reflexivity,
    apply eq_pathover,
    rewrite [ap_id,-ap_compose' (susp_functor' f⁻¹) (susp_functor' f)],
    krewrite [susp.elim_merid,susp.elim_merid], apply transpose,
    apply susp.merid_square (left_inv f a)
  end end


end susp

namespace susp
  variables {A B : Type} (f : A ≃ B)

  protected definition equiv : susp A ≃ susp B :=
  equiv.mk (susp_functor' f) _
end susp

namespace susp
  open pointed is_trunc
  variables {X X' Y Y' Z : Type*}

  definition susp_functor [constructor] (f : X →* Y) : susp X →* susp Y :=
  begin
    fconstructor,
    { exact susp_functor' f },
    { reflexivity }
  end

  notation `⅀→`:(max+5) := susp_functor

  definition is_equiv_susp_functor [constructor] (f : X →* Y) [Hf : is_equiv f]
    : is_equiv (susp_functor f) :=
  susp.is_equiv_functor f

  definition susp_pequiv [constructor] (f : X ≃* Y) : susp X ≃* susp Y :=
  pequiv_of_equiv (susp.equiv f) idp

  definition susp_functor_pcompose (g : Y →* Z) (f : X →* Y) :
    susp_functor (g ∘* f) ~* susp_functor g ∘* susp_functor f :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x,
      { reflexivity },
      { reflexivity },
      { apply eq_pathover, apply hdeg_square,
        refine !elim_merid ⬝ _ ⬝ (ap_compose (susp_functor g) _ _)⁻¹ᵖ,
        refine _ ⬝ ap02 _ !elim_merid⁻¹, exact !elim_merid⁻¹ }},
    { reflexivity },
  end

  definition susp_functor_phomotopy {f g : X →* Y} (p : f ~* g) :
    susp_functor f ~* susp_functor g :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x,
      { reflexivity },
      { reflexivity },
      { apply eq_pathover, apply hdeg_square, esimp, refine !elim_merid ⬝ _ ⬝ !elim_merid⁻¹ᵖ,
        exact ap merid (p a), }},
    { reflexivity },
  end

  notation `⅀⇒`:(max+5) := susp_functor_phomotopy

  definition susp_functor_pid (A : Type*) : susp_functor (pid A) ~* pid (susp A) :=
  begin
    fapply phomotopy.mk,
    { intro x, induction x,
      { reflexivity },
      { reflexivity },
      { apply eq_pathover_id_right, apply hdeg_square, apply elim_merid }},
    { reflexivity },
  end

  /- adjunction originally  ported from Coq-HoTT,
     but we proved some additional naturality conditions -/

  definition loop_susp_unit [constructor] (X : Type*) : X →* Ω(susp X) :=
  begin
    fconstructor,
    { intro x, exact merid x ⬝ (merid pt)⁻¹ },
    { apply con.right_inv },
  end

  definition loop_susp_unit_natural (f : X →* Y)
    : psquare (loop_susp_unit X) (loop_susp_unit Y) f (Ω→ (susp_functor f)) :=
  begin
    apply ptranspose,
    induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
    fapply phomotopy.mk,
    { intro x', symmetry,
      exact
        !ap1_gen_idp_left ⬝
        (!ap_con ⬝
        whisker_left _ !ap_inv) ⬝
        (!elim_merid ◾ (inverse2 !elim_merid)) },
    { rewrite [▸*, idp_con (con.right_inv _)],
      apply inv_con_eq_of_eq_con,
      refine _ ⬝ !con.assoc',
      rewrite inverse2_right_inv,
      refine _ ⬝ !con.assoc',
      rewrite [ap_con_right_inv],
      rewrite [ap1_gen_idp_left_con],
      rewrite [-ap_compose (concat idp)] },
  end

  definition loop_susp_counit [constructor] (X : Type*) : susp (Ω X) →* X :=
  begin
    fapply pmap.mk,
    { intro x, induction x, exact pt, exact pt, exact a },
    { reflexivity },
  end

  definition loop_susp_counit_natural (f : X →* Y)
    : psquare (loop_susp_counit X) (loop_susp_counit Y) (⅀→ (Ω→ f)) f :=
  begin
    induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
    fconstructor,
    { intro x', induction x' with p,
       { reflexivity },
      { reflexivity },
      { esimp, apply eq_pathover, apply hdeg_square,
        xrewrite [-ap_compose' f, -ap_compose' (susp.elim (f x) (f x) (λ (a : f x = f x), a)),▸*],
        xrewrite [+elim_merid, ap1_gen_idp_left] }},
    { reflexivity }
  end

  definition loop_susp_counit_unit (X : Type*)
    : ap1 (loop_susp_counit X) ∘* loop_susp_unit (Ω X) ~* pid (Ω X) :=
  begin
    induction X with X x, fconstructor,
    { intro p, esimp,
      refine !ap1_gen_idp_left ⬝
             (!ap_con ⬝
             whisker_left _ !ap_inv) ⬝
             (!elim_merid ◾ inverse2 !elim_merid) },
    { rewrite [▸*,inverse2_right_inv (elim_merid id idp)],
      refine !con.assoc ⬝ _,
      xrewrite [ap_con_right_inv (susp.elim x x (λa, a)) (merid idp),ap1_gen_idp_left_con,
        -ap_compose] }
  end

  definition loop_susp_unit_counit (X : Type*)
    : loop_susp_counit (susp X) ∘* susp_functor (loop_susp_unit X) ~* pid (susp X) :=
  begin
    induction X with X x, fconstructor,
    { intro x', induction x',
      { reflexivity },
      { exact merid pt },
      { apply eq_pathover,
        xrewrite [▸*, ap_id, -ap_compose' (susp.elim north north (λa, a)), +elim_merid,▸*],
        apply square_of_eq, exact !idp_con ⬝ !inv_con_cancel_right⁻¹ }},
    { reflexivity }
  end

  definition susp_elim [constructor] {X Y : Type*} (f : X →* Ω Y) : susp X →* Y :=
  loop_susp_counit Y ∘* susp_functor f

  definition loop_susp_intro [constructor] {X Y : Type*} (f : susp X →* Y) : X →* Ω Y :=
  ap1 f ∘* loop_susp_unit X

  definition susp_elim_susp_functor {A B C : Type*} (g : B →* Ω C) (f : A →* B) :
    susp_elim g ∘* susp_functor f ~* susp_elim (g ∘* f) :=
  begin
    refine !passoc ⬝* _, exact pwhisker_left _ !susp_functor_pcompose⁻¹*
  end

  definition susp_elim_phomotopy {A B : Type*} {f g : A →* Ω B} (p : f ~* g) : susp_elim f ~* susp_elim g :=
  pwhisker_left _ (susp_functor_phomotopy p)

  definition susp_elim_natural {X Y Z : Type*} (g : Y →* Z) (f : X →* Ω Y)
    : g ∘* susp_elim f ~* susp_elim (Ω→ g ∘* f) :=
  begin
    refine _ ⬝* pwhisker_left _ !susp_functor_pcompose⁻¹*,
    refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
    exact pwhisker_right _ !loop_susp_counit_natural
  end

  definition loop_susp_intro_natural {X Y Z : Type*} (g : susp Y →* Z) (f : X →* Y) :
    loop_susp_intro (g ∘* susp_functor f) ~* loop_susp_intro g ∘* f :=
  pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_susp_unit_natural ⬝*
  !passoc⁻¹*

  definition susp_adjoint_loop_right_inv {X Y : Type*} (g : X →* Ω Y) :
    loop_susp_intro (susp_elim g) ~* g :=
  begin
    refine !pwhisker_right !ap1_pcompose ⬝* _,
    refine !passoc ⬝* _,
    refine !pwhisker_left !loop_susp_unit_natural ⬝* _,
    refine !passoc⁻¹* ⬝* _,
    refine !pwhisker_right !loop_susp_counit_unit ⬝* _,
    apply pid_pcompose
  end

  definition susp_adjoint_loop_left_inv {X Y : Type*} (f : susp X →* Y) :
    susp_elim (loop_susp_intro f) ~* f :=
  begin
    refine !pwhisker_left !susp_functor_pcompose ⬝* _,
    refine !passoc⁻¹* ⬝* _,
    refine !pwhisker_right !loop_susp_counit_natural⁻¹* ⬝* _,
    refine !passoc ⬝* _,
    refine !pwhisker_left !loop_susp_unit_counit ⬝* _,
    apply pcompose_pid
  end

  definition susp_adjoint_loop_unpointed [constructor] (X Y : Type*) : susp X →* Y ≃ X →* Ω Y :=
  begin
    fapply equiv.MK,
    { exact loop_susp_intro },
    { exact susp_elim },
    { intro g, apply eq_of_phomotopy, exact susp_adjoint_loop_right_inv g },
    { intro f, apply eq_of_phomotopy, exact susp_adjoint_loop_left_inv f }
  end

  definition susp_functor_pconst_homotopy [unfold 3] {X Y : Type*} (x : susp X) :
    susp_functor (pconst X Y) x = pt :=
  begin
    induction x,
    { reflexivity },
    { exact (merid pt)⁻¹ },
    { apply eq_pathover, refine !elim_merid ⬝ph _ ⬝hp !ap_constant⁻¹, exact square_of_eq !con.right_inv⁻¹ }
  end

  definition susp_functor_pconst [constructor] (X Y : Type*) :
    susp_functor (pconst X Y) ~* pconst (susp X) (susp Y) :=
  begin
    fapply phomotopy.mk,
    { exact susp_functor_pconst_homotopy },
    { reflexivity }
  end

  definition susp_pfunctor [constructor] (X Y : Type*) : ppmap X Y →* ppmap (susp X) (susp Y) :=
  pmap.mk susp_functor (eq_of_phomotopy !susp_functor_pconst)

  definition susp_pelim [constructor] (X Y : Type*) : ppmap X (Ω Y) →* ppmap (susp X) Y :=
  ppcompose_left (loop_susp_counit Y) ∘* susp_pfunctor X (Ω Y)

  definition loop_susp_pintro [constructor] (X Y : Type*) : ppmap (susp X) Y →* ppmap X (Ω Y) :=
  ppcompose_right (loop_susp_unit X) ∘* pap1 (susp X) Y

  definition loop_susp_pintro_natural_left (f : X' →* X) :
    psquare (loop_susp_pintro X Y) (loop_susp_pintro X' Y)
            (ppcompose_right (susp_functor f)) (ppcompose_right f) :=
  !pap1_natural_left ⬝h* ppcompose_right_psquare (loop_susp_unit_natural f)

  definition loop_susp_pintro_natural_right (f : Y →* Y') :
    psquare (loop_susp_pintro X Y) (loop_susp_pintro X Y')
            (ppcompose_left f) (ppcompose_left (Ω→ f)) :=
  !pap1_natural_right ⬝h* !ppcompose_left_ppcompose_right⁻¹*

  definition is_equiv_loop_susp_pintro [constructor] (X Y : Type*) :
    is_equiv (loop_susp_pintro X Y) :=
  begin
    fapply adjointify,
    { exact susp_pelim X Y },
    { intro g, apply eq_of_phomotopy, exact susp_adjoint_loop_right_inv g },
    { intro f, apply eq_of_phomotopy, exact susp_adjoint_loop_left_inv f }
  end

  definition susp_adjoint_loop [constructor] (X Y : Type*) : ppmap (susp X) Y ≃* ppmap X (Ω Y) :=
  pequiv_of_pmap (loop_susp_pintro X Y) (is_equiv_loop_susp_pintro X Y)

  definition susp_adjoint_loop_natural_right (f : Y →* Y') :
    psquare (susp_adjoint_loop X Y) (susp_adjoint_loop X Y')
            (ppcompose_left f) (ppcompose_left (Ω→ f)) :=
  loop_susp_pintro_natural_right f

  definition susp_adjoint_loop_natural_left (f : X' →* X) :
    psquare (susp_adjoint_loop X Y) (susp_adjoint_loop X' Y)
            (ppcompose_right (susp_functor f)) (ppcompose_right f) :=
  loop_susp_pintro_natural_left f

  definition ap1_susp_elim {A : Type*} {X : Type*} (p : A →* Ω X) :
    Ω→(susp_elim p) ∘* loop_susp_unit A ~* p :=
  susp_adjoint_loop_right_inv p

  /- the underlying homotopies of susp_adjoint_loop_natural_* -/
  definition susp_adjoint_loop_nat_right (f : susp X →* Y) (g : Y →* Z)
    : susp_adjoint_loop X Z (g ∘* f) ~* ap1 g ∘* susp_adjoint_loop X Y f :=
  begin
    esimp [susp_adjoint_loop],
    refine _ ⬝* !passoc,
    apply pwhisker_right,
    apply ap1_pcompose
  end

  definition susp_adjoint_loop_nat_left (f : Y →* Ω Z) (g : X →* Y)
    : (susp_adjoint_loop X Z)⁻¹ᵉ (f ∘* g) ~* (susp_adjoint_loop Y Z)⁻¹ᵉ f ∘* susp_functor g :=
  begin
    esimp [susp_adjoint_loop],
    refine _ ⬝* !passoc⁻¹*,
    apply pwhisker_left,
    apply susp_functor_pcompose
  end

  /- iterated suspension -/
  definition iterate_susp (n : ℕ) (A : Type*) : Type* := iterate (λX, susp X) n A

  open is_conn trunc_index nat
  definition iterate_susp_succ (n : ℕ) (A : Type*) :
    iterate_susp (succ n) A = susp (iterate_susp n A) :=
  idp

  definition is_conn_iterate_susp [instance] (n : ℕ₋₂) (m : ℕ) (A : Type*)
    [H : is_conn n A] : is_conn (n + m) (iterate_susp m A) :=
  begin induction m with m IH, exact H, exact @is_conn_susp _ _ IH end

  -- Separate cases for n = 0, which comes up often
  definition is_conn_iterate_susp_zero [instance] (m : ℕ) (A : Type*)
    [H : is_conn 0 A] : is_conn m (iterate_susp m A) :=
  begin induction m with m IH, exact H, exact @is_conn_susp _ _ IH end

  definition iterate_susp_functor (n : ℕ) {A B : Type*} (f : A →* B) :
    iterate_susp n A →* iterate_susp n B :=
  begin
    induction n with n g,
    { exact f },
    { exact susp_functor g }
  end

  definition iterate_susp_succ_in (n : ℕ) (A : Type*) :
    iterate_susp (succ n) A ≃* iterate_susp n (susp A) :=
  begin
    induction n with n IH,
    { reflexivity},
    { exact susp_pequiv IH}
  end

  definition iterate_susp_adjoint_loopn [constructor] (X Y : Type*) (n : ℕ) :
    ppmap (iterate_susp n X) Y ≃* ppmap X (Ω[n] Y) :=
  begin
    revert X Y, induction n with n IH: intro X Y,
    { reflexivity },
    { refine !susp_adjoint_loop ⬝e* !IH ⬝e* _, apply ppmap_pequiv_ppmap_right,
      symmetry, apply loopn_succ_in }
  end

end susp