working on truncatedness of suspensions of sets

homotopy/susp_pset.hlean

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+/-
+Copyright (c) 2018 Ulrik Buchholtz. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Ulrik Buchholtz
+-/
+
+import algebra.group_theory hit.set_quotient types.list homotopy.vankampen
+       homotopy.susp .pushout ..algebra.free_group
+
+open eq pointed equiv is_equiv is_trunc set_quotient sum list susp trunc algebra
+     group pi pushout is_conn fiber unit function category paths
+
+-- special purpose lemmas
+definition tr_trunc_eq (A : Type) (a : A) {x y : A} (p : x = y) (q : x = a)
+  : transport (λ(z : A), trunc 0 (z = a)) p (tr q) = tr (p⁻¹ ⬝ q) :=
+by induction p; induction q; reflexivity
+
+namespace susp
+  section
+  universe variable u
+  parameters (A : pType.{u}) [H : is_set A]
+  include H
+
+  local notation `F` := Π₁⇒ (λ(a : A), star)
+
+  local abbreviation C : Groupoid := Groupoid_bpushout (@id A) F F
+  local abbreviation N : C := inl star
+  local abbreviation S : C := inr star
+
+--  local notation `N` := a
+
+  -- hom group of fundamental groupoid is fundamental group
+  -- the fundamental group of the suspension is the free group on A
+  -- could go via van Kampen, but would have to compose with opposite, which is not so well developed
+  -- definition fundamental_group_of_susp : π₁(⅀ A) ≃g free_group A :=
+  -- sorry
+
+  /-
+  van Kampen instead?
+  
+     game plan:
+1. lift to 1-connected cover
+2. apply flattening lemma
+3. provide equivalences F A ≃ ∥N = N∥ ≃ ∥S = N∥
+4. move to is_contr
+5. induction induction induction!
+  -/
+
+  definition pglueNS (a : A) : hom N S :=
+  class_of [ bpushout_prehom_index.DE (@id A) F F a ]
+
+  definition pglueSN (a : A) : hom S N :=
+  class_of [ bpushout_prehom_index.ED (@id A) F F a ]
+
+  definition f : A × hom N N → hom S N :=
+  prod.rec (λ a p, p ∘ pglueSN a)
+
+  definition g : A × trunc 0 (@susp.north A = @susp.north A) → trunc 0 (@susp.south A = @susp.north A) :=
+  prod.rec (λ a p, tconcat (tr (merid a)⁻¹) p)
+
+  --set_option pp.notation false
+  --set_option pp.implicit true
+
+
+  definition foo : (Σ(z : susp A), trunc 0 (z = susp.north)) ≃ pushout prod.pr2 g :=
+  begin
+    apply equiv.trans !pushout.flattening',
+    fapply pushout.equiv,
+    { apply sigma.equiv_prod },
+    { apply sigma.sigma_unit_left },
+    { apply sigma.sigma_unit_left },
+    { intro z, induction z with a p, induction p with p, reflexivity },
+    { intro z, induction z with a p, induction p with p, apply tr_trunc_eq }
+  end
+
+  definition bar : pushout prod.pr2 g ≃ pushout prod.pr2 f :=
+  begin
+    fapply pushout.equiv,
+    { apply prod.prod_equiv_prod_right, apply vankampen },
+    { apply vankampen },
+    { apply vankampen },
+    { intro z, induction z with a p, reflexivity },
+    { intro z, induction z with a p, 
+      change (encode (@id A) (λ(z : A), star) (λ(z : A), star) (tconcat (tr (merid a)⁻¹) p))
+        = (encode (@id A) (λ(z : A), star) (λ(z : A), star) p ∘ pglueSN a),
+      revert p, fapply @trunc.rec 0 (@susp.north A = @susp.north A),
+      { intro p, apply is_trunc_succ, apply is_trunc_eq, apply is_set_code }, intro p,
+      apply trans (encode_tcon (@id A) (λ(z : A), star) (λ(z : A), star) (tr (merid a)⁻¹) (tr p)),
+      apply ap (λ h, encode (@id A) (λ(z : A), star) (λ(z : A), star) (tr p) ∘ h),
+      apply encode_decode_singleton }
+  end
+
+  definition is_contr_susp_fiber_tr : is_contr (Σ(z : susp A), trunc 0 (z = susp.north)) := sorry
+
+  definition pfiber_susp_equiv_sigma : pfiber (ptr 1 (⅀ A)) ≃ (Σ(z : susp A), trunc 0 (z = susp.north)) := 
+  begin
+    apply equiv.trans !fiber.sigma_char,
+    apply sigma.sigma_equiv_sigma_right,
+    intro z, apply tr_eq_tr_equiv
+  end
+
+  definition is_trunc_susp_of_is_set : is_trunc 1 (susp A) :=
+  begin
+    apply is_trunc_of_is_equiv_tr,
+    apply is_equiv_of_is_contr_fun,
+    fapply @is_conn.elim -1 (ptrunc 1 (⅀ A)),
+    change is_contr (pfiber (ptr 1 (⅀ A))),
+    apply is_contr_equiv_closed_rev pfiber_susp_equiv_sigma,
+    apply is_contr_susp_fiber_tr
+  end
+
+  end
+
+end susp