876aa20ad6
Also prove a theorem similar to Lemma 7.3.1 There are still some sorry's in hit.suspension
111 lines
3.9 KiB
Text
111 lines
3.9 KiB
Text
/-
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Copyright (c) 2015 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Declaration of suspension
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-/
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import .pushout types.pointed
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open pushout unit eq equiv equiv.ops pointed
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definition suspension (A : Type) : Type := pushout (λ(a : A), star.{0}) (λ(a : A), star.{0})
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namespace suspension
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variable {A : Type}
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definition north (A : Type) : suspension A :=
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inl _ _ star
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definition south (A : Type) : suspension A :=
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inr _ _ star
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definition merid (a : A) : north A = south A :=
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glue _ _ a
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protected definition rec {P : suspension A → Type} (PN : P !north) (PS : P !south)
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(Pm : Π(a : A), PN =[merid a] PS) (x : suspension A) : P x :=
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begin
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fapply pushout.rec_on _ _ x,
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{ intro u, cases u, exact PN},
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{ intro u, cases u, exact PS},
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{ exact Pm},
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end
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protected definition rec_on [reducible] {P : suspension A → Type} (y : suspension A)
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(PN : P !north) (PS : P !south) (Pm : Π(a : A), PN =[merid a] PS) : P y :=
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suspension.rec PN PS Pm y
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theorem rec_merid {P : suspension A → Type} (PN : P !north) (PS : P !south)
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(Pm : Π(a : A), PN =[merid a] PS) (a : A)
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: apdo (suspension.rec PN PS Pm) (merid a) = Pm a :=
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!rec_glue
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protected definition elim {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS)
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(x : suspension A) : P :=
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suspension.rec PN PS (λa, pathover_of_eq (Pm a)) x
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protected definition elim_on [reducible] {P : Type} (x : suspension A)
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(PN : P) (PS : P) (Pm : A → PN = PS) : P :=
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suspension.elim PN PS Pm x
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theorem elim_merid {P : Type} (PN : P) (PS : P) (Pm : A → PN = PS) (a : A)
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: ap (suspension.elim PN PS Pm) (merid a) = Pm a :=
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begin
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apply eq_of_fn_eq_fn_inv !(pathover_constant (merid a)),
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rewrite [▸*,-apdo_eq_pathover_of_eq_ap,↑suspension.elim,rec_merid],
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end
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protected definition elim_type (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(x : suspension A) : Type :=
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suspension.elim PN PS (λa, ua (Pm a)) x
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protected definition elim_type_on [reducible] (x : suspension A)
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(PN : Type) (PS : Type) (Pm : A → PN ≃ PS) : Type :=
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suspension.elim_type PN PS Pm x
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theorem elim_type_merid (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(a : A) : transport (suspension.elim_type PN PS Pm) (merid a) = Pm a :=
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by rewrite [tr_eq_cast_ap_fn,↑suspension.elim_type,elim_merid];apply cast_ua_fn
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end suspension
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attribute suspension.north suspension.south [constructor]
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attribute suspension.rec suspension.elim [unfold-c 6] [recursor 6]
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attribute suspension.elim_type [unfold-c 5]
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attribute suspension.rec_on suspension.elim_on [unfold-c 3]
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attribute suspension.elim_type_on [unfold-c 2]
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namespace suspension
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definition pointed_suspension [instance] [constructor] (A : Type) : pointed (suspension A) :=
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pointed.mk !north
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definition suspension_adjoint_loop (A B : Pointed)
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: map₊ (pointed.mk' (suspension A)) B ≃ map₊ A (Ω B) :=
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begin
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fapply equiv.MK,
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{ intro f, fapply pointed_map.mk,
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intro a, refine !respect_pt⁻¹ ⬝ ap f (merid a ⬝ (merid pt)⁻¹) ⬝ !respect_pt,
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refine ap _ !con.right_inv ⬝ !con.left_inv},
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{ intro g, fapply pointed_map.mk,
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{ esimp, intro a, induction a,
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exact pt,
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exact pt,
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exact g a} ,
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{ reflexivity}},
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{ intro g, fapply pointed_map_eq,
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intro a, esimp [respect_pt], refine !idp_con ⬝ !ap_con ⬝ ap _ !ap_inv
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⬝ ap _ !elim_merid ⬝ ap _ !elim_merid ⬝ ap _ (respect_pt g) ⬝ _, exact idp,
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-- rewrite [ap_con,ap_inv,+elim_merid,idp_con,respect_pt],
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esimp, exact sorry},
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{ intro f, fapply pointed_map_eq,
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{ esimp, intro a, induction a; all_goals esimp,
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exact !respect_pt⁻¹,
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exact !respect_pt⁻¹ ⬝ ap f (merid pt),
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apply pathover_eq, exact sorry},
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{ esimp, exact !con.left_inv⁻¹}},
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end
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end suspension
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