112 lines
4 KiB
Text
112 lines
4 KiB
Text
/-
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Copyright (c) 2016 Jakob von Raumer. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jakob von Raumer
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The Cofiber Type
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-/
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import hit.pushout function .susp types.unit
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open eq pushout unit pointed is_trunc is_equiv susp unit equiv
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definition cofiber {A B : Type} (f : A → B) := pushout f (λ (a : A), ⋆)
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namespace cofiber
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section
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parameters {A B : Type} (f : A → B)
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definition cod : B → cofiber f := inl
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definition base : cofiber f := inr ⋆
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parameter {f}
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protected definition glue (a : A) : cofiber.cod f (f a) = cofiber.base f :=
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pushout.glue a
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protected definition rec {P : cofiber f → Type} (Pcod : Π (b : B), P (cod b)) (Pbase : P base)
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(Pglue : Π (a : A), pathover P (Pcod (f a)) (glue a) Pbase) :
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(Π y, P y) :=
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begin
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intro y, induction y, exact Pcod x, induction x, exact Pbase, exact Pglue x
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end
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protected definition rec_on {P : cofiber f → Type} (y : cofiber f)
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(Pcod : Π (b : B), P (cod b)) (Pbase : P base)
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(Pglue : Π (a : A), pathover P (Pcod (f a)) (glue a) Pbase) : P y :=
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cofiber.rec Pcod Pbase Pglue y
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protected theorem rec_glue {P : cofiber f → Type} (Pcod : Π (b : B), P (cod b)) (Pbase : P base)
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(Pglue : Π (a : A), pathover P (Pcod (f a)) (glue a) Pbase) (a : A)
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: apd (cofiber.rec Pcod Pbase Pglue) (cofiber.glue a) = Pglue a :=
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!pushout.rec_glue
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protected definition elim {P : Type} (Pcod : B → P) (Pbase : P)
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(Pglue : Π (x : A), Pcod (f x) = Pbase) (y : cofiber f) : P :=
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pushout.elim Pcod (λu, Pbase) Pglue y
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protected definition elim_on {P : Type} (y : cofiber f) (Pcod : B → P) (Pbase : P)
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(Pglue : Π (x : A), Pcod (f x) = Pbase) : P :=
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cofiber.elim Pcod Pbase Pglue y
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protected theorem elim_glue {P : Type} (Pcod : B → P) (Pbase : P)
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(Pglue : Π (x : A), Pcod (f x) = Pbase) (a : A)
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: ap (cofiber.elim Pcod Pbase Pglue) (cofiber.glue a) = Pglue a :=
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!pushout.elim_glue
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end
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end cofiber
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attribute cofiber.base cofiber.cod [constructor]
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attribute cofiber.rec cofiber.elim [recursor 8] [unfold 8]
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attribute cofiber.rec_on cofiber.elim_on [unfold 5]
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-- pointed version
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definition pcofiber [constructor] {A B : Type*} (f : A →* B) : Type* :=
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pointed.MK (cofiber f) !cofiber.base
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notation `ℂ` := pcofiber
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namespace cofiber
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variables {A B : Type*} (f : A →* B)
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definition is_contr_cofiber_of_equiv [H : is_equiv f] : is_contr (cofiber f) :=
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begin
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fapply is_contr.mk, exact cofiber.base f,
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intro a, induction a with b a,
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{ exact !glue⁻¹ ⬝ ap inl (right_inv f b) },
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{ reflexivity },
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{ apply eq_pathover_constant_left_id_right, apply move_top_of_left,
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refine _ ⬝pv natural_square_tr cofiber.glue (left_inv f a) ⬝vp !ap_constant,
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refine ap02 inl _ ⬝ !ap_compose⁻¹, exact adj f a },
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end
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definition pcod [constructor] (f : A →* B) : B →* pcofiber f :=
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pmap.mk (cofiber.cod f) (ap inl (respect_pt f)⁻¹ ⬝ cofiber.glue pt)
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definition pcod_pcompose [constructor] (f : A →* B) : pcod f ∘* f ~* pconst A (ℂ f) :=
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begin
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fapply phomotopy.mk,
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{ intro a, exact cofiber.glue a },
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{ exact !con_inv_cancel_left⁻¹ ⬝ idp ◾ (!ap_inv⁻¹ ◾ idp) }
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end
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definition pcofiber_punit (A : Type*) : pcofiber (pconst A punit) ≃* susp A :=
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begin
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fapply pequiv_of_pmap,
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{ fapply pmap.mk, intro x, induction x, exact north, exact south, exact merid x,
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exact (merid pt)⁻¹ },
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{ esimp, fapply adjointify,
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{ intro s, induction s, exact inl ⋆, exact inr ⋆, apply glue a },
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{ intro s, induction s, do 2 reflexivity, esimp,
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apply eq_pathover, refine _ ⬝hp !ap_id⁻¹, apply hdeg_square,
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refine !(ap_compose (pushout.elim _ _ _)) ⬝ _,
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refine ap _ !elim_merid ⬝ _, apply elim_glue },
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{ intro c, induction c with u, induction u, reflexivity,
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reflexivity, esimp, apply eq_pathover, apply hdeg_square,
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refine _ ⬝ !ap_id⁻¹, refine !(ap_compose (pushout.elim _ _ _)) ⬝ _,
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refine ap02 _ !elim_glue ⬝ _, apply elim_merid }},
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end
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end cofiber
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