move some stuff to more appropriate places (before big move to HoTT library)
This commit is contained in:
Floris van Doorn
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9ad673682d
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9a3eed11bb
7 changed files with 123 additions and 91 deletions
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@ -6,6 +6,10 @@ import .module_exact_couple
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namespace left_module
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/- The Atiyah-Hirzebruch spectral sequence (with local coefficents) -/
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/- The Serre Spectral Sequence -/
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end left_module
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@ -54,7 +54,7 @@ begin
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{ intro f, induction f, apply eq_of_homotopy, intro x, esimp, induction x with a b: reflexivity }
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end
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/- currently we prove it using univalence -/
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/- currently we prove it using univalence, which means we cannot apply it to lift. TODO: change -/
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definition has_choice_equiv_closed.{u} (n : ℕ₋₂) {A B : Type.{u}} (f : A ≃ B) (hA : has_choice n B)
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: has_choice n A :=
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begin
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@ -1,4 +1,4 @@
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-- WIP
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import ..move_to_lib
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open function eq
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@ -1,7 +1,7 @@
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/-
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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, Ulrik Buchholtz
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Authors: Floris van Doorn
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The Wedge Sum of a family of Pointed Types
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-/
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@ -1,82 +1,9 @@
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import ..pointed
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open susp eq pointed function is_equiv
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variables {X X' Y Y' Z : Type*}
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-- move
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definition pap1 [constructor] (X Y : Type*) : ppmap X Y →* ppmap (Ω X) (Ω Y) :=
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pmap.mk ap1 (eq_of_phomotopy !ap1_pconst)
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definition ap1_gen_const {A B : Type} {a₁ a₂ : A} (b : B) (p : a₁ = a₂) :
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ap1_gen (const A b) idp idp p = idp :=
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ap1_gen_idp_left (const A b) p ⬝ ap_constant p b
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definition ap1_gen_compose_const_left
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{A B C : Type} (c : C) (f : A → B) {a₁ a₂ : A} (p : a₁ = a₂) :
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ap1_gen_compose (const B c) f idp idp idp idp p ⬝
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ap1_gen_const c (ap1_gen f idp idp p) =
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ap1_gen_const c p :=
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begin induction p, reflexivity end
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definition ap1_gen_compose_const_right
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{A B C : Type} (g : B → C) (b : B) {a₁ a₂ : A} (p : a₁ = a₂) :
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ap1_gen_compose g (const A b) idp idp idp idp p ⬝
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ap (ap1_gen g idp idp) (ap1_gen_const b p) =
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ap1_gen_const (g b) p :=
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begin induction p, reflexivity end
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definition ap1_pcompose_pconst_left {A B C : Type*} (f : A →* B) :
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phsquare (ap1_pcompose (pconst B C) f)
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(ap1_pconst A C)
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(ap1_phomotopy (pconst_pcompose f))
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(pwhisker_right (Ω→ f) (ap1_pconst B C) ⬝* pconst_pcompose (Ω→ f)) :=
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begin
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induction A with A a₀, induction B with B b₀, induction C with C c₀, induction f with f f₀,
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esimp at *, induction f₀,
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refine idp ◾** !trans_refl ⬝ _ ⬝ !refl_trans⁻¹ ⬝ !ap1_phomotopy_refl⁻¹ ◾** idp,
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fapply phomotopy_eq,
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{ exact ap1_gen_compose_const_left c₀ f },
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{ reflexivity }
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end
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definition ap1_pcompose_pconst_right {A B C : Type*} (g : B →* C) :
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phsquare (ap1_pcompose g (pconst A B))
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(ap1_pconst A C)
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(ap1_phomotopy (pcompose_pconst g))
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(pwhisker_left (Ω→ g) (ap1_pconst A B) ⬝* pcompose_pconst (Ω→ g)) :=
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begin
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induction A with A a₀, induction B with B b₀, induction C with C c₀, induction g with g g₀,
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esimp at *, induction g₀,
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refine idp ◾** !trans_refl ⬝ _ ⬝ !refl_trans⁻¹ ⬝ !ap1_phomotopy_refl⁻¹ ◾** idp,
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fapply phomotopy_eq,
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{ exact ap1_gen_compose_const_right g b₀ },
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{ reflexivity }
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end
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definition pap1_natural_left [constructor] (f : X' →* X) :
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psquare (pap1 X Y) (pap1 X' Y) (ppcompose_right f) (ppcompose_right (Ω→ f)) :=
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begin
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fapply phomotopy_mk_ppmap,
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{ intro g, exact !ap1_pcompose⁻¹* },
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{ refine idp ◾** (ap phomotopy_of_eq (!ap1_eq_of_phomotopy ◾ idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝
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!phomotopy_of_eq_of_phomotopy) ⬝ _ ⬝ (ap phomotopy_of_eq (!pcompose_right_eq_of_phomotopy ◾
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idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝ !phomotopy_of_eq_of_phomotopy)⁻¹,
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apply symm_trans_eq_of_eq_trans, exact (ap1_pcompose_pconst_left f)⁻¹ }
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end
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definition pap1_natural_right [constructor] (f : Y →* Y') :
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psquare (pap1 X Y) (pap1 X Y') (ppcompose_left f) (ppcompose_left (Ω→ f)) :=
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begin
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fapply phomotopy_mk_ppmap,
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{ intro g, exact !ap1_pcompose⁻¹* },
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{ refine idp ◾** (ap phomotopy_of_eq (!ap1_eq_of_phomotopy ◾ idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝
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!phomotopy_of_eq_of_phomotopy) ⬝ _ ⬝ (ap phomotopy_of_eq (!pcompose_left_eq_of_phomotopy ◾
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idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝ !phomotopy_of_eq_of_phomotopy)⁻¹,
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apply symm_trans_eq_of_eq_trans, exact (ap1_pcompose_pconst_right f)⁻¹ }
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end
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namespace susp
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variables {X X' Y Y' Z : Type*}
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definition susp_functor_pconst_homotopy [unfold 3] {X Y : Type*} (x : psusp X) :
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psusp_functor (pconst X Y) x = pt :=
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begin
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@ -899,7 +899,41 @@ namespace function
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is_surjective f' :=
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is_surjective_homotopy_closed p⁻¹ʰᵗʸ H
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end function
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definition is_equiv_ap1_gen_of_is_embedding {A B : Type} (f : A → B) [is_embedding f]
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{a a' : A} {b b' : B} (q : f a = b) (q' : f a' = b') : is_equiv (ap1_gen f q q') :=
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begin
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induction q, induction q',
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exact is_equiv.homotopy_closed _ (ap1_gen_idp_left f)⁻¹ʰᵗʸ,
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end
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definition is_equiv_ap1_of_is_embedding {A B : Type*} (f : A →* B) [is_embedding f] :
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is_equiv (Ω→ f) :=
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is_equiv_ap1_gen_of_is_embedding f (respect_pt f) (respect_pt f)
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definition loop_pequiv_loop_of_is_embedding [constructor] {A B : Type*} (f : A →* B)
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[is_embedding f] : Ω A ≃* Ω B :=
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pequiv_of_pmap (Ω→ f) (is_equiv_ap1_of_is_embedding f)
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definition loopn_pequiv_loopn_of_is_embedding [constructor] (n : ℕ) [H : is_succ n]
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{A B : Type*} (f : A →* B) [is_embedding f] : Ω[n] A ≃* Ω[n] B :=
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begin
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induction H with n,
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exact !loopn_succ_in ⬝e*
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loopn_pequiv_loopn n (loop_pequiv_loop_of_is_embedding f) ⬝e*
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!loopn_succ_in⁻¹ᵉ*
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end
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definition homotopy_group_isomorphism_of_is_embedding (n : ℕ) [H : is_succ n] {A B : Type*}
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(f : A →* B) [H2 : is_embedding f] : πg[n] A ≃g πg[n] B :=
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begin
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apply isomorphism.mk (homotopy_group_homomorphism n f),
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induction H with n,
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apply is_equiv_of_equiv_of_homotopy
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(ptrunc_pequiv_ptrunc 0 (loopn_pequiv_loopn_of_is_embedding (n+1) f)),
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exact sorry
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end
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end function open function
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namespace fiber
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open pointed
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@ -1201,6 +1235,9 @@ namespace is_conn
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definition component_incl [constructor] (A : Type*) : component A →* A :=
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pmap.mk pr1 idp
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definition is_embedding_component_incl [instance] (A : Type*) : is_embedding (component_incl A) :=
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is_embedding_pr1 _
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definition component_intro [constructor] {A B : Type*} (f : A →* B) (H : merely_constant f) :
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A →* component B :=
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begin
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@ -1220,17 +1257,7 @@ namespace is_conn
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-- exact subtype_eq !respect_pt
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-- end
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/- move!-/
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lemma is_equiv_ap1_of_is_embedding {A B : Type*} (f : A →* B) [is_embedding f] :
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is_equiv (Ω→ f) :=
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sorry
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definition loop_pequiv_loop_of_is_embedding [constructor] {A B : Type*} (f : A →* B) [is_embedding f] :
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Ω A ≃* Ω B :=
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pequiv_of_pmap (Ω→ f) (is_equiv_ap1_of_is_embedding f)
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definition loop_component (A : Type*) : Ω (component A) ≃* Ω A :=
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have is_embedding (component_incl A), from is_embedding_pr1 _,
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loop_pequiv_loop_of_is_embedding (component_incl A)
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lemma loopn_component (n : ℕ) (A : Type*) : Ω[n+1] (component A) ≃* Ω[n+1] A :=
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@ -1240,9 +1267,10 @@ namespace is_conn
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-- isomorphism_of_equiv (trunc_equiv_trunc 0 (loop_component A)) _
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lemma homotopy_group_component (n : ℕ) (A : Type*) : πg[n+1] (component A) ≃g πg[n+1] A :=
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sorry
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homotopy_group_isomorphism_of_is_embedding (n+1) (component_incl A)
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definition is_trunc_component [instance] (n : ℕ₋₂) (A : Type*) [is_trunc n A] : is_trunc n (component A) :=
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definition is_trunc_component [instance] (n : ℕ₋₂) (A : Type*) [is_trunc n A] :
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is_trunc n (component A) :=
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begin
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apply @is_trunc_sigma, intro a, cases n with n,
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{ apply is_contr_of_inhabited_prop, exact tr !is_prop.elim },
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@ -1262,7 +1290,8 @@ namespace is_conn
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{ exact sorry }
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end
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definition ptrunc_component (n : ℕ₋₂) (A : Type*) : ptrunc n (component A) ≃* component (ptrunc n A) :=
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definition ptrunc_component (n : ℕ₋₂) (A : Type*) :
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ptrunc n (component A) ≃* component (ptrunc n A) :=
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begin
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cases n with n, exact sorry,
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cases n with n, exact sorry,
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@ -1004,4 +1004,76 @@ namespace pointed
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-- definition phomotopy_eq_equiv_phomotopy2 : p = q ≃ p ~*2 q :=
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-- sorry
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variables {X X' Y Y' Z : Type*}
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definition pap1 [constructor] (X Y : Type*) : ppmap X Y →* ppmap (Ω X) (Ω Y) :=
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pmap.mk ap1 (eq_of_phomotopy !ap1_pconst)
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definition ap1_gen_const {A B : Type} {a₁ a₂ : A} (b : B) (p : a₁ = a₂) :
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ap1_gen (const A b) idp idp p = idp :=
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ap1_gen_idp_left (const A b) p ⬝ ap_constant p b
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definition ap1_gen_compose_const_left
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{A B C : Type} (c : C) (f : A → B) {a₁ a₂ : A} (p : a₁ = a₂) :
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ap1_gen_compose (const B c) f idp idp idp idp p ⬝
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ap1_gen_const c (ap1_gen f idp idp p) =
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ap1_gen_const c p :=
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begin induction p, reflexivity end
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definition ap1_gen_compose_const_right
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{A B C : Type} (g : B → C) (b : B) {a₁ a₂ : A} (p : a₁ = a₂) :
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ap1_gen_compose g (const A b) idp idp idp idp p ⬝
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ap (ap1_gen g idp idp) (ap1_gen_const b p) =
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ap1_gen_const (g b) p :=
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begin induction p, reflexivity end
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definition ap1_pcompose_pconst_left {A B C : Type*} (f : A →* B) :
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phsquare (ap1_pcompose (pconst B C) f)
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(ap1_pconst A C)
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(ap1_phomotopy (pconst_pcompose f))
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(pwhisker_right (Ω→ f) (ap1_pconst B C) ⬝* pconst_pcompose (Ω→ f)) :=
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begin
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induction A with A a₀, induction B with B b₀, induction C with C c₀, induction f with f f₀,
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esimp at *, induction f₀,
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refine idp ◾** !trans_refl ⬝ _ ⬝ !refl_trans⁻¹ ⬝ !ap1_phomotopy_refl⁻¹ ◾** idp,
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fapply phomotopy_eq,
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{ exact ap1_gen_compose_const_left c₀ f },
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{ reflexivity }
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end
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definition ap1_pcompose_pconst_right {A B C : Type*} (g : B →* C) :
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phsquare (ap1_pcompose g (pconst A B))
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(ap1_pconst A C)
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(ap1_phomotopy (pcompose_pconst g))
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(pwhisker_left (Ω→ g) (ap1_pconst A B) ⬝* pcompose_pconst (Ω→ g)) :=
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begin
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induction A with A a₀, induction B with B b₀, induction C with C c₀, induction g with g g₀,
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esimp at *, induction g₀,
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refine idp ◾** !trans_refl ⬝ _ ⬝ !refl_trans⁻¹ ⬝ !ap1_phomotopy_refl⁻¹ ◾** idp,
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fapply phomotopy_eq,
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{ exact ap1_gen_compose_const_right g b₀ },
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{ reflexivity }
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end
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definition pap1_natural_left [constructor] (f : X' →* X) :
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psquare (pap1 X Y) (pap1 X' Y) (ppcompose_right f) (ppcompose_right (Ω→ f)) :=
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begin
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fapply phomotopy_mk_ppmap,
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{ intro g, exact !ap1_pcompose⁻¹* },
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{ refine idp ◾** (ap phomotopy_of_eq (!ap1_eq_of_phomotopy ◾ idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝
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!phomotopy_of_eq_of_phomotopy) ⬝ _ ⬝ (ap phomotopy_of_eq (!pcompose_right_eq_of_phomotopy ◾
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idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝ !phomotopy_of_eq_of_phomotopy)⁻¹,
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apply symm_trans_eq_of_eq_trans, exact (ap1_pcompose_pconst_left f)⁻¹ }
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end
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definition pap1_natural_right [constructor] (f : Y →* Y') :
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psquare (pap1 X Y) (pap1 X Y') (ppcompose_left f) (ppcompose_left (Ω→ f)) :=
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begin
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fapply phomotopy_mk_ppmap,
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{ intro g, exact !ap1_pcompose⁻¹* },
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{ refine idp ◾** (ap phomotopy_of_eq (!ap1_eq_of_phomotopy ◾ idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝
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!phomotopy_of_eq_of_phomotopy) ⬝ _ ⬝ (ap phomotopy_of_eq (!pcompose_left_eq_of_phomotopy ◾
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idp ⬝ !eq_of_phomotopy_trans⁻¹) ⬝ !phomotopy_of_eq_of_phomotopy)⁻¹,
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apply symm_trans_eq_of_eq_trans, exact (ap1_pcompose_pconst_right f)⁻¹ }
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end
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end pointed
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