add stuff about Postnikov towers, EM-spaces and components
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Floris van Doorn
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2 changed files with 184 additions and 23 deletions
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@ -499,6 +499,65 @@ namespace EM
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equivalence.mk (EM_cfunctor (n+2)) (is_equivalence_EM_cfunctor n)
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end category
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/- move -/
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-- switch arguments in homotopy_group_trunc_of_le
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lemma ghomotopy_group_trunc_of_le (k n : ℕ) (A : Type*) [Hk : is_succ k] (H : k ≤[ℕ] n)
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: πg[k] (ptrunc n A) ≃g πg[k] A :=
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begin
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exact sorry
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end
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lemma homotopy_group_isomorphism_of_ptrunc_pequiv {A B : Type*}
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(n k : ℕ) (H : n+1 ≤[ℕ] k) (f : ptrunc k A ≃* ptrunc k B) : πg[n+1] A ≃g πg[n+1] B :=
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(ghomotopy_group_trunc_of_le _ k A H)⁻¹ᵍ ⬝g
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homotopy_group_isomorphism_of_pequiv n f ⬝g
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ghomotopy_group_trunc_of_le _ k B H
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open trunc_index
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lemma minus_two_add_plus_two (n : ℕ₋₂) : -2+2+n = n :=
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by induction n with n p; reflexivity; exact ap succ p
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definition is_trunc_succ_of_is_trunc_loop (n : ℕ₋₂) (A : Type*) (H : is_trunc (n.+1) (Ω A))
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(H2 : is_conn 0 A) : is_trunc (n.+2) A :=
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begin
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apply is_trunc_succ_of_is_trunc_loop, apply minus_one_le_succ,
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refine is_conn.elim -1 _ _, exact H
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end
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lemma is_trunc_of_is_trunc_loopn (m n : ℕ) (A : Type*) (H : is_trunc n (Ω[m] A))
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(H2 : is_conn m A) : is_trunc (m + n) A :=
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begin
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revert A H H2; induction m with m IH: intro A H H2,
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{ rewrite [nat.zero_add], exact H },
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rewrite [succ_add],
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apply is_trunc_succ_of_is_trunc_loop,
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{ apply IH,
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{ apply is_trunc_equiv_closed _ !loopn_succ_in },
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apply is_conn_loop },
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exact is_conn_of_le _ (zero_le_of_nat (succ m))
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end
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lemma is_trunc_of_is_set_loopn (m : ℕ) (A : Type*) (H : is_set (Ω[m] A))
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(H2 : is_conn m A) : is_trunc m A :=
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is_trunc_of_is_trunc_loopn m 0 A H H2
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definition pequiv_EMadd1_of_loopn_pequiv_EM1 {G : AbGroup} {X : Type*} (n : ℕ) (e : Ω[n] X ≃* EM1 G)
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[H1 : is_conn n X] : X ≃* EMadd1 G n :=
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begin
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symmetry, apply EMadd1_pequiv,
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refine isomorphism_of_eq (ap (λx, πg[x+1] X) !zero_add⁻¹) ⬝g homotopy_group_add X 0 n ⬝g _ ⬝g
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!fundamental_group_EM1,
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exact homotopy_group_isomorphism_of_pequiv 0 e,
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refine is_trunc_of_is_trunc_loopn n 1 X _ _,
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apply is_trunc_equiv_closed_rev 1 e
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end
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definition EM1_pequiv_EM1 {G H : Group} (φ : G ≃g H) : EM1 G ≃* EM1 H :=
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sorry
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definition EMadd1_pequiv_EMadd1 (n : ℕ) {G H : AbGroup} (φ : G ≃g H) : EMadd1 G n ≃* EMadd1 H n :=
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sorry
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/- Eilenberg MacLane spaces are the fibers of the Postnikov system of a type -/
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definition postnikov_map [constructor] (A : Type*) (n : ℕ₋₂) : ptrunc (n.+1) A →* ptrunc n A :=
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@ -507,30 +566,40 @@ namespace EM
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open fiber EM.ops
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definition loopn_succ_pfiber_postnikov_map (A : Type*) (k : ℕ) (n : ℕ₋₂) :
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Ω[k+1] (pfiber (postnikov_map A (n.+1))) ≃* Ω[k] (pfiber (postnikov_map A n)) :=
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Ω[k+1] (pfiber (postnikov_map A (n.+1))) ≃* Ω[k] (pfiber (postnikov_map (Ω A) n)) :=
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begin
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exact sorry
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end
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definition loopn_pfiber_postnikov_map (A : Type*) (n : ℕ) :
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Ω[n+1] (pfiber (postnikov_map A n)) ≃* ptrunc 0 A :=
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definition pfiber_postnikov_map_zero (A : Type*) :
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pfiber (postnikov_map A 0) ≃* EM1 (πg[1] A) :=
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begin
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induction n with n IH,
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{ exact loopn_succ_pfiber_postnikov_map A 0 -1 ⬝e* !pfiber_pequiv_of_is_prop },
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exact loopn_succ_pfiber_postnikov_map A (n+1) n ⬝e* IH
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symmetry, apply EM1_pequiv,
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{ refine !homotopy_group_component⁻¹ᵍ ⬝g _,
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apply homotopy_group_isomorphism_of_ptrunc_pequiv, exact le.refl 1, symmetry,
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refine !ptrunc_pequiv ⬝e* _,
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exact sorry
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},
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{ apply @is_conn_fun_trunc_elim, apply is_conn_fun_tr }
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end
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definition loopn_pfiber_postnikov_map (A : Type*) (n : ℕ) :
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Ω[n] (pfiber (postnikov_map A n)) ≃* EM1 (πg[n+1] A) :=
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begin
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revert A, induction n with n IH: intro A,
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{ apply pfiber_postnikov_map_zero },
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exact loopn_succ_pfiber_postnikov_map A n n ⬝e* IH (Ω A) ⬝e*
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EM1_pequiv_EM1 !ghomotopy_group_succ_in⁻¹ᵍ
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end
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definition pfiber_postnikov_map_succ (A : Type*) (n : ℕ) :
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pfiber (postnikov_map A (n+1)) ≃* EMadd1 (πag[n+2] A) (n+1) :=
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begin
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symmetry, apply EMadd1_pequiv,
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{ },
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{ apply @is_conn_fun_trunc_elim, apply is_conn_fun_tr }
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apply pequiv_EMadd1_of_loopn_pequiv_EM1,
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{ exact sorry },
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{ apply is_conn_fun_trunc_elim, apply is_conn_fun_tr }
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end
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definition pfiber_postnikov_map_zero (A : Type*) : pfiber (postnikov_map A 0) ≃* EM1 (πg[1] A) :=
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begin
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exact sorry
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end
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end EM
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@ -954,6 +954,7 @@ namespace fiber
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Ω→ (ppoint f) ∘* (loop_pfiber f)⁻¹ᵉ* ~* ppoint (Ω→ f) :=
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(phomotopy_pinv_right_of_phomotopy (ppoint_loop_pfiber f))⁻¹*
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-- rename to pfiber_pequiv_...
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lemma pfiber_equiv_of_phomotopy_ppoint {A B : Type*} {f g : A →* B} (h : f ~* g)
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: ppoint g ∘* pfiber_equiv_of_phomotopy h ~* ppoint f :=
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begin
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@ -1005,19 +1006,13 @@ namespace fiber
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[is_trunc n A] [is_trunc (n.+1) B] : is_trunc n (pfiber f) :=
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is_trunc_fiber n f pt
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definition fiber_equiv_of_is_prop [constructor] {A B : Type} (f : A → B) (b : B) [is_prop B] :
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definition fiber_equiv_of_is_contr [constructor] {A B : Type} (f : A → B) (b : B) [is_contr B] :
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fiber f b ≃ A :=
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begin
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fapply equiv.MK,
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{ intro f, cases f with a H, exact a },
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{ intro a, apply fiber.mk a, apply is_prop.elim },
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{ intro a, reflexivity },
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{ intro f, cases f with a H, apply ap (fiber.mk a) !is_prop.elim }
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end
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!fiber.sigma_char ⬝e !sigma_equiv_of_is_contr_right
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definition pfiber_pequiv_of_is_prop [constructor] {A B : Type*} (f : A →* B) [is_prop B] :
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definition pfiber_pequiv_of_is_contr [constructor] {A B : Type*} (f : A →* B) [is_contr B] :
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pfiber f ≃* A :=
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pequiv_of_equiv (fiber_equiv_of_is_prop f pt) idp
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pequiv_of_equiv (fiber_equiv_of_is_contr f pt) idp
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end fiber
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@ -1181,6 +1176,103 @@ namespace is_conn
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cases A with A a, exact H k H2
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end
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/- move! -/
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open sigma.ops pointed
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definition merely_constant {A B : Type} (f : A → B) : Type :=
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Σb, Πa, merely (f a = b)
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definition merely_constant_pmap {A B : Type*} {f : A →* B} (H : merely_constant f) (a : A) :
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merely (f a = pt) :=
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tconcat (tconcat (H.2 a) (tinverse (H.2 pt))) (tr (respect_pt f))
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definition merely_constant_of_is_conn {A B : Type*} (f : A →* B) [is_conn 0 A] : merely_constant f :=
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⟨pt, is_conn.elim -1 _ (tr (respect_pt f))⟩
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open sigma
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definition component [constructor] (A : Type*) : Type* :=
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pType.mk (Σ(a : A), merely (pt = a)) ⟨pt, tr idp⟩
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lemma is_conn_component [instance] (A : Type*) : is_conn 0 (component A) :=
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is_contr.mk (tr pt)
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begin
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intro x, induction x with x, induction x with a p, induction p with p, induction p, reflexivity
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end
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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 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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fapply pmap.mk,
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{ intro a, refine ⟨f a, _⟩, exact tinverse (merely_constant_pmap H a) },
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exact subtype_eq !respect_pt
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end
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definition component_functor [constructor] {A B : Type*} (f : A →* B) : component A →* component B :=
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component_intro (f ∘* component_incl A) !merely_constant_of_is_conn
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-- definition component_elim [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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-- fapply pmap.mk,
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-- { intro a, refine ⟨f a, _⟩, exact tinverse (merely_constant_pmap H a) },
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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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!loopn_succ_in ⬝e* loopn_pequiv_loopn n (loop_component A) ⬝e* !loopn_succ_in⁻¹ᵉ*
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-- lemma fundamental_group_component (A : Type*) : π₁ (component A) ≃g π₁ A :=
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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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definition is_trunc_component [instance] (n : ℕ₋₂) (A : Type*) [is_trunc n A] : 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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{ apply is_trunc_succ_of_is_prop },
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end
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definition ptrunc_component' (n : ℕ₋₂) (A : Type*) :
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ptrunc (n.+2) (component A) ≃* component (ptrunc (n.+2) A) :=
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begin
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fapply pequiv.MK,
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{ exact ptrunc.elim (n.+2) (component_functor !ptr) },
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{ intro x, cases x with x p, induction x with a,
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refine tr ⟨a, _⟩,
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note q := trunc_functor -1 !tr_eq_tr_equiv p,
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exact trunc_trunc_equiv_left _ !minus_one_le_succ q },
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{ exact sorry },
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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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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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exact ptrunc_component' n A
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end
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definition pfiber_pequiv_component_of_is_contr [constructor] {A B : Type*} (f : A →* B) [is_contr B]
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/- extra condition, something like trunc_functor 0 f is an embedding -/ : pfiber f ≃* component A :=
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sorry
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end is_conn
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namespace circle
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