feat(chain_complex): give the construction of the LES of homotopy groups
This commit defines "type_chain_complex" which is a typal variant of a chain complex, where the exactness condition is formulated without a propositional truncation in it. The fiber sequence of a pointed map is an instance of this structure. It also defines "chain_complex" which is the usual notion of a chain complex: a sequence of pointed sets with pointed maps between them, such that the kernel and image of consecutive maps coincide. The biggest part of this commit is the definition of the long exact sequence of homotopy groups of a pointed map. The definition uses the fiber sequence of a pointed map.
This commit is contained in:
Floris van Doorn
parent
efd5d25039
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5c9355c4c1
2 changed files with 640 additions and 136 deletions
443
homotopy/LES_of_homotopy_groups.hlean
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443
homotopy/LES_of_homotopy_groups.hlean
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/-
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Copyright (c) 2016 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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We define the fiber sequence of a pointed map f : X →* Y. We follow the proof in section 8.4 of
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the book closely. First we define a sequence fiber_sequence as in Definition 8.4.3.
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It has types X(n) : Type*
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X(0) := Y,
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X(1) := X,
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X(n+1) := pfiber (f(n))
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with functions f(n) : X(n+1) →* X(n)
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f(0) := f
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f(n+1) := ppoint f(n)
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We prove that this is an exact sequence.
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Then we prove Lemma 8.4.3, by showing that X(n+3) ≃* Ω(X(n)) and that this equivalence sends
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the map f(n+3) to -Ω(f(n)), i.e. the composition of Ω(f(n)) with path inversion.
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This is the hardest part of this formalization, because we need to show that they are the same
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as pointed maps (we define a pointed homotopy between them).
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Using this equivalence we get a boundary_map : Ω(Y) → pfiber f and we can define a new
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fiber sequence X'(n) : Type*
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X'(0) := Y,
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X'(1) := X,
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X'(2) := pfiber f
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X'(n+3) := Ω(X'(n))
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and maps f'(n) : X'(n+1) →* X'(n)
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f'(0) := f
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f'(1) := ppoint f
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f'(2) := boundary_map f
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f'(3) := -Ω(f)
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f'(4) := -Ω(ppoint f)
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f'(5) := -Ω(boundary_map f)
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f'(n+6) := Ω²(f'(n))
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We can show that these sequences are equivalent, hence the sequence (X',f') is an exact
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sequence. Now we get the fiber sequence by taking the set-truncation of this sequence.
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-/
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import .chain_complex algebra.homotopy_group
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open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc equiv.ops nat trunc algebra
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namespace chain_complex
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definition fiber_sequence_helper [constructor] (v : Σ(X Y : Type*), X →* Y)
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: Σ(Z X : Type*), Z →* X :=
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⟨pfiber v.2.2, v.1, ppoint v.2.2⟩
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definition fiber_sequence_helpern (v : Σ(X Y : Type*), X →* Y) (n : ℕ)
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: Σ(Z X : Type*), Z →* X :=
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iterate fiber_sequence_helper n v
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universe variable u
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variables {X Y : pType.{u}} (f : X →* Y) (n : ℕ)
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include f
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definition fiber_sequence_carrier : Type* :=
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(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.1
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definition fiber_sequence_fun
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: fiber_sequence_carrier f (n + 1) →* fiber_sequence_carrier f n :=
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(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.2
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/- Definition 8.4.3 -/
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definition fiber_sequence : left_type_chain_complex.{u} :=
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begin
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fconstructor,
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{ exact fiber_sequence_carrier f},
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{ exact fiber_sequence_fun f},
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{ intro n x, cases n with n,
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{ exact point_eq x},
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{ exact point_eq x}}
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end
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definition is_exact_fiber_sequence : is_exact_lt (fiber_sequence f) :=
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λn x p, fiber.mk (fiber.mk x p) rfl
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/- (generalization of) Lemma 8.4.4(i)(ii) -/
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definition fiber_sequence_carrier_equiv
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: fiber_sequence_carrier f (n+3) ≃ Ω(fiber_sequence_carrier f n) :=
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calc
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fiber_sequence_carrier f (n+3) ≃ fiber (fiber_sequence_fun f (n+1)) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f (n+1) x = pt
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: fiber.sigma_char
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... ≃ Σ(x : fiber (fiber_sequence_fun f n) pt), fiber_sequence_fun f _ x = pt
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: erfl
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
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fiber_sequence_fun f _ (fiber.mk v.1 v.2) = pt
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: by exact sigma_equiv_sigma !fiber.sigma_char (λa, erfl)
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
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v.1 = pt
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: erfl
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), x = pt),
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fiber_sequence_fun f _ v.1 = pt
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: sigma_assoc_comm_equiv
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... ≃ fiber_sequence_fun f _ !center.1 = pt
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: @(sigma_equiv_of_is_contr_left _) !is_contr_sigma_eq'
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... ≃ fiber_sequence_fun f _ pt = pt
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: erfl
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... ≃ pt = pt
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: by exact !equiv_eq_closed_left !respect_pt
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... ≃ Ω(fiber_sequence_carrier f n) : erfl
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/- computation rule -/
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definition fiber_sequence_carrier_equiv_eq
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(x : fiber_sequence_carrier f (n+1)) (p : fiber_sequence_fun f n x = pt)
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(q : fiber_sequence_fun f (n+1) (fiber.mk x p) = pt)
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: fiber_sequence_carrier_equiv f n (fiber.mk (fiber.mk x p) q)
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= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun f n) q⁻¹ ⬝ p :=
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begin
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refine _ ⬝ !con.assoc⁻¹,
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apply whisker_left,
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refine transport_eq_Fl _ _ ⬝ _,
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apply whisker_right,
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refine inverse2 !ap_inv ⬝ !inv_inv ⬝ _,
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refine ap_compose (fiber_sequence_fun f n) pr₁ _ ⬝
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ap02 (fiber_sequence_fun f n) !ap_pr1_center_eq_sigma_eq',
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end
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definition fiber_sequence_carrier_equiv_inv_eq
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(p : Ω(fiber_sequence_carrier f n)) : (fiber_sequence_carrier_equiv f n)⁻¹ᵉ p =
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fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun f n) ⬝ p)) idp :=
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begin
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apply inv_eq_of_eq,
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refine _ ⬝ !fiber_sequence_carrier_equiv_eq⁻¹, esimp,
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exact !inv_con_cancel_left⁻¹
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end
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definition fiber_sequence_carrier_pequiv
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: fiber_sequence_carrier f (n+3) ≃* Ω(fiber_sequence_carrier f n) :=
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pequiv_of_equiv (fiber_sequence_carrier_equiv f n)
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begin
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esimp,
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apply con.left_inv
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end
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definition fiber_sequence_carrier_pequiv_eq
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(x : fiber_sequence_carrier f (n+1)) (p : fiber_sequence_fun f n x = pt)
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(q : fiber_sequence_fun f (n+1) (fiber.mk x p) = pt)
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: fiber_sequence_carrier_pequiv f n (fiber.mk (fiber.mk x p) q)
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= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun f n) q⁻¹ ⬝ p :=
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fiber_sequence_carrier_equiv_eq f n x p q
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definition fiber_sequence_carrier_pequiv_inv_eq
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(p : Ω(fiber_sequence_carrier f n)) : (fiber_sequence_carrier_pequiv f n)⁻¹ᵉ* p =
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fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun f n) ⬝ p)) idp :=
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fiber_sequence_carrier_equiv_inv_eq f n p
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attribute pequiv._trans_of_to_pmap [unfold 3]
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/- Lemma 8.4.4(iii) -/
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definition fiber_sequence_fun_eq_helper
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(p : Ω(fiber_sequence_carrier f (n + 1))) :
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fiber_sequence_carrier_pequiv f n
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(fiber_sequence_fun f (n + 3)
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((fiber_sequence_carrier_pequiv f (n + 1))⁻¹ᵉ* p)) =
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ap1 (fiber_sequence_fun f n) p⁻¹ :=
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begin
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refine ap (λx, fiber_sequence_carrier_pequiv f n (fiber_sequence_fun f (n + 3) x))
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(fiber_sequence_carrier_pequiv_inv_eq f (n+1) p) ⬝ _,
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/- the following three lines are rewriting some reflexivities: -/
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-- replace (n + 3) with (n + 2 + 1),
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-- refine ap (fiber_sequence_carrier_pequiv f n)
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-- (fiber_sequence_fun_eq1 f (n+2) _ idp) ⬝ _,
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refine fiber_sequence_carrier_pequiv_eq f n pt (respect_pt (fiber_sequence_fun f n)) _ ⬝ _,
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esimp,
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apply whisker_right,
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apply whisker_left,
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apply ap02, apply inverse2, apply idp_con,
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end
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theorem fiber_sequence_carrier_pequiv_eq_point_eq_idp :
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fiber_sequence_carrier_pequiv_eq f n
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(Point (fiber_sequence_carrier f (n+1)))
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(respect_pt (fiber_sequence_fun f n))
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(respect_pt (fiber_sequence_fun f (n + 1))) = idp :=
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begin
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apply con_inv_eq_idp,
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refine ap (λx, whisker_left _ (_ ⬝ x)) _ ⬝ _,
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{ reflexivity},
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{ reflexivity},
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esimp,
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refine ap (whisker_left _)
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(transport_eq_Fl_idp_left (fiber_sequence_fun f n)
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(respect_pt (fiber_sequence_fun f n))) ⬝ _,
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apply whisker_left_idp_con_eq_assoc
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end
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theorem fiber_sequence_fun_phomotopy_helper :
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(fiber_sequence_carrier_pequiv f n ∘*
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fiber_sequence_fun f (n + 3)) ∘*
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(fiber_sequence_carrier_pequiv f (n + 1))⁻¹ᵉ* ~*
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ap1 (fiber_sequence_fun f n) ∘* pinverse :=
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begin
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fapply phomotopy.mk,
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{ exact (fiber_sequence_fun_eq_helper f n)},
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{ esimp, rewrite [idp_con], refine _ ⬝ whisker_left _ !idp_con⁻¹,
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apply whisker_right,
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apply whisker_left,
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exact fiber_sequence_carrier_pequiv_eq_point_eq_idp f n}
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end
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theorem fiber_sequence_fun_eq : Π(x : fiber_sequence_carrier f (n + 4)),
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fiber_sequence_carrier_pequiv f n (fiber_sequence_fun f (n + 3) x) =
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ap1 (fiber_sequence_fun f n) (fiber_sequence_carrier_pequiv f (n + 1) x)⁻¹ :=
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homotopy_of_inv_homotopy
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(pequiv.to_equiv (fiber_sequence_carrier_pequiv f (n + 1)))
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(fiber_sequence_fun_eq_helper f n)
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theorem fiber_sequence_fun_phomotopy :
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fiber_sequence_carrier_pequiv f n ∘*
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fiber_sequence_fun f (n + 3) ~*
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(ap1 (fiber_sequence_fun f n) ∘* pinverse) ∘* fiber_sequence_carrier_pequiv f (n + 1) :=
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begin
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apply phomotopy_of_pinv_right_phomotopy,
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apply fiber_sequence_fun_phomotopy_helper
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end
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definition boundary_map : Ω Y →* pfiber f :=
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fiber_sequence_fun f 2 ∘* (fiber_sequence_carrier_pequiv f 0)⁻¹ᵉ*
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/- Now we are ready to define the long exact sequence of homotopy groups.
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First we define its carrier -/
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definition homotopy_groups : ℕ → Type*
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| 0 := Y
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| 1 := X
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| 2 := pfiber f
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| (k+3) := Ω (homotopy_groups k)
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definition homotopy_groups_add3 [unfold_full] :
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homotopy_groups f (n+3) = Ω (homotopy_groups f n) :=
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proof idp qed
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definition homotopy_groups_mul3
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: Πn, homotopy_groups f (3 * n) = Ω[n] Y :> Type*
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| 0 := proof rfl qed
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| (k+1) := proof ap (λX, Ω X) (homotopy_groups_mul3 k) qed
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definition homotopy_groups_mul3add1
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: Πn, homotopy_groups f (3 * n + 1) = Ω[n] X :> Type*
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| 0 := proof rfl qed
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| (k+1) := proof ap (λX, Ω X) (homotopy_groups_mul3add1 k) qed
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definition homotopy_groups_mul3add2
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: Πn, homotopy_groups f (3 * n + 2) = Ω[n] (pfiber f) :> Type*
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| 0 := proof rfl qed
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| (k+1) := proof ap (λX, Ω X) (homotopy_groups_mul3add2 k) qed
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/- The maps between the homotopy groups -/
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definition homotopy_groups_fun
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: Π(n : ℕ), homotopy_groups f (n+1) →* homotopy_groups f n
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| 0 := proof f qed
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| 1 := proof ppoint f qed
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| 2 := proof boundary_map f qed
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| 3 := proof ap1 f ∘* pinverse qed
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| 4 := proof ap1 (ppoint f) ∘* pinverse qed
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| 5 := proof ap1 (boundary_map f) ∘* pinverse qed
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| (k+6) := proof ap1 (ap1 (homotopy_groups_fun k)) qed
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definition homotopy_groups_fun_add6 [unfold_full] :
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homotopy_groups_fun f (n + 6) = ap1 (ap1 (homotopy_groups_fun f n)) :=
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proof idp qed
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/- this is a simpler defintion of the functions, but which are the same as the previous ones
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(there is a pointed homotopy) -/
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definition homotopy_groups_fun'
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: Π(n : ℕ), homotopy_groups f (n+1) →* homotopy_groups f n
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| 0 := proof f qed
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| 1 := proof ppoint f qed
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| 2 := proof boundary_map f qed
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| (k+3) := proof ap1 (homotopy_groups_fun' k) ∘* pinverse qed
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definition homotopy_groups_fun'_add3 [unfold_full] :
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homotopy_groups_fun' f (n+3) = ap1 (homotopy_groups_fun' f n) ∘* pinverse :=
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proof idp qed
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theorem homotopy_groups_fun_eq
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: Π(n : ℕ), homotopy_groups_fun f n ~* homotopy_groups_fun' f n
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| 0 := proof phomotopy.rfl qed
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| 1 := proof phomotopy.rfl qed
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| 2 := proof phomotopy.rfl qed
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| 3 := proof phomotopy.rfl qed
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| 4 := proof phomotopy.rfl qed
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| 5 := proof phomotopy.rfl qed
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| (k+6) :=
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begin
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rewrite [homotopy_groups_fun_add6 f k],
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replace (k + 6) with (k + 3 + 3),
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rewrite [homotopy_groups_fun'_add3 f (k+3)],
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rewrite [homotopy_groups_fun'_add3 f k],
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refine _ ⬝* pwhisker_right _ !ap1_compose⁻¹*,
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refine _ ⬝* !passoc⁻¹*,
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refine !comp_pid⁻¹* ⬝* _,
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refine pconcat2 _ _,
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/- Currently ap1_phomotopy is defined using function extensionality -/
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{ apply ap1_phomotopy, apply pap ap1, apply homotopy_groups_fun_eq},
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{ refine _ ⬝* (pwhisker_right _ ap1_pinverse)⁻¹*, fapply phomotopy.mk,
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{ intro q, esimp, exact !inv_inv⁻¹},
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{ reflexivity}}
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end
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definition fiber_sequence_pequiv_homotopy_groups :
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Πn, fiber_sequence_carrier f n ≃* homotopy_groups f n
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| 0 := proof pequiv.rfl qed
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| 1 := proof pequiv.rfl qed
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| 2 := proof pequiv.rfl qed
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| (k+3) :=
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begin
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refine fiber_sequence_carrier_pequiv f k ⬝e* _,
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apply loop_space_pequiv,
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exact fiber_sequence_pequiv_homotopy_groups k
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end
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definition fiber_sequence_pequiv_homotopy_groups_add3
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: fiber_sequence_pequiv_homotopy_groups f (n + 3) =
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ap1 (fiber_sequence_pequiv_homotopy_groups f n) ∘* fiber_sequence_carrier_pequiv f n :=
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by reflexivity
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definition fiber_sequence_pequiv_homotopy_groups_3_phomotopy
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: fiber_sequence_pequiv_homotopy_groups f 3 ~* fiber_sequence_carrier_pequiv f 0 :=
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begin
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refine fiber_sequence_pequiv_homotopy_groups_add3 f 0 ⬝p* _,
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refine pwhisker_right _ ap1_id ⬝* _,
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apply pid_comp
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end
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theorem fiber_sequence_phomotopy_homotopy_groups' :
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Π(n : ℕ),
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fiber_sequence_pequiv_homotopy_groups f n ∘* fiber_sequence_fun f n ~*
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homotopy_groups_fun' f n ∘* fiber_sequence_pequiv_homotopy_groups f (n + 1)
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| 0 := by reflexivity
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| 1 := by reflexivity
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| 2 :=
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begin
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refine !pid_comp ⬝* _,
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replace homotopy_groups_fun' f 2 with boundary_map f,
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refine _ ⬝* pwhisker_left _ (fiber_sequence_pequiv_homotopy_groups_3_phomotopy f)⁻¹*,
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apply phomotopy_of_pinv_right_phomotopy,
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reflexivity
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end
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| (k+3) :=
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begin
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replace (k + 3 + 1) with (k + 1 + 3),
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rewrite [fiber_sequence_pequiv_homotopy_groups_add3 f k,
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fiber_sequence_pequiv_homotopy_groups_add3 f (k+1)],
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refine !passoc ⬝* _,
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refine pwhisker_left _ (fiber_sequence_fun_phomotopy f k) ⬝* _,
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refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
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apply pwhisker_right,
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rewrite [homotopy_groups_fun'_add3],
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refine _ ⬝* !passoc⁻¹*,
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refine _ ⬝* pwhisker_left _ !ap1_compose_pinverse,
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refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
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apply pwhisker_right,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose,
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apply ap1_phomotopy,
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exact fiber_sequence_phomotopy_homotopy_groups' k
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end
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|
||||
theorem fiber_sequence_phomotopy_homotopy_groups (n : ℕ)
|
||||
(x : fiber_sequence_carrier f (n + 1)) :
|
||||
fiber_sequence_pequiv_homotopy_groups f n (fiber_sequence_fun f n x) =
|
||||
homotopy_groups_fun f n (fiber_sequence_pequiv_homotopy_groups f (n + 1) x) :=
|
||||
begin
|
||||
refine fiber_sequence_phomotopy_homotopy_groups' f n x ⬝ _,
|
||||
exact (homotopy_groups_fun_eq f n _)⁻¹
|
||||
end
|
||||
|
||||
/- the long exact sequence of homotopy groups -/
|
||||
definition LES_of_homotopy_groups [constructor] : left_chain_complex :=
|
||||
trunc_left_chain_complex
|
||||
(transfer_left_type_chain_complex
|
||||
(fiber_sequence f)
|
||||
(homotopy_groups_fun f)
|
||||
(fiber_sequence_pequiv_homotopy_groups f)
|
||||
(fiber_sequence_phomotopy_homotopy_groups f))
|
||||
|
||||
/- the fiber sequence is exact -/
|
||||
definition is_exact_LES_of_homotopy_groups : is_exact_l (LES_of_homotopy_groups f) :=
|
||||
begin
|
||||
intro n,
|
||||
apply is_exact_at_l_trunc,
|
||||
apply is_exact_at_lt_transfer,
|
||||
apply is_exact_fiber_sequence
|
||||
end
|
||||
|
||||
/- for a numeral, the carrier of the fiber sequence is definitionally what we want
|
||||
(as pointed sets) -/
|
||||
example : LES_of_homotopy_groups f 6 = π*[2] Y :> Set* := by reflexivity
|
||||
example : LES_of_homotopy_groups f 7 = π*[2] X :> Set* := by reflexivity
|
||||
example : LES_of_homotopy_groups f 8 = π*[2] (pfiber f) :> Set* := by reflexivity
|
||||
|
||||
/- for a numeral, the functions of the fiber sequence is definitionally what we want
|
||||
(as pointed function). All these functions have at most one "pinverse" in them, and these
|
||||
inverses are inside the π→*[2*k].
|
||||
-/
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 6 = π→*[2] f
|
||||
:> (_ →* _) := by reflexivity
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 7 = π→*[2] (ppoint f)
|
||||
:> (_ →* _) := by reflexivity
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 8 = π→*[2] (boundary_map f)
|
||||
:> (_ →* _) := by reflexivity
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 9 = π→*[2] (ap1 f ∘* pinverse)
|
||||
:> (_ →* _) := by reflexivity
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 10 = π→*[2] (ap1 (ppoint f) ∘* pinverse)
|
||||
:> (_ →* _) := by reflexivity
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 11 = π→*[2] (ap1 (boundary_map f) ∘* pinverse)
|
||||
:> (_ →* _) := by reflexivity
|
||||
example : lcc_to_fn (LES_of_homotopy_groups f) 12 = π→*[4] f
|
||||
:> (_ →* _) := by reflexivity
|
||||
|
||||
/- the carrier of the fiber sequence is what we want for natural numbers of the form
|
||||
3n, 3n+1 and 3n+2 -/
|
||||
definition LES_of_homotopy_groups_mul3 (n : ℕ) : LES_of_homotopy_groups f (3 * n) = π*[n] Y :> Set* :=
|
||||
begin
|
||||
apply ptrunctype_eq_of_pType_eq,
|
||||
exact ap (ptrunc 0) (homotopy_groups_mul3 f n)
|
||||
end
|
||||
|
||||
definition LES_of_homotopy_groups_mul3add1 (n : ℕ) : LES_of_homotopy_groups f (3 * n + 1) = π*[n] X :> Set* :=
|
||||
begin
|
||||
apply ptrunctype_eq_of_pType_eq,
|
||||
exact ap (ptrunc 0) (homotopy_groups_mul3add1 f n)
|
||||
end
|
||||
|
||||
definition LES_of_homotopy_groups_mul3add2 (n : ℕ)
|
||||
: LES_of_homotopy_groups f (3 * n + 2) = π*[n] (pfiber f) :> Set* :=
|
||||
begin
|
||||
apply ptrunctype_eq_of_pType_eq,
|
||||
exact ap (ptrunc 0) (homotopy_groups_mul3add2 f n)
|
||||
end
|
||||
|
||||
definition group_LES_of_homotopy_groups (n : ℕ) : group (LES_of_homotopy_groups f (n + 3)) :=
|
||||
group_homotopy_group 0 (homotopy_groups f n)
|
||||
|
||||
definition comm_group_LES_of_homotopy_groups (n : ℕ) : comm_group (LES_of_homotopy_groups f (n + 6)) :=
|
||||
comm_group_homotopy_group 0 (homotopy_groups f n)
|
||||
|
||||
-- TODO: the pointed maps are what we want for 3n, 3n+1 and 3n+2
|
||||
-- TODO: they are group homomorphisms for n+3
|
||||
end chain_complex
|
||||
|
|
@ -1,48 +1,172 @@
|
|||
import types.pointed types.int types.fiber
|
||||
/-
|
||||
Copyright (c) 2016 Floris van Doorn. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Floris van Doorn
|
||||
|
||||
open algebra nat int pointed unit sigma fiber sigma.ops eq equiv prod is_trunc equiv.ops
|
||||
-/
|
||||
|
||||
import types.int types.pointed2 types.trunc
|
||||
|
||||
open eq pointed int unit is_equiv equiv is_trunc trunc equiv.ops
|
||||
|
||||
namespace eq
|
||||
definition transport_eq_Fl_idp_left {A B : Type} {a : A} {b : B} (f : A → B) (q : f a = b)
|
||||
: transport_eq_Fl idp q = !idp_con⁻¹ :=
|
||||
by induction q; reflexivity
|
||||
|
||||
definition whisker_left_idp_con_eq_assoc
|
||||
{A : Type} {a₁ a₂ a₃ : A} (p : a₁ = a₂) (q : a₂ = a₃)
|
||||
: whisker_left p (idp_con q)⁻¹ = con.assoc p idp q :=
|
||||
by induction q; reflexivity
|
||||
|
||||
end eq open eq
|
||||
|
||||
namespace chain_complex
|
||||
|
||||
structure chain_complex : Type :=
|
||||
-- are chain complexes with the "set"-requirement removed interesting?
|
||||
structure type_chain_complex : Type :=
|
||||
(car : ℤ → Type*)
|
||||
(fn : Π{n : ℤ}, car (n + 1) →* car n)
|
||||
(is_chain_complex : Π{n : ℤ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
|
||||
(fn : Π(n : ℤ), car (n + 1) →* car n)
|
||||
(is_chain_complex : Π{n : ℤ} (x : car ((n + 1) + 1)), fn n (fn (n+1) x) = pt)
|
||||
|
||||
structure left_type_chain_complex : Type := -- chain complex on the naturals with maps going down
|
||||
(car : ℕ → Type*)
|
||||
(fn : Π(n : ℕ), car (n + 1) →* car n)
|
||||
(is_chain_complex : Π{n : ℕ} (x : car ((n + 1) + 1)), fn n (fn (n+1) x) = pt)
|
||||
|
||||
structure right_type_chain_complex : Type := -- chain complex on the naturals with maps going up
|
||||
(car : ℕ → Type*)
|
||||
(fn : Π(n : ℕ), car n →* car (n + 1))
|
||||
(is_chain_complex : Π{n : ℕ} (x : car n), fn (n+1) (fn n x) = pt)
|
||||
|
||||
definition tcc_to_car [unfold 1] [coercion] := @type_chain_complex.car
|
||||
definition tcc_to_fn [unfold 1] := @type_chain_complex.fn
|
||||
definition tcc_is_chain_complex [unfold 1] := @type_chain_complex.is_chain_complex
|
||||
definition ltcc_to_car [unfold 1] [coercion] := @left_type_chain_complex.car
|
||||
definition ltcc_to_fn [unfold 1] := @left_type_chain_complex.fn
|
||||
definition ltcc_is_chain_complex [unfold 1] := @left_type_chain_complex.is_chain_complex
|
||||
definition rtcc_to_car [unfold 1] [coercion] := @right_type_chain_complex.car
|
||||
definition rtcc_to_fn [unfold 1] := @right_type_chain_complex.fn
|
||||
definition rtcc_is_chain_complex [unfold 1] := @right_type_chain_complex.is_chain_complex
|
||||
|
||||
-- important: these notions are shifted by one! (this is to avoid transports)
|
||||
definition is_exact_at_t [reducible] (X : type_chain_complex) (n : ℤ) : Type :=
|
||||
Π(x : X (n + 1)), tcc_to_fn X n x = pt → fiber (tcc_to_fn X (n+1)) x
|
||||
definition is_exact_at_lt [reducible] (X : left_type_chain_complex) (n : ℕ) : Type :=
|
||||
Π(x : X (n + 1)), ltcc_to_fn X n x = pt → fiber (ltcc_to_fn X (n+1)) x
|
||||
definition is_exact_at_rt [reducible] (X : right_type_chain_complex) (n : ℕ) : Type :=
|
||||
Π(x : X (n + 1)), rtcc_to_fn X (n+1) x = pt → fiber (rtcc_to_fn X n) x
|
||||
|
||||
definition is_exact_t [reducible] (X : type_chain_complex) : Type :=
|
||||
Π(n : ℤ), is_exact_at_t X n
|
||||
definition is_exact_lt [reducible] (X : left_type_chain_complex) : Type :=
|
||||
Π(n : ℕ), is_exact_at_lt X n
|
||||
definition is_exact_rt [reducible] (X : right_type_chain_complex) : Type :=
|
||||
Π(n : ℕ), is_exact_at_rt X n
|
||||
|
||||
definition type_chain_complex_from_left (X : left_type_chain_complex) : type_chain_complex :=
|
||||
type_chain_complex.mk (int.rec X (λn, punit))
|
||||
begin
|
||||
intro n, fconstructor,
|
||||
{ induction n with n n,
|
||||
{ exact @ltcc_to_fn X n},
|
||||
{ esimp, intro x, exact star}},
|
||||
{ induction n with n n,
|
||||
{ apply respect_pt},
|
||||
{ reflexivity}}
|
||||
end
|
||||
begin
|
||||
intro n, induction n with n n,
|
||||
{ exact ltcc_is_chain_complex X},
|
||||
{ esimp, intro x, reflexivity}
|
||||
end
|
||||
|
||||
definition is_exact_t_from_left {X : left_type_chain_complex} {n : ℕ} (H : is_exact_at_lt X n)
|
||||
: is_exact_at_t (type_chain_complex_from_left X) n :=
|
||||
H
|
||||
|
||||
definition transfer_left_type_chain_complex [constructor] (X : left_type_chain_complex)
|
||||
{Y : ℕ → Type*} (g : Π{n : ℕ}, Y (n + 1) →* Y n) (e : Π{n}, X n ≃* Y n)
|
||||
(p : Π{n} (x : X (n + 1)), e (ltcc_to_fn X n x) = g (e x)) : left_type_chain_complex :=
|
||||
left_type_chain_complex.mk Y @g
|
||||
begin
|
||||
intro n, apply equiv_rect (equiv_of_pequiv e), intro x,
|
||||
refine ap g (p x)⁻¹ ⬝ _,
|
||||
refine (p _)⁻¹ ⬝ _,
|
||||
refine ap e (ltcc_is_chain_complex X _) ⬝ _,
|
||||
apply respect_pt
|
||||
end
|
||||
|
||||
definition is_exact_at_lt_transfer {X : left_type_chain_complex} {Y : ℕ → Type*}
|
||||
{g : Π{n : ℕ}, Y (n + 1) →* Y n} (e : Π{n}, X n ≃* Y n)
|
||||
(p : Π{n} (x : X (n + 1)), e (ltcc_to_fn X n x) = g (e x)) {n : ℕ}
|
||||
(H : is_exact_at_lt X n) : is_exact_at_lt (transfer_left_type_chain_complex X @g @e @p) n :=
|
||||
begin
|
||||
intro y q, esimp at *,
|
||||
assert H2 : ltcc_to_fn X n (e⁻¹ᵉ* y) = pt,
|
||||
{ refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
|
||||
refine ap _ q ⬝ _,
|
||||
exact respect_pt e⁻¹ᵉ*},
|
||||
cases (H _ H2) with x r,
|
||||
refine fiber.mk (e x) _,
|
||||
refine (p x)⁻¹ ⬝ _,
|
||||
refine ap e r ⬝ _,
|
||||
apply right_inv
|
||||
end
|
||||
|
||||
definition trunc_left_type_chain_complex [constructor] (X : left_type_chain_complex)
|
||||
(k : trunc_index) : left_type_chain_complex :=
|
||||
left_type_chain_complex.mk
|
||||
(λn, ptrunc k (X n))
|
||||
(λn, ptrunc_functor k (ltcc_to_fn X n))
|
||||
begin
|
||||
intro n x, esimp at *,
|
||||
refine trunc.rec _ x, -- why doesn't induction work here?
|
||||
clear x, intro x, esimp,
|
||||
exact ap tr (ltcc_is_chain_complex X x)
|
||||
end
|
||||
|
||||
/- actual (set) chain complexes -/
|
||||
|
||||
structure chain_complex : Type :=
|
||||
(car : ℤ → Set*)
|
||||
(fn : Π(n : ℤ), car (n + 1) →* car n)
|
||||
(is_chain_complex : Π{n : ℤ} (x : car ((n + 1) + 1)), fn n (fn (n+1) x) = pt)
|
||||
|
||||
structure left_chain_complex : Type := -- chain complex on the naturals with maps going down
|
||||
(car : ℕ → Type*)
|
||||
(fn : Π{n : ℕ}, car (n + 1) →* car n)
|
||||
(is_chain_complex : Π{n : ℕ} (x : car ((n + 1) + 1)), fn (fn x) = pt)
|
||||
(car : ℕ → Set*)
|
||||
(fn : Π(n : ℕ), car (n + 1) →* car n)
|
||||
(is_chain_complex : Π{n : ℕ} (x : car ((n + 1) + 1)), fn n (fn (n+1) x) = pt)
|
||||
|
||||
structure right_chain_complex : Type := -- chain complex on the naturals with maps going up
|
||||
(car : ℕ → Type*)
|
||||
(fn : Π{n : ℕ}, car n →* car (n + 1))
|
||||
(is_chain_complex : Π{n : ℕ} (x : car n), fn (fn x) = pt)
|
||||
(car : ℕ → Set*)
|
||||
(fn : Π(n : ℕ), car n →* car (n + 1))
|
||||
(is_chain_complex : Π{n : ℕ} (x : car n), fn (n+1) (fn n x) = pt)
|
||||
|
||||
definition cc_to_car [coercion] := @chain_complex.car
|
||||
definition cc_to_fn := @chain_complex.fn
|
||||
definition cc_is_chain_complex := @chain_complex.is_chain_complex
|
||||
definition lcc_to_car [coercion] := @left_chain_complex.car
|
||||
definition lcc_to_fn := @left_chain_complex.fn
|
||||
definition lcc_is_chain_complex := @left_chain_complex.is_chain_complex
|
||||
definition rcc_to_car [coercion] := @right_chain_complex.car
|
||||
definition rcc_to_fn := @right_chain_complex.fn
|
||||
definition rcc_is_chain_complex := @right_chain_complex.is_chain_complex
|
||||
definition cc_to_car [unfold 1] [coercion] := @chain_complex.car
|
||||
definition cc_to_fn [unfold 1] := @chain_complex.fn
|
||||
definition cc_is_chain_complex [unfold 1] := @chain_complex.is_chain_complex
|
||||
definition lcc_to_car [unfold 1] [coercion] := @left_chain_complex.car
|
||||
definition lcc_to_fn [unfold 1] := @left_chain_complex.fn
|
||||
definition lcc_is_chain_complex [unfold 1] := @left_chain_complex.is_chain_complex
|
||||
definition rcc_to_car [unfold 1] [coercion] := @right_chain_complex.car
|
||||
definition rcc_to_fn [unfold 1] := @right_chain_complex.fn
|
||||
definition rcc_is_chain_complex [unfold 1] := @right_chain_complex.is_chain_complex
|
||||
|
||||
-- note: these notions are shifted by one!
|
||||
-- important: these notions are shifted by one! (this is to avoid transports)
|
||||
definition is_exact_at [reducible] (X : chain_complex) (n : ℤ) : Type :=
|
||||
Π(x : X (n + 1)), cc_to_fn X x = pt → Σ(y : X ((n + 1) + 1)), cc_to_fn X y = x
|
||||
Π(x : X (n + 1)), cc_to_fn X n x = pt → image (cc_to_fn X (n+1)) x
|
||||
definition is_exact_at_l [reducible] (X : left_chain_complex) (n : ℕ) : Type :=
|
||||
Π(x : X (n + 1)), lcc_to_fn X x = pt → Σ(y : X ((n + 1) + 1)), lcc_to_fn X y = x
|
||||
Π(x : X (n + 1)), lcc_to_fn X n x = pt → image (lcc_to_fn X (n+1)) x
|
||||
definition is_exact_at_r [reducible] (X : right_chain_complex) (n : ℕ) : Type :=
|
||||
Π(x : X (n + 1)), rcc_to_fn X x = pt → Σ(y : X n), rcc_to_fn X y = x
|
||||
Π(x : X (n + 1)), rcc_to_fn X (n+1) x = pt → image (rcc_to_fn X n) x
|
||||
|
||||
definition is_exact [reducible] (X : chain_complex) : Type := Π(n : ℤ), is_exact_at X n
|
||||
definition is_exact_l [reducible] (X : left_chain_complex) : Type := Π(n : ℕ), is_exact_at_l X n
|
||||
definition is_exact_r [reducible] (X : right_chain_complex) : Type := Π(n : ℕ), is_exact_at_r X n
|
||||
|
||||
definition chain_complex_from_left (X : left_chain_complex) : chain_complex :=
|
||||
chain_complex.mk (int.rec X (λn, Unit))
|
||||
chain_complex.mk (int.rec X (λn, punit))
|
||||
begin
|
||||
intro n, fconstructor,
|
||||
{ induction n with n n,
|
||||
|
|
@ -62,121 +186,58 @@ namespace chain_complex
|
|||
: is_exact_at (chain_complex_from_left X) n :=
|
||||
H
|
||||
|
||||
-- move to pointed
|
||||
definition pfiber [constructor] {X Y : Type*} (f : X →* Y) : Type* :=
|
||||
pointed.MK (fiber f pt) (fiber.mk pt !respect_pt)
|
||||
|
||||
definition pequiv_of_equiv [constructor] {A B : Type*} (f : A ≃ B) (H : f pt = pt) : A ≃* B :=
|
||||
pequiv.mk' (pmap.mk f H)
|
||||
|
||||
definition fiber_sequence_helper [constructor] (v : Σ(X Y : Type*), X →* Y)
|
||||
: Σ(Z X : Type*), Z →* X :=
|
||||
⟨pfiber v.2.2, v.1, pmap.mk point rfl⟩
|
||||
|
||||
definition fiber_sequence_carrier {X Y : Type*} (f : X →* Y) (n : ℕ) : Type* :=
|
||||
nat.cases_on n Y (λk, (iterate fiber_sequence_helper k ⟨X, Y, f⟩).1)
|
||||
|
||||
definition fiber_sequence_fun {X Y : Type*} (f : X →* Y) (n : ℕ)
|
||||
: fiber_sequence_carrier f (n + 1) →* fiber_sequence_carrier f n :=
|
||||
nat.cases_on n f proof (λk, pmap.mk point rfl) qed
|
||||
|
||||
/- Definition 8.4.3 -/
|
||||
definition fiber_sequence.{u} {X Y : Pointed.{u}} (f : X →* Y) : left_chain_complex.{u} :=
|
||||
begin
|
||||
fconstructor,
|
||||
{ exact fiber_sequence_carrier f},
|
||||
{ exact fiber_sequence_fun f},
|
||||
{ intro n x, cases n with n,
|
||||
{ exact point_eq x},
|
||||
{ exact point_eq x}}
|
||||
end
|
||||
|
||||
definition is_exact_fiber_sequence {X Y : Type*} (f : X →* Y) : is_exact_l (fiber_sequence f) :=
|
||||
begin
|
||||
intro n x p, cases n with n,
|
||||
{ exact ⟨fiber.mk x p, rfl⟩},
|
||||
{ exact ⟨fiber.mk x p, rfl⟩}
|
||||
end
|
||||
|
||||
-- move to types.sigma
|
||||
definition sigma_assoc_comm_equiv [constructor] {A : Type} (B C : A → Type)
|
||||
: (Σ(v : Σa, B a), C v.1) ≃ (Σ(u : Σa, C a), B u.1) :=
|
||||
calc (Σ(v : Σa, B a), C v.1)
|
||||
≃ (Σa (b : B a), C a) : !sigma_assoc_equiv⁻¹
|
||||
... ≃ (Σa, B a × C a) : sigma_equiv_sigma_id (λa, !equiv_prod)
|
||||
... ≃ (Σa, C a × B a) : sigma_equiv_sigma_id (λa, !prod_comm_equiv)
|
||||
... ≃ (Σa (c : C a), B a) : sigma_equiv_sigma_id (λa, !equiv_prod)
|
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... ≃ (Σ(u : Σa, C a), B u.1) : sigma_assoc_equiv
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attribute is_equiv_sigma_functor is_equiv.is_equiv_id pequiv.mk' [constructor]
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attribute sigma.eta [unfold 3]
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-- set_option pp.notation false
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/- Lemma 8.4.4(i) -/
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definition fiber_sequence_carrier_equiv0.{u} {X Y : Pointed.{u}} (f : X →* Y)
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: fiber_sequence_carrier f 3 ≃* Ω Y :=
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pequiv_of_equiv
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(calc
|
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fiber_sequence_carrier f 3 ≃ fiber (fiber_sequence_fun f 1) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f 2), fiber_sequence_fun f 1 x = pt : fiber.sigma_char
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... ≃ Σ(v : fiber f pt), fiber_sequence_fun f 1 v = pt : erfl
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... ≃ Σ(v : Σ(x : X), f x = pt), fiber_sequence_fun f 1 (fiber.mk v.1 v.2) = pt
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: sigma_equiv_sigma_left !fiber.sigma_char
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... ≃ Σ(v : Σ(x : X), f x = pt), v.1 = pt : erfl
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... ≃ Σ(v : Σ(x : X), x = pt), f v.1 = pt : sigma_assoc_comm_equiv
|
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... ≃ f !center.1 = pt : sigma_equiv_of_is_contr_left _
|
||||
... ≃ f pt = pt : erfl
|
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... ≃ pt = pt : by exact !equiv_eq_closed_left !respect_pt
|
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... ≃ Ω Y : erfl)
|
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definition transfer_left_chain_complex [constructor] (X : left_chain_complex) {Y : ℕ → Set*}
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(g : Π{n : ℕ}, Y (n + 1) →* Y n) (e : Π{n}, X n ≃* Y n)
|
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(p : Π{n} (x : X (n + 1)), e (lcc_to_fn X n x) = g (e x)) : left_chain_complex :=
|
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left_chain_complex.mk Y @g
|
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begin
|
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change (respect_pt f)⁻¹ ⬝
|
||||
((center_eq ⟨Pointed.Point X, refl (Pointed.Point X)⟩)⁻¹ ▸ respect_pt f) = idp,
|
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rewrite tr_constant,
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apply con.left_inv
|
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intro n, apply equiv_rect (equiv_of_pequiv e), intro x,
|
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refine ap g (p x)⁻¹ ⬝ _,
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refine (p _)⁻¹ ⬝ _,
|
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refine ap e (lcc_is_chain_complex X _) ⬝ _,
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||||
apply respect_pt
|
||||
end
|
||||
|
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/- (generalization of) Lemma 8.4.4(ii) -/
|
||||
definition fiber_sequence_carrier_equiv1.{u} {X Y : Pointed.{u}} (f : X →* Y) (n : ℕ)
|
||||
: fiber_sequence_carrier f (n+4) ≃* Ω(fiber_sequence_carrier f (n+1)) :=
|
||||
pequiv_of_equiv
|
||||
(calc
|
||||
fiber_sequence_carrier f (n+4) ≃ fiber (fiber_sequence_fun f (n+2)) pt : erfl
|
||||
... ≃ Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f (n+2) x = pt
|
||||
: fiber.sigma_char
|
||||
... ≃ Σ(x : fiber (fiber_sequence_fun f (n+1)) pt), fiber_sequence_fun f _ x = pt
|
||||
: erfl
|
||||
... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
|
||||
fiber_sequence_fun f _ (fiber.mk v.1 v.2) = pt
|
||||
: by exact sigma_equiv_sigma !fiber.sigma_char (λa, erfl)
|
||||
... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
|
||||
v.1 = pt
|
||||
: erfl
|
||||
... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), x = pt),
|
||||
fiber_sequence_fun f _ v.1 = pt
|
||||
: sigma_assoc_comm_equiv
|
||||
... ≃ fiber_sequence_fun f _ !center.1 = pt
|
||||
: @(sigma_equiv_of_is_contr_left _) !is_contr_sigma_eq'
|
||||
... ≃ fiber_sequence_fun f _ pt = pt
|
||||
: erfl
|
||||
... ≃ pt = pt
|
||||
: by exact !equiv_eq_closed_left !respect_pt
|
||||
... ≃ Ω(fiber_sequence_carrier f (n+1)) : erfl)
|
||||
begin reflexivity end
|
||||
|
||||
/- Lemma 8.4.4 (i)(ii), combined -/
|
||||
definition fiber_sequence_carrier_equiv {X Y : Type*} (f : X →* Y) (n : ℕ)
|
||||
: fiber_sequence_carrier f (n+3) ≃* Ω(fiber_sequence_carrier f n) :=
|
||||
nat.cases_on n (fiber_sequence_carrier_equiv0 f) (fiber_sequence_carrier_equiv1 f)
|
||||
exit
|
||||
/- Lemma 8.4.4(iii) -/
|
||||
definition fiber_sequence_function0 {X Y : Type*} (f : X →* Y)
|
||||
: Π(x : fiber_sequence_carrier f 4), ap1 f (fiber_sequence_carrier_equiv f 1 x)⁻¹ᵖ =
|
||||
fiber_sequence_carrier_equiv f 0 (fiber_sequence_fun f 3 x) :=
|
||||
take (x : fiber (fiber_sequence_fun f 2) pt),
|
||||
obtain (v : fiber (fiber_sequence_fun f 1) pt) (q : _), from x,
|
||||
definition transfer_is_exact_at_l (X : left_chain_complex) {Y : ℕ → Set*}
|
||||
(g : Π{n : ℕ}, Y (n + 1) →* Y n) (e : Π{n}, X n ≃* Y n)
|
||||
(p : Π{n} (x : X (n + 1)), e (lcc_to_fn X n x) = g (e x))
|
||||
{n : ℕ} (H : is_exact_at_l X n) : is_exact_at_l (transfer_left_chain_complex X @g @e @p) n :=
|
||||
begin
|
||||
unfold [fiber_sequence_carrier_equiv,fiber_sequence_carrier_equiv0,fiber_sequence_carrier_equiv1,equiv.trans, equiv.symm, pequiv._trans_of_to_pmap],
|
||||
esimp [sigma_assoc_equiv, equiv.symm, equiv.trans], unfold [fiber_sequence_fun, fiber_sequence_carrier]
|
||||
intro y q, esimp at *,
|
||||
assert H2 : lcc_to_fn X n (e⁻¹ᵉ* y) = pt,
|
||||
{ refine (inv_commute (λn, equiv_of_pequiv e) _ _ @p _)⁻¹ᵖ ⬝ _,
|
||||
refine ap _ q ⬝ _,
|
||||
exact respect_pt e⁻¹ᵉ*},
|
||||
induction (H _ H2) with x,
|
||||
induction x with x r,
|
||||
refine image.mk (e x) _,
|
||||
refine (p x)⁻¹ ⬝ _,
|
||||
refine ap e r ⬝ _,
|
||||
apply right_inv
|
||||
end
|
||||
|
||||
definition trunc_left_chain_complex [constructor] (X : left_type_chain_complex)
|
||||
: left_chain_complex :=
|
||||
left_chain_complex.mk
|
||||
(λn, ptrunc 0 (X n))
|
||||
(λn, ptrunc_functor 0 (ltcc_to_fn X n))
|
||||
begin
|
||||
intro n x, esimp at *,
|
||||
refine @trunc.rec _ _ _ (λH, !is_trunc_eq) _ x,
|
||||
clear x, intro x, esimp,
|
||||
exact ap tr (ltcc_is_chain_complex X x)
|
||||
end
|
||||
|
||||
definition is_exact_at_l_trunc (X : left_type_chain_complex) {n : ℕ}
|
||||
(H : is_exact_at_lt X n) : is_exact_at_l (trunc_left_chain_complex X) n :=
|
||||
begin
|
||||
intro x p, esimp at *,
|
||||
induction x with x, esimp at *,
|
||||
note q := !tr_eq_tr_equiv p,
|
||||
induction q with q,
|
||||
induction H x q with y r,
|
||||
refine image.mk (tr y) _,
|
||||
esimp, exact ap tr r
|
||||
end
|
||||
|
||||
end chain_complex
|
||||
|
|
|
|||
Loading…
Reference in a new issue