37fbe56b8c
The maps on every level are just the functorial action of the homotopy groups (possibly composed by a cast), but there are no compositions with path inversion. There are also some updates in various files after changes in the HoTT library.
389 lines
17 KiB
Text
389 lines
17 KiB
Text
import .LES_of_homotopy_groups
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open eq pointed is_trunc trunc_index trunc group is_equiv equiv algebra prod fin fiber nat
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succ_str chain_complex
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/--------------
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PART 3'
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--------------/
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namespace chain_complex'
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section
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universe variable u
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parameters {X Y : pType.{u}} (f : X →* Y)
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definition homotopy_groups2 [reducible] : +3ℕ → Type*
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| (n, fin.mk 0 H) := Ω[n] Y
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| (n, fin.mk 1 H) := Ω[n] X
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| (n, fin.mk k H) := Ω[n] (pfiber f)
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definition homotopy_groups2_add1 (n : ℕ) : Π(x : fin (succ 2)),
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homotopy_groups2 (n+1, x) = Ω(homotopy_groups2 (n, x))
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| (fin.mk 0 H) := by reflexivity
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| (fin.mk 1 H) := by reflexivity
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| (fin.mk 2 H) := by reflexivity
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| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition homotopy_groups_fun2 : Π(n : +3ℕ), homotopy_groups2 (S n) →* homotopy_groups2 n
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| (n, fin.mk 0 H) := proof Ω→[n] f qed
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| (n, fin.mk 1 H) := proof Ω→[n] (ppoint f) qed
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| (n, fin.mk 2 H) :=
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proof Ω→[n] (boundary_map f) ∘* pcast (loop_space_succ_eq_in Y n) qed
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| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition homotopy_groups_fun2_add1_0 (n : ℕ) (H : 0 < succ 2)
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: homotopy_groups_fun2 (n+1, fin.mk 0 H) ~*
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cast proof idp qed ap1 (homotopy_groups_fun2 (n, fin.mk 0 H)) :=
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by reflexivity
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definition homotopy_groups_fun2_add1_1 (n : ℕ) (H : 1 < succ 2)
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: homotopy_groups_fun2 (n+1, fin.mk 1 H) ~*
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cast proof idp qed ap1 (homotopy_groups_fun2 (n, fin.mk 1 H)) :=
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by reflexivity
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definition homotopy_groups_fun2_add1_2 (n : ℕ) (H : 2 < succ 2)
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: homotopy_groups_fun2 (n+1, fin.mk 2 H) ~*
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cast proof idp qed ap1 (homotopy_groups_fun2 (n, fin.mk 2 H)) :=
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begin
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esimp,
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refine _ ⬝* !ap1_compose⁻¹*,
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exact pwhisker_left _ !pcast_ap_loop_space,
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end
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exit
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definition nat_of_str_3S [unfold 2] [reducible]
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: Π(x : stratified +ℕ 2), nat_of_str x + 1 = nat_of_str (@S (stratified +ℕ 2) x)
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| (n, fin.mk 0 H) := by reflexivity
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| (n, fin.mk 1 H) := by reflexivity
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| (n, fin.mk 2 H) := by reflexivity
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| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition homotopy_groups2_pequiv : Π(x : +3ℕ),
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homotopy_groups (nat_of_str x) ≃* homotopy_groups2 x
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| (0, (fin.mk 0 H)) := by reflexivity
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| (0, (fin.mk 1 H)) := by reflexivity
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| (0, (fin.mk 2 H)) := by reflexivity
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| ((n+1), (fin.mk 0 H)) :=
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begin
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-- uncomment the next two lines to have prettier subgoals
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-- esimp, replace (succ 5 * (n + 1) + 0) with (6*n+3+3),
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-- rewrite [+homotopy_groups_add3, homotopy_groups2_add1],
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apply loop_pequiv_loop,
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rexact homotopy_groups2_pequiv (n, fin.mk 0 H)
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end
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| ((n+1), (fin.mk 1 H)) :=
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begin
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apply loop_pequiv_loop,
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rexact homotopy_groups2_pequiv (n, fin.mk 1 H)
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end
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| ((n+1), (fin.mk 2 H)) :=
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begin
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apply loop_pequiv_loop,
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rexact homotopy_groups2_pequiv (n, fin.mk 2 H)
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end
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| (n, (fin.mk (k+3) H)) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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-- | (n, x) := homotopy_groups2_pequiv' n x
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/- all cases where n>0 are basically the same -/
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definition homotopy_groups_fun2_phomotopy (x : +6ℕ) :
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homotopy_groups2_pequiv x ∘* homotopy_groups_fun (nat_of_str x) ~*
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(homotopy_groups_fun2 x ∘* homotopy_groups2_pequiv (S x))
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∘* pcast (ap (homotopy_groups f) (nat_of_str_6S x)) :=
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begin
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cases x with n x, cases x with k H,
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cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 1,
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cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 2,
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{ /-k=0-/
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induction n with n IH,
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{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
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reflexivity},
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{ refine _ ⬝* !comp_pid⁻¹*,
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refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_0)⁻¹*,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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exact IH ⬝* !comp_pid}},
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{ /-k=1-/
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induction n with n IH,
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{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
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reflexivity},
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{ refine _ ⬝* !comp_pid⁻¹*,
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refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_1)⁻¹*,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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exact IH ⬝* !comp_pid}},
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{ /-k=2-/
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induction n with n IH,
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{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
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refine _ ⬝* !comp_pid⁻¹*,
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reflexivity},
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{ refine _ ⬝* !comp_pid⁻¹*,
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refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_2)⁻¹*,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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exact IH ⬝* !comp_pid}},
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{ /-k=3-/
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induction n with n IH,
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{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
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reflexivity},
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{ refine _ ⬝* !comp_pid⁻¹*,
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refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_3)⁻¹*,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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exact IH ⬝* !comp_pid}},
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{ /-k=4-/
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induction n with n IH,
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{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
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reflexivity},
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{ refine _ ⬝* !comp_pid⁻¹*,
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refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_4)⁻¹*,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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exact IH ⬝* !comp_pid}},
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{ /-k=5-/
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induction n with n IH,
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{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
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refine !comp_pid⁻¹* ⬝* pconcat2 _ _,
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{ exact !comp_pid⁻¹*},
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{ refine cast (ap (λx, _ ~* loop_pequiv_loop x) !loopn_pequiv_loopn_rfl)⁻¹ _,
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refine cast (ap (λx, _ ~* x) !loopn_pequiv_loopn_rfl)⁻¹ _, reflexivity}},
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{ refine _ ⬝* !comp_pid⁻¹*,
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refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_5)⁻¹*,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
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exact IH ⬝* !comp_pid}},
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{ /-k=k'+6-/ exfalso, apply lt_le_antisymm H, apply le_add_left}
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end
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definition type_LES_of_homotopy_groups2 [constructor] : type_chain_complex +6ℕ :=
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transfer_type_chain_complex2
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(type_LES_of_homotopy_groups f)
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!fin_prod_nat_equiv_nat
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nat_of_str_6S
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@homotopy_groups_fun2
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@homotopy_groups2_pequiv
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begin
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intro m x,
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refine homotopy_groups_fun2_phomotopy m x ⬝ _,
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apply ap (homotopy_groups_fun2 m), apply ap (homotopy_groups2_pequiv (S m)),
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esimp, exact ap010 cast !ap_compose⁻¹ x
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end
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definition is_exact_type_LES_of_homotopy_groups2 : is_exact_t (type_LES_of_homotopy_groups2) :=
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begin
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intro n,
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apply is_exact_at_transfer2,
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apply is_exact_type_LES_of_homotopy_groups
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end
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definition LES_of_homotopy_groups2 [constructor] : chain_complex +6ℕ :=
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trunc_chain_complex type_LES_of_homotopy_groups2
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/--------------
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PART 4'
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--------------/
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definition homotopy_groups3 [reducible] : +6ℕ → Set*
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| (n, fin.mk 0 H) := π*[2*n] Y
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| (n, fin.mk 1 H) := π*[2*n] X
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| (n, fin.mk 2 H) := π*[2*n] (pfiber f)
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| (n, fin.mk 3 H) := π*[2*n + 1] Y
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| (n, fin.mk 4 H) := π*[2*n + 1] X
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| (n, fin.mk k H) := π*[2*n + 1] (pfiber f)
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definition homotopy_groups3eq2 [reducible]
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: Π(n : +6ℕ), ptrunc 0 (homotopy_groups2 n) ≃* homotopy_groups3 n
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| (n, fin.mk 0 H) := by reflexivity
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| (n, fin.mk 1 H) := by reflexivity
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| (n, fin.mk 2 H) := by reflexivity
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| (n, fin.mk 3 H) := by reflexivity
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| (n, fin.mk 4 H) := by reflexivity
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| (n, fin.mk 5 H) := by reflexivity
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| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition homotopy_groups_fun3 : Π(n : +6ℕ), homotopy_groups3 (S n) →* homotopy_groups3 n
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| (n, fin.mk 0 H) := proof π→*[2*n] f qed
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| (n, fin.mk 1 H) := proof π→*[2*n] (ppoint f) qed
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| (n, fin.mk 2 H) :=
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proof π→*[2*n] (boundary_map f) ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n))) qed
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| (n, fin.mk 3 H) := proof π→*[2*n + 1] f ∘* tinverse qed
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| (n, fin.mk 4 H) := proof π→*[2*n + 1] (ppoint f) ∘* tinverse qed
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| (n, fin.mk 5 H) :=
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proof (π→*[2*n + 1] (boundary_map f) ∘* tinverse)
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∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n+1))) qed
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| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition homotopy_groups_fun3eq2 [reducible]
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: Π(n : +6ℕ), homotopy_groups3eq2 n ∘* ptrunc_functor 0 (homotopy_groups_fun2 n) ~*
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homotopy_groups_fun3 n ∘* homotopy_groups3eq2 (S n)
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| (n, fin.mk 0 H) := by reflexivity
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| (n, fin.mk 1 H) := by reflexivity
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| (n, fin.mk 2 H) :=
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begin
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refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
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refine !ptrunc_functor_pcompose ⬝* _,
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apply pwhisker_left, apply ptrunc_functor_pcast,
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end
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| (n, fin.mk 3 H) :=
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begin
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refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
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refine !ptrunc_functor_pcompose ⬝* _,
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apply pwhisker_left, apply ptrunc_functor_pinverse
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end
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| (n, fin.mk 4 H) :=
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begin
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refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
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refine !ptrunc_functor_pcompose ⬝* _,
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apply pwhisker_left, apply ptrunc_functor_pinverse
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end
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| (n, fin.mk 5 H) :=
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begin
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refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
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refine !ptrunc_functor_pcompose ⬝* _,
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apply pconcat2,
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{ refine !ptrunc_functor_pcompose ⬝* _,
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apply pwhisker_left, apply ptrunc_functor_pinverse},
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{ apply ptrunc_functor_pcast}
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end
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| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition LES_of_homotopy_groups3 [constructor] : chain_complex +6ℕ :=
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transfer_chain_complex
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LES_of_homotopy_groups2
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homotopy_groups_fun3
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homotopy_groups3eq2
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homotopy_groups_fun3eq2
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definition is_exact_LES_of_homotopy_groups3 : is_exact (LES_of_homotopy_groups3) :=
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begin
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intro n,
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apply is_exact_at_transfer,
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apply is_exact_at_trunc,
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apply is_exact_type_LES_of_homotopy_groups2
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end
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end
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open is_trunc
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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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/- the carrier of the fiber sequence is definitionally what we want (as pointed sets) -/
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example : LES_of_homotopy_groups3 f (str_of_nat 6) = π*[2] Y :> Set* := by reflexivity
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example : LES_of_homotopy_groups3 f (str_of_nat 7) = π*[2] X :> Set* := by reflexivity
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example : LES_of_homotopy_groups3 f (str_of_nat 8) = π*[2] (pfiber f) :> Set* := by reflexivity
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example : LES_of_homotopy_groups3 f (str_of_nat 9) = π*[3] Y :> Set* := by reflexivity
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example : LES_of_homotopy_groups3 f (str_of_nat 10) = π*[3] X :> Set* := by reflexivity
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example : LES_of_homotopy_groups3 f (str_of_nat 11) = π*[3] (pfiber f) :> Set* := by reflexivity
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definition LES_of_homotopy_groups3_0 : LES_of_homotopy_groups3 f (n, 0) = π*[2*n] Y :=
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by reflexivity
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definition LES_of_homotopy_groups3_1 : LES_of_homotopy_groups3 f (n, 1) = π*[2*n] X :=
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by reflexivity
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definition LES_of_homotopy_groups3_2 : LES_of_homotopy_groups3 f (n, 2) = π*[2*n] (pfiber f) :=
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by reflexivity
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definition LES_of_homotopy_groups3_3 : LES_of_homotopy_groups3 f (n, 3) = π*[2*n + 1] Y :=
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by reflexivity
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definition LES_of_homotopy_groups3_4 : LES_of_homotopy_groups3 f (n, 4) = π*[2*n + 1] X :=
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by reflexivity
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definition LES_of_homotopy_groups3_5 : LES_of_homotopy_groups3 f (n, 5) = π*[2*n + 1] (pfiber f):=
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by reflexivity
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/- the functions of the fiber sequence is definitionally what we want (as pointed function).
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-/
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definition LES_of_homotopy_groups_fun3_0 :
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cc_to_fn (LES_of_homotopy_groups3 f) (n, 0) = π→*[2*n] f :=
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by reflexivity
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definition LES_of_homotopy_groups_fun3_1 :
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cc_to_fn (LES_of_homotopy_groups3 f) (n, 1) = π→*[2*n] (ppoint f) :=
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by reflexivity
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definition LES_of_homotopy_groups_fun3_2 : cc_to_fn (LES_of_homotopy_groups3 f) (n, 2) =
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π→*[2*n] (boundary_map f) ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n))) :=
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by reflexivity
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definition LES_of_homotopy_groups_fun3_3 :
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cc_to_fn (LES_of_homotopy_groups3 f) (n, 3) = π→*[2*n + 1] f ∘* tinverse :=
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by reflexivity
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definition LES_of_homotopy_groups_fun3_4 :
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cc_to_fn (LES_of_homotopy_groups3 f) (n, 4) = π→*[2*n + 1] (ppoint f) ∘* tinverse :=
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by reflexivity
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definition LES_of_homotopy_groups_fun3_5 : cc_to_fn (LES_of_homotopy_groups3 f) (n, 5) =
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(π→*[2*n + 1] (boundary_map f) ∘* tinverse) ∘*
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pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n+1))) :=
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by reflexivity
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definition group_LES_of_homotopy_groups3_0 :
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Π(k : ℕ) (H : k + 3 < succ 5), group (LES_of_homotopy_groups3 f (0, fin.mk (k+3) H))
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| 0 H := begin rexact group_homotopy_group 0 Y end
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| 1 H := begin rexact group_homotopy_group 0 X end
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| 2 H := begin rexact group_homotopy_group 0 (pfiber f) end
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| (k+3) H := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition comm_group_LES_of_homotopy_groups3 (n : ℕ) : Π(x : fin (succ 5)),
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comm_group (LES_of_homotopy_groups3 f (n + 1, x))
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| (fin.mk 0 H) := proof comm_group_homotopy_group (2*n) Y qed
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| (fin.mk 1 H) := proof comm_group_homotopy_group (2*n) X qed
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| (fin.mk 2 H) := proof comm_group_homotopy_group (2*n) (pfiber f) qed
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| (fin.mk 3 H) := proof comm_group_homotopy_group (2*n+1) Y qed
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| (fin.mk 4 H) := proof comm_group_homotopy_group (2*n+1) X qed
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| (fin.mk 5 H) := proof comm_group_homotopy_group (2*n+1) (pfiber f) qed
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| (fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition CommGroup_LES_of_homotopy_groups3 (n : +6ℕ) : CommGroup.{u} :=
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CommGroup.mk (LES_of_homotopy_groups3 f (pr1 n + 1, pr2 n))
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(comm_group_LES_of_homotopy_groups3 f (pr1 n) (pr2 n))
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definition homomorphism_LES_of_homotopy_groups_fun3 : Π(k : +6ℕ),
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CommGroup_LES_of_homotopy_groups3 f (S k) →g CommGroup_LES_of_homotopy_groups3 f k
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| (k, fin.mk 0 H) :=
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proof homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 0))
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(phomotopy_group_functor_mul _ _) qed
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| (k, fin.mk 1 H) :=
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proof homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 1))
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(phomotopy_group_functor_mul _ _) qed
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| (k, fin.mk 2 H) :=
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begin
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apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 2)),
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exact abstract begin rewrite [LES_of_homotopy_groups_fun3_2],
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refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1)] boundary_map f) _ _ _,
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{ apply group_homotopy_group ((2 * k) + 1)},
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{ apply phomotopy_group_functor_mul},
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{ rewrite [▸*, -ap_compose', ▸*],
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apply is_homomorphism_cast_loop_space_succ_eq_in} end end
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end
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| (k, fin.mk 3 H) :=
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begin
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apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 3)),
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exact abstract begin rewrite [LES_of_homotopy_groups_fun3_3],
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refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1) + 1] f) tinverse _ _,
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{ apply group_homotopy_group (2 * (k+1))},
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{ apply phomotopy_group_functor_mul},
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{ apply is_homomorphism_inverse} end end
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end
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| (k, fin.mk 4 H) :=
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begin
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apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 4)),
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exact abstract begin rewrite [LES_of_homotopy_groups_fun3_4],
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refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1) + 1] (ppoint f)) tinverse _ _,
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{ apply group_homotopy_group (2 * (k+1))},
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{ apply phomotopy_group_functor_mul},
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{ apply is_homomorphism_inverse} end end
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||
end
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| (k, fin.mk 5 H) :=
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begin
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||
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 5)),
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exact abstract begin rewrite [LES_of_homotopy_groups_fun3_5],
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refine @is_homomorphism_compose _ _ _ _ _ _
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(π→*[2 * (k + 1) + 1] (boundary_map f) ∘ tinverse) _ _ _,
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{ refine @is_homomorphism_compose _ _ _ _ _ _
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||
(π→*[2 * (k + 1) + 1] (boundary_map f)) tinverse _ _,
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||
{ apply group_homotopy_group (2 * (k+1))},
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{ apply phomotopy_group_functor_mul},
|
||
{ apply is_homomorphism_inverse}},
|
||
{ rewrite [▸*, -ap_compose', ▸*],
|
||
apply is_homomorphism_cast_loop_space_succ_eq_in} end end
|
||
end
|
||
| (k, fin.mk (l+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
--TODO: the maps 3, 4 and 5 are anti-homomorphisms.
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||
|
||
end chain_complex'
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