63ec1b8d37
Note: the Serre spectral sequence only works for unreduced cohomology, so we need some results for that For reduced homology we might get a similar result if we replace the sigma in the RHS by a dependent version of the smash product
580 lines
25 KiB
Text
580 lines
25 KiB
Text
-- Authors: Floris van Doorn
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import homotopy.EM algebra.category.functor.equivalence types.pointed2 ..pointed_pi ..pointed
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..move_to_lib
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open eq equiv is_equiv algebra group nat pointed EM.ops is_trunc trunc susp function is_conn
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/- TODO: try to fix the speed of this file -/
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namespace EM
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definition EMadd1_functor_succ [unfold_full] {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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EMadd1_functor φ (succ n) ~* ptrunc_functor (n+2) (psusp_functor (EMadd1_functor φ n)) :=
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by reflexivity
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definition EM1_functor_gid (G : Group) : EM1_functor (gid G) ~* !pid :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x,
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{ reflexivity },
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{ apply eq_pathover_id_right, apply hdeg_square, apply elim_pth, },
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{ apply @is_prop.elim, apply is_trunc_pathover }},
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{ reflexivity },
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end
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definition EMadd1_functor_gid (G : AbGroup) (n : ℕ) : EMadd1_functor (gid G) n ~* !pid :=
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begin
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induction n with n p,
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{ apply EM1_functor_gid },
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{ refine !EMadd1_functor_succ ⬝* _,
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refine ptrunc_functor_phomotopy _ (psusp_functor_phomotopy p ⬝* !psusp_functor_pid) ⬝* _,
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apply ptrunc_functor_pid }
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end
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definition EM_functor_gid (G : AbGroup) (n : ℕ) : EM_functor (gid G) n ~* !pid :=
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begin
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cases n with n,
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{ apply pmap_of_homomorphism_gid },
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{ apply EMadd1_functor_gid }
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end
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definition EM1_functor_gcompose {G H K : Group} (ψ : H →g K) (φ : G →g H) :
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EM1_functor (ψ ∘g φ) ~* EM1_functor ψ ∘* EM1_functor φ :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x,
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{ reflexivity },
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{ apply eq_pathover, apply hdeg_square, esimp,
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refine !elim_pth ⬝ _ ⬝ (ap_compose (EM1_functor ψ) _ _)⁻¹,
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refine _ ⬝ ap02 _ !elim_pth⁻¹, exact !elim_pth⁻¹ },
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{ apply @is_prop.elim, apply is_trunc_pathover }},
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{ reflexivity },
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end
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definition EMadd1_functor_gcompose {G H K : AbGroup} (ψ : H →g K) (φ : G →g H) (n : ℕ) :
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EMadd1_functor (ψ ∘g φ) n ~* EMadd1_functor ψ n ∘* EMadd1_functor φ n :=
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begin
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induction n with n p,
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{ apply EM1_functor_gcompose },
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{ refine !EMadd1_functor_succ ⬝* _,
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refine ptrunc_functor_phomotopy _ (psusp_functor_phomotopy p ⬝* !psusp_functor_pcompose) ⬝* _,
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apply ptrunc_functor_pcompose }
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end
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definition EM_functor_gcompose {G H K : AbGroup} (ψ : H →g K) (φ : G →g H) (n : ℕ) :
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EM_functor (ψ ∘g φ) n ~* EM_functor ψ n ∘* EM_functor φ n :=
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begin
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cases n with n,
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{ apply pmap_of_homomorphism_gcompose },
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{ apply EMadd1_functor_gcompose }
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end
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definition EM1_functor_phomotopy {G H : Group} {φ ψ : G →g H} (p : φ ~ ψ) :
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EM1_functor φ ~* EM1_functor ψ :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x,
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{ reflexivity },
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{ apply eq_pathover, apply hdeg_square, esimp,
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refine !elim_pth ⬝ _ ⬝ !elim_pth⁻¹, exact ap pth (p g) },
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{ apply @is_prop.elim, apply is_trunc_pathover }},
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{ reflexivity },
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end
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definition EMadd1_functor_phomotopy {G H : AbGroup} {φ ψ : G →g H} (p : φ ~ ψ) (n : ℕ) :
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EMadd1_functor φ n ~* EMadd1_functor ψ n :=
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begin
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induction n with n q,
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{ exact EM1_functor_phomotopy p },
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{ exact ptrunc_functor_phomotopy _ (psusp_functor_phomotopy q) }
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end
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definition EM_functor_phomotopy {G H : AbGroup} {φ ψ : G →g H} (p : φ ~ ψ) (n : ℕ) :
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EM_functor φ n ~* EM_functor ψ n :=
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begin
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cases n with n,
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{ exact pmap_of_homomorphism_phomotopy p },
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{ exact EMadd1_functor_phomotopy p n }
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end
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definition EM_equiv_EM [constructor] {G H : AbGroup} (φ : G ≃g H) (n : ℕ) : K G n ≃* K H n :=
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begin
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fapply pequiv.MK',
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{ exact EM_functor φ n },
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{ exact EM_functor φ⁻¹ᵍ n },
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{ intro x, refine (EM_functor_gcompose φ⁻¹ᵍ φ n)⁻¹* x ⬝ _,
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refine _ ⬝ EM_functor_gid G n x,
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refine EM_functor_phomotopy _ n x,
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rexact left_inv φ },
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{ intro x, refine (EM_functor_gcompose φ φ⁻¹ᵍ n)⁻¹* x ⬝ _,
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refine _ ⬝ EM_functor_gid H n x,
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refine EM_functor_phomotopy _ n x,
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rexact right_inv φ }
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end
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definition is_equiv_EM_functor [constructor] {G H : AbGroup} (φ : G →g H) [H2 : is_equiv φ]
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(n : ℕ) : is_equiv (EM_functor φ n) :=
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to_is_equiv (EM_equiv_EM (isomorphism.mk φ H2) n)
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definition fundamental_group_EM1' (G : Group) : G ≃g π₁ (EM1 G) :=
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(fundamental_group_EM1 G)⁻¹ᵍ
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definition ghomotopy_group_EMadd1' (G : AbGroup) (n : ℕ) : G ≃g πg[n+1] (EMadd1 G n) :=
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begin
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change G ≃g π₁ (Ω[n] (EMadd1 G n)),
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refine _ ⬝g homotopy_group_isomorphism_of_pequiv 0 (loopn_EMadd1_pequiv_EM1 G n),
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apply fundamental_group_EM1'
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end
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definition homotopy_group_functor_EM1_functor {G H : Group} (φ : G →g H) :
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π→g[1] (EM1_functor φ) ∘ fundamental_group_EM1' G ~ fundamental_group_EM1' H ∘ φ :=
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begin
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intro g, apply ap tr, exact !idp_con ⬝ !elim_pth,
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end
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section
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definition ghomotopy_group_EMadd1'_0 (G : AbGroup) :
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ghomotopy_group_EMadd1' G 0 ~ fundamental_group_EM1' G :=
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begin
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refine _ ⬝hty id_compose _,
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unfold [ghomotopy_group_EMadd1'],
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apply hwhisker_right (fundamental_group_EM1' G),
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refine _ ⬝hty trunc_functor_id _ _,
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exact trunc_functor_homotopy _ ap1_pid,
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end
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definition loopn_EMadd1_pequiv_EM1_succ (G : AbGroup) (n : ℕ) :
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loopn_EMadd1_pequiv_EM1 G (succ n) ~* (loopn_succ_in (EMadd1 G (succ n)) n)⁻¹ᵉ* ∘*
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Ω→[n] (loop_EMadd1 G n) ∘* loopn_EMadd1_pequiv_EM1 G n :=
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by reflexivity
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-- definition is_trunc_EMadd1' [instance] (G : AbGroup) (n : ℕ) : is_trunc (succ n) (EMadd1 G n) :=
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-- is_trunc_EMadd1 G n
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definition loop_EMadd1_succ (G : AbGroup) (n : ℕ) :
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loop_EMadd1 G (n+1) ~* (loop_ptrunc_pequiv (n+1+1) (psusp (EMadd1 G (n+1))))⁻¹ᵉ* ∘*
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freudenthal_pequiv (EMadd1 G (n+1)) (add_mul_le_mul_add n 1 1) ∘*
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(ptrunc_pequiv (n+1+1) (EMadd1 G (n+1)))⁻¹ᵉ* :=
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by reflexivity
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definition ap1_EMadd1_natural {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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Ω→ (EMadd1_functor φ (succ n)) ∘* loop_EMadd1 G n ~* loop_EMadd1 H n ∘* EMadd1_functor φ n :=
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begin
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refine pwhisker_right _ (ap1_phomotopy !EMadd1_functor_succ) ⬝* _,
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induction n with n IH,
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{ refine pwhisker_left _ !hopf.to_pmap_delooping_pinv ⬝* _ ⬝*
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pwhisker_right _ !hopf.to_pmap_delooping_pinv⁻¹*,
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refine !loop_psusp_unit_natural⁻¹* ⬝h* _,
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apply ap1_psquare,
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apply ptr_natural },
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{ refine pwhisker_left _ !loop_EMadd1_succ ⬝* _ ⬝* pwhisker_right _ !loop_EMadd1_succ⁻¹*,
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refine _ ⬝h* !ap1_ptrunc_functor,
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refine (@(ptrunc_pequiv_natural (n+1+1) _) _ _)⁻¹ʰ* ⬝h* _,
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refine pwhisker_left _ !to_pmap_freudenthal_pequiv ⬝* _ ⬝*
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pwhisker_right _ !to_pmap_freudenthal_pequiv⁻¹*,
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apply ptrunc_functor_psquare,
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exact !loop_psusp_unit_natural⁻¹* }
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end
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definition apn_EMadd1_pequiv_EM1_natural {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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Ω→[n] (EMadd1_functor φ n) ∘* loopn_EMadd1_pequiv_EM1 G n ~*
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loopn_EMadd1_pequiv_EM1 H n ∘* EM1_functor φ :=
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begin
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induction n with n IH,
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{ reflexivity },
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{ refine pwhisker_left _ !loopn_EMadd1_pequiv_EM1_succ ⬝* _,
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refine _ ⬝* pwhisker_right _ !loopn_EMadd1_pequiv_EM1_succ⁻¹*,
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refine _ ⬝h* !loopn_succ_in_inv_natural,
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exact IH ⬝h* (apn_psquare n !ap1_EMadd1_natural) }
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end
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definition homotopy_group_functor_EMadd1_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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π→g[n+1] (EMadd1_functor φ n) ∘ ghomotopy_group_EMadd1' G n ~
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ghomotopy_group_EMadd1' H n ∘ φ :=
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begin
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refine hwhisker_left _ (to_fun_isomorphism_trans _ _) ⬝hty _ ⬝hty
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hwhisker_right _ (to_fun_isomorphism_trans _ _)⁻¹ʰᵗʸ,
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refine _ ⬝htyh (homotopy_group_homomorphism_psquare 1 (apn_EMadd1_pequiv_EM1_natural φ n)),
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apply homotopy_group_functor_EM1_functor
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end
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definition homotopy_group_functor_EMadd1_functor' {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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φ ∘ (ghomotopy_group_EMadd1' G n)⁻¹ᵍ ~
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(ghomotopy_group_EMadd1' H n)⁻¹ᵍ ∘ π→g[n+1] (EMadd1_functor φ n) :=
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(homotopy_group_functor_EMadd1_functor φ n)⁻¹ʰᵗʸʰ
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definition EM1_pmap_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G → Ω X)
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(eY : H → Ω Y) (rX : Πg h, eX (g * h) = eX g ⬝ eX h) (rY : Πg h, eY (g * h) = eY g ⬝ eY h)
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[H1 : is_conn 0 X] [H2 : is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y]
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(φ : G →g H) (p : hsquare eX eY φ (Ω→ f)) :
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psquare (EM1_pmap eX rX) (EM1_pmap eY rY) (EM1_functor φ) f :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x using EM.set_rec,
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{ exact respect_pt f },
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{ apply eq_pathover,
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refine ap_compose f _ _ ⬝ph _ ⬝hp (ap_compose (EM1_pmap eY rY) _ _)⁻¹,
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refine ap02 _ !elim_pth ⬝ph _ ⬝hp ap02 _ !elim_pth⁻¹,
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refine _ ⬝hp !elim_pth⁻¹, apply transpose, exact square_of_eq_bot (p g) }},
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{ exact !idp_con⁻¹ }
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end
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definition EM1_pequiv'_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G ≃* Ω X)
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(eY : H ≃* Ω Y) (rX : Πg h, eX (g * h) = eX g ⬝ eX h) (rY : Πg h, eY (g * h) = eY g ⬝ eY h)
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[H1 : is_conn 0 X] [H2 : is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y]
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(φ : G →g H) (p : Ω→ f ∘ eX ~ eY ∘ φ) :
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f ∘* EM1_pequiv' eX rX ~* EM1_pequiv' eY rY ∘* EM1_functor φ :=
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EM1_pmap_natural f eX eY rX rY φ p
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definition EM1_pequiv_natural {G H : Group} {X Y : Type*} (f : X →* Y) (eX : G ≃g π₁ X)
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(eY : H ≃g π₁ Y) [H1 : is_conn 0 X] [H2 : is_trunc 1 X] [is_conn 0 Y] [is_trunc 1 Y]
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(φ : G →g H) (p : π→g[1] f ∘ eX ~ eY ∘ φ) :
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f ∘* EM1_pequiv eX ~* EM1_pequiv eY ∘* EM1_functor φ :=
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EM1_pequiv'_natural f _ _ _ _ φ
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begin
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assert p' : ptrunc_functor 0 (Ω→ f) ∘* pequiv_of_isomorphism eX ~*
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pequiv_of_isomorphism eY ∘* pmap_of_homomorphism φ, exact phomotopy_of_homotopy p,
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exact p' ⬝h* (ptrunc_pequiv_natural 0 (Ω→ f)),
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end
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definition EM1_pequiv_type_natural {X Y : Type*} (f : X →* Y) [H1 : is_conn 0 X] [H2 : is_trunc 1 X]
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[H3 : is_conn 0 Y] [H4 : is_trunc 1 Y] :
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f ∘* EM1_pequiv_type X ~* EM1_pequiv_type Y ∘* EM1_functor (π→g[1] f) :=
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begin refine EM1_pequiv_natural f _ _ _ _, reflexivity end
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definition EM_up_natural {G H : AbGroup} (φ : G →g H) {X Y : Type*} (f : X →* Y) {n : ℕ}
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(eX : Ω[succ (succ n)] X ≃* G) (eY : Ω[succ (succ n)] Y ≃* H) (p : φ ∘ eX ~ eY ∘ Ω→[succ (succ n)] f)
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: φ ∘ EM_up eX ~ EM_up eY ∘ Ω→[succ n] (Ω→ f) :=
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begin
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refine _ ⬝htyh p,
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exact to_homotopy (phinverse (loopn_succ_in_natural (succ n) f)⁻¹*)
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end
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definition EMadd1_pmap_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ) (eX : Ω[succ n] X ≃* G)
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(eY : Ω[succ n] Y ≃* H) (rX : Πp q, eX (p ⬝ q) = eX p * eX q) (rY : Πp q, eY (p ⬝ q) = eY p * eY q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] [H3 : is_conn n Y] [H4 : is_trunc (n.+1) Y]
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(φ : G →g H) (p : φ ∘ eX ~ eY ∘ Ω→[succ n] f) :
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f ∘* EMadd1_pmap n eX rX ~* EMadd1_pmap n eY rY ∘* EMadd1_functor φ n :=
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begin
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revert X Y f eX eY rX rY H1 H2 H3 H4 p, induction n with n IH: intros,
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{ apply EM1_pmap_natural, exact @hhinverse _ _ _ _ _ _ eX eY p },
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{ do 2 rewrite [EMadd1_pmap_succ], refine _ ⬝* pwhisker_left _ !EMadd1_functor_succ⁻¹*,
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refine (ptrunc_elim_pcompose ((succ n).+1) _ _)⁻¹* ⬝* _ ⬝*
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(ptrunc_elim_ptrunc_functor ((succ n).+1) _ _)⁻¹*,
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apply ptrunc_elim_phomotopy,
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refine _ ⬝* !psusp_elim_psusp_functor⁻¹*,
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refine _ ⬝* psusp_elim_phomotopy (IH _ _ _ _ _ (is_homomorphism_EM_up eX rX) _ (@is_conn_loop _ _ H1)
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(@is_trunc_loop _ _ H2) _ _ (EM_up_natural φ f eX eY p)),
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apply psusp_elim_natural }
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end
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definition EMadd1_pequiv'_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ) (eX : Ω[succ n] X ≃* G)
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(eY : Ω[succ n] Y ≃* H) (rX : Πp q, eX (p ⬝ q) = eX p * eX q) (rY : Πp q, eY (p ⬝ q) = eY p * eY q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] [is_conn n Y] [is_trunc (n.+1) Y]
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(φ : G →g H) (p : φ ∘ eX ~ eY ∘ Ω→[succ n] f) :
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f ∘* EMadd1_pequiv' n eX rX ~* EMadd1_pequiv' n eY rY ∘* EMadd1_functor φ n :=
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begin rexact EMadd1_pmap_natural f n eX eY rX rY φ p end
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definition EMadd1_pequiv_natural_local_instance {X : Type*} (n : ℕ) [H : is_trunc (n.+1) X] :
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is_set (Ω[succ n] X) :=
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@(is_set_loopn (succ n) X) H
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local attribute EMadd1_pequiv_natural_local_instance [instance]
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definition EMadd1_pequiv_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ) (eX : πg[n+1] X ≃g G)
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(eY : πg[n+1] Y ≃g H) [H1 : is_conn n X] [H2 : is_trunc (n.+1) X] [H3 : is_conn n Y]
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[H4 : is_trunc (n.+1) Y] (φ : G →g H) (p : φ ∘ eX ~ eY ∘ π→g[n+1] f) :
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f ∘* EMadd1_pequiv n eX ~* EMadd1_pequiv n eY ∘* EMadd1_functor φ n :=
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EMadd1_pequiv'_natural f n
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((ptrunc_pequiv 0 (Ω[succ n] X))⁻¹ᵉ* ⬝e* pequiv_of_isomorphism eX)
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((ptrunc_pequiv 0 (Ω[succ n] Y))⁻¹ᵉ* ⬝e* pequiv_of_isomorphism eY)
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_ _ φ (hhconcat (to_homotopy (phinverse (ptrunc_pequiv_natural 0 (Ω→[succ n] f)))) p)
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definition EMadd1_pequiv_succ_natural {G H : AbGroup} {X Y : Type*} (f : X →* Y) (n : ℕ)
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(eX : πag[n+2] X ≃g G) (eY : πag[n+2] Y ≃g H) [is_conn (n.+1) X] [is_trunc (n.+2) X]
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[is_conn (n.+1) Y] [is_trunc (n.+2) Y] (φ : G →g H) (p : φ ∘ eX ~ eY ∘ π→g[n+2] f) :
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f ∘* EMadd1_pequiv_succ n eX ~* EMadd1_pequiv_succ n eY ∘* EMadd1_functor φ (n+1) :=
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@(EMadd1_pequiv_natural f (succ n) eX eY) _ _ _ _ φ p
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definition EMadd1_pequiv_type_natural {X Y : Type*} (f : X →* Y) (n : ℕ)
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[H1 : is_conn (n+1) X] [H2 : is_trunc (n+1+1) X] [H3 : is_conn (n+1) Y] [H4 : is_trunc (n+1+1) Y] :
|
||
f ∘* EMadd1_pequiv_type X n ~* EMadd1_pequiv_type Y n ∘* EMadd1_functor (π→g[n+2] f) (succ n) :=
|
||
EMadd1_pequiv_succ_natural f n !isomorphism.refl !isomorphism.refl (π→g[n+2] f)
|
||
proof λa, idp qed
|
||
|
||
/- The Eilenberg-MacLane space K(G,n) with the same homotopy group as X on level n -/
|
||
definition EM_type (X : Type*) : ℕ → Type*
|
||
| 0 := ptrunc 0 X
|
||
| 1 := EM1 (π₁ X)
|
||
| (n+2) := EMadd1 (πag[n+2] X) (n+1)
|
||
|
||
definition EM_type_pequiv.{u} {X Y : pType.{u}} (n : ℕ) [Hn : is_succ n] (e : πg[n] Y ≃g πg[n] X)
|
||
[H1 : is_conn (n.-1) X] [H2 : is_trunc n X] : EM_type Y n ≃* X :=
|
||
begin
|
||
induction Hn with n, cases n with n,
|
||
{ have is_conn 0 X, from H1,
|
||
have is_trunc 1 X, from H2,
|
||
exact EM1_pequiv e },
|
||
{ have is_conn (n+1) X, from H1,
|
||
have is_trunc ((n+1).+1) X, from H2,
|
||
exact EMadd1_pequiv (n+1) e⁻¹ᵍ }
|
||
end
|
||
|
||
-- definition EM1_functor_equiv' (X Y : Type*) [H1 : is_conn 0 X] [H2 : is_trunc 1 X]
|
||
-- [H3 : is_conn 0 Y] [H4 : is_trunc 1 Y] : (X →* Y) ≃ (π₁ X →g π₁ Y) :=
|
||
-- begin
|
||
-- fapply equiv.MK,
|
||
-- { intro f, exact π→g[1] f },
|
||
-- { intro φ, exact EM1_pequiv_type Y ∘* EM1_functor φ ∘* (EM1_pequiv_type X)⁻¹ᵉ* },
|
||
-- { intro φ, apply homomorphism_eq,
|
||
-- refine homotopy_group_homomorphism_pcompose _ _ _ ⬝hty _,
|
||
-- refine hwhisker_left _ (homotopy_group_homomorphism_pcompose _ _ _) ⬝hty _,
|
||
-- refine (hassoc _ _ _)⁻¹ʰᵗʸ ⬝hty _, exact sorry },
|
||
-- { intro f, apply eq_of_phomotopy, refine !passoc⁻¹* ⬝* _, apply pinv_right_phomotopy_of_phomotopy,
|
||
-- exact sorry }
|
||
-- end
|
||
|
||
-- definition EMadd1_functor_equiv' (n : ℕ) (X Y : Type*) [H1 : is_conn (n+1) X] [H2 : is_trunc (n+1+1) X]
|
||
-- [H3 : is_conn (n+1) Y] [H4 : is_trunc (n+1+1) Y] : (X →* Y) ≃ (πag[n+2] X →g πag[n+2] Y) :=
|
||
-- begin
|
||
-- fapply equiv.MK,
|
||
-- { intro f, exact π→g[n+2] f },
|
||
-- { intro φ, exact EMadd1_pequiv_type Y n ∘* EMadd1_functor φ (n+1) ∘* (EMadd1_pequiv_type X n)⁻¹ᵉ* },
|
||
-- { intro φ, apply homomorphism_eq,
|
||
-- refine homotopy_group_homomorphism_pcompose _ _ _ ⬝hty _,
|
||
-- refine hwhisker_left _ (homotopy_group_homomorphism_pcompose _ _ _) ⬝hty _,
|
||
-- intro g, exact sorry },
|
||
-- { intro f, apply eq_of_phomotopy, refine !passoc⁻¹* ⬝* _, apply pinv_right_phomotopy_of_phomotopy,
|
||
-- exact !EMadd1_pequiv_type_natural⁻¹* }
|
||
-- end
|
||
|
||
-- definition EM_functor_equiv (n : ℕ) (G H : AbGroup) : (G →g H) ≃ (EMadd1 G (n+1) →* EMadd1 H (n+1)) :=
|
||
-- begin
|
||
-- fapply equiv.MK,
|
||
-- { intro φ, exact EMadd1_functor φ (n+1) },
|
||
-- { intro f, exact ghomotopy_group_EMadd1 H (n+1) ∘g π→g[n+2] f ∘g (ghomotopy_group_EMadd1 G (n+1))⁻¹ᵍ },
|
||
-- { intro f, apply homomorphism_eq, },
|
||
-- { }
|
||
-- end
|
||
|
||
|
||
-- definition EMadd1_pmap {G : AbGroup} {X : Type*} (n : ℕ)
|
||
-- (e : Ω[succ n] X ≃* G)
|
||
-- (r : Πp q, e (p ⬝ q) = e p * e q)
|
||
-- [H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n →* X :=
|
||
-- begin
|
||
-- revert X e r H1 H2, induction n with n f: intro X e r H1 H2,
|
||
-- { exact EM1_pmap e⁻¹ᵉ* (equiv.inv_preserve_binary e concat mul r) },
|
||
-- rewrite [EMadd1_succ],
|
||
-- exact ptrunc.elim ((succ n).+1)
|
||
-- (psusp.elim (f _ (EM_up e) (is_mul_hom_EM_up e r) _ _)),
|
||
-- end
|
||
|
||
-- definition is_set_pmap_ptruncconntype {n : ℕ₋₂} (X Y : (n.+1)-Type*[n]) : is_set (X →* Y) :=
|
||
-- begin
|
||
-- apply is_trunc_succ_intro,
|
||
-- intro f g,
|
||
-- apply @(is_trunc_equiv_closed_rev -1 (pmap_eq_equiv f g)),
|
||
-- apply is_prop.mk,
|
||
-- exact sorry
|
||
-- end
|
||
|
||
|
||
end
|
||
|
||
section category
|
||
/- category -/
|
||
structure ptruncconntype' (n : ℕ₋₂) : Type :=
|
||
(A : Type*)
|
||
(H1 : is_conn n A)
|
||
(H2 : is_trunc (n+1) A)
|
||
|
||
attribute ptruncconntype'.A [coercion]
|
||
attribute ptruncconntype'.H1 ptruncconntype'.H2 [instance]
|
||
|
||
definition EM1_pequiv_ptruncconntype' (X : ptruncconntype' 0) : EM1 (πg[1] X) ≃* X :=
|
||
@(EM1_pequiv_type X) _ (ptruncconntype'.H2 X)
|
||
|
||
definition EMadd1_pequiv_ptruncconntype' {n : ℕ} (X : ptruncconntype' (n+1)) :
|
||
EMadd1 (πag[n+2] X) (succ n) ≃* X :=
|
||
@(EMadd1_pequiv_type X n) _ (ptruncconntype'.H2 X)
|
||
|
||
open trunc_index
|
||
definition is_set_pmap_ptruncconntype {n : ℕ₋₂} (X Y : ptruncconntype' n) : is_set (X →* Y) :=
|
||
begin
|
||
cases n with n, { exact _ },
|
||
cases Y with Y H1 H2, cases Y with Y y₀,
|
||
exact is_trunc_pmap_of_is_conn X n -1 (ptrunctype.mk Y _ y₀),
|
||
end
|
||
|
||
open category functor nat_trans
|
||
|
||
definition precategory_ptruncconntype'.{u} [constructor] (n : ℕ₋₂) :
|
||
precategory.{u+1 u} (ptruncconntype' n) :=
|
||
begin
|
||
fapply precategory.mk,
|
||
{ exact λX Y, X →* Y },
|
||
{ exact is_set_pmap_ptruncconntype },
|
||
{ exact λX Y Z g f, g ∘* f },
|
||
{ exact λX, pid X },
|
||
{ intros, apply eq_of_phomotopy, exact !passoc⁻¹* },
|
||
{ intros, apply eq_of_phomotopy, apply pid_pcompose },
|
||
{ intros, apply eq_of_phomotopy, apply pcompose_pid }
|
||
end
|
||
|
||
definition cptruncconntype' [constructor] (n : ℕ₋₂) : Precategory :=
|
||
precategory.Mk (precategory_ptruncconntype' n)
|
||
|
||
notation `cType*[`:95 n `]`:0 := cptruncconntype' n
|
||
|
||
definition tEM1 [constructor] (G : Group) : ptruncconntype' 0 :=
|
||
ptruncconntype'.mk (EM1 G) _ !is_trunc_EM1
|
||
|
||
definition tEM [constructor] (G : AbGroup) (n : ℕ) : ptruncconntype' (n.-1) :=
|
||
ptruncconntype'.mk (EM G n) _ !is_trunc_EM
|
||
|
||
definition EM1_cfunctor : Grp ⇒ cType*[0] :=
|
||
functor.mk
|
||
(λG, tEM1 G)
|
||
(λG H φ, EM1_functor φ)
|
||
begin intro, fapply eq_of_phomotopy, apply EM1_functor_gid end
|
||
begin intros, fapply eq_of_phomotopy, apply EM1_functor_gcompose end
|
||
|
||
definition EM_cfunctor (n : ℕ) : AbGrp ⇒ cType*[n.-1] :=
|
||
functor.mk
|
||
(λG, tEM G n)
|
||
(λG H φ, EM_functor φ n)
|
||
begin intro, fapply eq_of_phomotopy, apply EM_functor_gid end
|
||
begin intros, fapply eq_of_phomotopy, apply EM_functor_gcompose end
|
||
|
||
definition homotopy_group_cfunctor : cType*[0] ⇒ Grp :=
|
||
functor.mk
|
||
(λX, πg[1] X)
|
||
(λX Y f, π→g[1] f)
|
||
begin intro, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_pid end
|
||
begin intros, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_compose end
|
||
|
||
definition ab_homotopy_group_cfunctor (n : ℕ) : cType*[n+2.-1] ⇒ AbGrp :=
|
||
functor.mk
|
||
(λX, πag[n+2] X)
|
||
(λX Y f, π→g[n+2] f)
|
||
begin intro, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_pid end
|
||
begin intros, apply homomorphism_eq, exact to_homotopy !homotopy_group_functor_compose end
|
||
|
||
|
||
definition is_equivalence_EM1_cfunctor.{u} : is_equivalence EM1_cfunctor.{u} :=
|
||
begin
|
||
fapply is_equivalence.mk,
|
||
{ exact homotopy_group_cfunctor.{u} },
|
||
{ fapply natural_iso.mk,
|
||
{ fapply nat_trans.mk,
|
||
{ intro G, exact (fundamental_group_EM1' G)⁻¹ᵍ },
|
||
{ intro G H φ, apply homomorphism_eq, exact hhinverse (homotopy_group_functor_EM1_functor φ) }},
|
||
{ intro G, fapply iso.is_iso.mk,
|
||
{ exact fundamental_group_EM1' G },
|
||
{ apply homomorphism_eq,
|
||
exact to_right_inv (equiv_of_isomorphism (fundamental_group_EM1' G)), },
|
||
{ apply homomorphism_eq,
|
||
exact to_left_inv (equiv_of_isomorphism (fundamental_group_EM1' G)), }}},
|
||
{ fapply natural_iso.mk,
|
||
{ fapply nat_trans.mk,
|
||
{ intro X, exact EM1_pequiv_ptruncconntype' X },
|
||
{ intro X Y f, apply eq_of_phomotopy, apply EM1_pequiv_type_natural }},
|
||
{ intro X, fapply iso.is_iso.mk,
|
||
{ exact (EM1_pequiv_ptruncconntype' X)⁻¹ᵉ* },
|
||
{ apply eq_of_phomotopy, apply pleft_inv },
|
||
{ apply eq_of_phomotopy, apply pright_inv }}}
|
||
end
|
||
|
||
definition is_equivalence_EM_cfunctor (n : ℕ) : is_equivalence (EM_cfunctor (n+2)) :=
|
||
begin
|
||
fapply is_equivalence.mk,
|
||
{ exact ab_homotopy_group_cfunctor n },
|
||
{ fapply natural_iso.mk,
|
||
{ fapply nat_trans.mk,
|
||
{ intro G, exact (ghomotopy_group_EMadd1' G (n+1))⁻¹ᵍ },
|
||
{ intro G H φ, apply homomorphism_eq, exact homotopy_group_functor_EMadd1_functor' φ (n+1) }},
|
||
{ intro G, fapply iso.is_iso.mk,
|
||
{ exact ghomotopy_group_EMadd1' G (n+1) },
|
||
{ apply homomorphism_eq,
|
||
exact to_right_inv (equiv_of_isomorphism (ghomotopy_group_EMadd1' G (n+1))), },
|
||
{ apply homomorphism_eq,
|
||
exact to_left_inv (equiv_of_isomorphism (ghomotopy_group_EMadd1' G (n+1))), }}},
|
||
{ fapply natural_iso.mk,
|
||
{ fapply nat_trans.mk,
|
||
{ intro X, exact EMadd1_pequiv_ptruncconntype' X },
|
||
{ intro X Y f, apply eq_of_phomotopy, apply EMadd1_pequiv_type_natural }},
|
||
{ intro X, fapply iso.is_iso.mk,
|
||
{ exact (EMadd1_pequiv_ptruncconntype' X)⁻¹ᵉ* },
|
||
{ apply eq_of_phomotopy, apply pleft_inv },
|
||
{ apply eq_of_phomotopy, apply pright_inv }}}
|
||
end
|
||
|
||
definition Grp_equivalence_cptruncconntype'.{u} [constructor] : Grp.{u} ≃c cType*[0] :=
|
||
equivalence.mk EM1_cfunctor.{u} is_equivalence_EM1_cfunctor.{u}
|
||
|
||
definition AbGrp_equivalence_cptruncconntype' [constructor] (n : ℕ) : AbGrp ≃c cType*[n+2.-1] :=
|
||
equivalence.mk (EM_cfunctor (n+2)) (is_equivalence_EM_cfunctor n)
|
||
end category
|
||
|
||
/- move -/
|
||
|
||
open trunc_index
|
||
definition is_trunc_succ_of_is_trunc_loop (n : ℕ₋₂) (A : Type*) (H : is_trunc (n.+1) (Ω A))
|
||
(H2 : is_conn 0 A) : is_trunc (n.+2) A :=
|
||
begin
|
||
apply is_trunc_succ_of_is_trunc_loop, apply minus_one_le_succ,
|
||
refine is_conn.elim -1 _ _, exact H
|
||
end
|
||
|
||
lemma is_trunc_of_is_trunc_loopn (m n : ℕ) (A : Type*) (H : is_trunc n (Ω[m] A))
|
||
(H2 : is_conn m A) : is_trunc (m + n) A :=
|
||
begin
|
||
revert A H H2; induction m with m IH: intro A H H2,
|
||
{ rewrite [nat.zero_add], exact H },
|
||
rewrite [succ_add],
|
||
apply is_trunc_succ_of_is_trunc_loop,
|
||
{ apply IH,
|
||
{ apply is_trunc_equiv_closed _ !loopn_succ_in },
|
||
apply is_conn_loop },
|
||
exact is_conn_of_le _ (zero_le_of_nat (succ m))
|
||
end
|
||
|
||
lemma is_trunc_of_is_set_loopn (m : ℕ) (A : Type*) (H : is_set (Ω[m] A))
|
||
(H2 : is_conn m A) : is_trunc m A :=
|
||
is_trunc_of_is_trunc_loopn m 0 A H H2
|
||
|
||
definition pequiv_EMadd1_of_loopn_pequiv_EM1 {G : AbGroup} {X : Type*} (n : ℕ) (e : Ω[n] X ≃* EM1 G)
|
||
[H1 : is_conn n X] : X ≃* EMadd1 G n :=
|
||
begin
|
||
symmetry, apply EMadd1_pequiv,
|
||
refine isomorphism_of_eq (ap (λx, πg[x+1] X) !zero_add⁻¹) ⬝g homotopy_group_add X 0 n ⬝g _ ⬝g
|
||
!fundamental_group_EM1,
|
||
exact homotopy_group_isomorphism_of_pequiv 0 e,
|
||
refine is_trunc_of_is_trunc_loopn n 1 X _ _,
|
||
apply is_trunc_equiv_closed_rev 1 e
|
||
end
|
||
|
||
definition EM1_pequiv_EM1 {G H : Group} (φ : G ≃g H) : EM1 G ≃* EM1 H :=
|
||
pequiv.MK (EM1_functor φ) (EM1_functor φ⁻¹ᵍ)
|
||
abstract (EM1_functor_gcompose φ⁻¹ᵍ φ)⁻¹* ⬝* EM1_functor_phomotopy proof left_inv φ qed ⬝*
|
||
EM1_functor_gid G end
|
||
abstract (EM1_functor_gcompose φ φ⁻¹ᵍ)⁻¹* ⬝* EM1_functor_phomotopy proof right_inv φ qed ⬝*
|
||
EM1_functor_gid H end
|
||
|
||
definition EMadd1_pequiv_EMadd1 (n : ℕ) {G H : AbGroup} (φ : G ≃g H) : EMadd1 G n ≃* EMadd1 H n :=
|
||
pequiv.MK (EMadd1_functor φ n) (EMadd1_functor φ⁻¹ᵍ n)
|
||
abstract (EMadd1_functor_gcompose φ⁻¹ᵍ φ n)⁻¹* ⬝* EMadd1_functor_phomotopy proof left_inv φ qed n ⬝*
|
||
EMadd1_functor_gid G n end
|
||
abstract (EMadd1_functor_gcompose φ φ⁻¹ᵍ n)⁻¹* ⬝* EMadd1_functor_phomotopy proof right_inv φ qed n ⬝*
|
||
EMadd1_functor_gid H n end
|
||
|
||
definition EM_pequiv_EM (n : ℕ) {G H : AbGroup} (φ : G ≃g H) : K G n ≃* K H n :=
|
||
begin
|
||
cases n with n,
|
||
{ exact pequiv_of_isomorphism φ },
|
||
{ exact EMadd1_pequiv_EMadd1 n φ }
|
||
end
|
||
|
||
end EM
|