define deloopable types, define cup product
The cup product on Eilenberg Maclane spaces is now defined, but no properties are proven yet
This commit is contained in:
Floris van Doorn
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db8402e1af
6 changed files with 164 additions and 17 deletions
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@ -38,6 +38,7 @@ namespace group
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{ intro a, exact con.left_inv (f a) },
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{ exact !con_left_inv_idp }},
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end
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-- inf_pgroup_pequiv_closed (loop_pppi_pequiv B) _
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definition inf_group_ppi [constructor] {A : Type*} (B : A → Type*) : inf_group (Π*a, Ω (B a)) :=
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@inf_group_of_inf_pgroup _ (inf_pgroup_pppi B)
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@ -128,6 +129,10 @@ namespace group
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gtrunc (gloop (Π*(a : A), B a)) ≃g gtrunc (gppi_loop B) :=
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proof trunc_ppi_loop_isomorphism_gen (ppi_const B) qed
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definition gppi_loop_homomorphism_right [constructor] {A : Type*} {B B' : A → Type*}
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(g : Πa, B a →* B' a) : gppi_loop B →∞g gppi_loop B' :=
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gloop_ppi_isomorphism B' ∘∞g Ωg→ (pppi_compose_left g) ∘∞g (gloop_ppi_isomorphism B)⁻¹ᵍ⁸
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/- We first define the group structure on A →* Ω B (except for truncatedness).
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Instead of Ω B, we could also choose any infinity group. However, we need various 2-coherences,
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@ -183,12 +188,30 @@ namespace group
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definition gpmap_loop [reducible] [constructor] (A B : Type*) : InfGroup :=
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InfGroup.mk (A →* Ω B) !inf_group_ppi
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definition gpmap_loopn [constructor] (n : ℕ) [H : is_succ n] (A B : Type*) : InfGroup :=
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InfGroup.mk (A →** Ω[n] B) (by induction H with n; exact inf_group_ppi (λa, Ω[n] B))
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definition gloop_pmap_isomorphism (A B : Type*) : Ωg (A →** B) ≃∞g gpmap_loop A B :=
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gloop_ppi_isomorphism _
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definition gloopn_pmap_isomorphism (n : ℕ) [H : is_succ n] (A B : Type*) :
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Ωg[n] (A →** B) ≃∞g gpmap_loopn n A B :=
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begin
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induction H with n, induction n with n IH,
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{ exact gloop_pmap_isomorphism A B },
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{ rexact Ωg≃ (pequiv_of_inf_isomorphism IH) ⬝∞g gloop_pmap_isomorphism A (Ω[succ n] B) }
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end
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definition gpmap_loop' [reducible] [constructor] (A : Type*) {B C : Type*} (e : Ω C ≃* B) :
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InfGroup :=
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InfGroup.mk (A →* B)
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(@inf_group_of_inf_pgroup _ (inf_pgroup_pequiv_closed (ppmap_pequiv_ppmap_right e)
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!inf_pgroup_pppi))
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definition gpmap_loop_homomorphism_right [constructor] (A : Type*) {B B' : Type*}
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(g : B →* B') : gpmap_loop A B →∞g gpmap_loop A B' :=
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gppi_loop_homomorphism_right (λa, g)
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definition Group_trunc_pmap [reducible] [constructor] (A B : Type*) : Group :=
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Group.mk (trunc 0 (A →* Ω B)) (@group_trunc _ !inf_group_ppi)
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@ -85,8 +85,6 @@ begin
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exact pfiber_postnikov_map A n
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end
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set_option formatter.hide_full_terms false
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definition pfiber_postnikov_map_pred' (A : spectrum) (n k l : ℤ) (p : n + k = l) :
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pfiber (postnikov_map_pred (A k) (maxm2 l)) ≃* EM_spectrum (πₛ[n] A) l :=
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begin
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@ -156,7 +156,7 @@ namespace EM
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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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definition loop_EMadd1_natural {G H : AbGroup} (φ : G →g H) (n : ℕ) :
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psquare (loop_EMadd1 G n) (loop_EMadd1 H n)
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(EMadd1_functor φ n) (Ω→ (EMadd1_functor φ (succ n))) :=
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begin
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@ -181,7 +181,7 @@ namespace EM
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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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exact IH ⬝h* (apn_psquare n !loop_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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@ -301,15 +301,18 @@ namespace EM
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proof λa, idp qed
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definition EMadd1_pmap_equiv (n : ℕ) (X Y : Type*) [is_conn (n+1) X] [is_trunc (n+1+1) X]
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[is_conn (n+1) Y] [is_trunc (n+1+1) Y] : (X →* Y) ≃ πag[n+2] X →g πag[n+2] Y :=
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[H4 : is_trunc (n+1+1) Y] : (X →* Y) ≃ πag[n+2] X →g πag[n+2] Y :=
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begin
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have H4' : is_trunc ((n+1).+1) Y, from H4,
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have is_set (Ωg[succ (n+1)] Y), from is_set_loopn (n+1+1) Y,
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fapply equiv.MK,
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{ exact π→g[n+2] },
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{ exact λφ, (EMadd1_pequiv_type Y n ∘* EMadd1_functor φ (n+1)) ∘* (EMadd1_pequiv_type X n)⁻¹ᵉ* },
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{ intro φ, refine (_ ∘* EMadd1_functor φ (n+1)) ∘* (EMadd1_pequiv_type X n)⁻¹ᵉ*,
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exact EMadd1_pmap (n+1) (gtrunc_isomorphism (Ωg[succ (n+1)] Y)) },
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{ intro φ, apply homomorphism_eq,
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refine homotopy_group_functor_pcompose (n+2) _ _ ⬝hty _, exact sorry }, -- easy but tedious
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{ intro f, apply eq_of_phomotopy, refine (phomotopy_pinv_right_of_phomotopy _)⁻¹*,
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apply EMadd1_pequiv_type_natural }
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exact sorry /-apply EMadd1_pequiv_type_natural-/ }
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end
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definition EM1_functor_trivial_homomorphism [constructor] (G H : Group) :
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@ -343,19 +346,39 @@ namespace EM
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{ refine !loopn_succ_in ⬝e* Ω≃[n] (loop_EMadd1 G (m + n))⁻¹ᵉ* ⬝e* e }
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end
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definition loopn_EMadd1_add_of_eq (G : AbGroup) {n m k : ℕ} (p : m + n = k) : Ω[n] (EMadd1 G k) ≃* EMadd1 G m :=
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by induction p; exact loopn_EMadd1_add G n m
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definition EM1_phomotopy {G : Group} {X : Type*} [is_trunc 1 X] {f g : EM1 G →* X} (h : Ω→ f ~ Ω→ g) :
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f ~* g :=
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begin
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fapply phomotopy.mk,
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{ intro x, induction x using EM.set_rec with a,
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{ exact respect_pt f ⬝ (respect_pt g)⁻¹ },
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{ apply eq_pathover, apply move_top_of_right, apply move_bot_of_right,
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apply move_top_of_left', apply move_bot_of_left, apply hdeg_square, exact h (pth a) }},
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{ apply inv_con_cancel_right }
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end
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/- properties about EM -/
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definition deloopable_EM [instance] [constructor] (G : AbGroup) (n : ℕ) : deloopable (EM G n) :=
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deloopable.mk (EMadd1 G n) (loop_EM G n)
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definition gEM (G : AbGroup) (n : ℕ) : InfGroup :=
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InfGroup_equiv_closed (Ωg (EMadd1 G n)) (loop_EM G n)
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InfGroup_of_deloopable (EM G n)
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definition gloop_EM1 [constructor] (G : Group) : Ωg (EM1 G) ≃∞g InfGroup_of_Group G :=
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inf_isomorphism_of_equiv (EM.base_eq_base_equiv G) groupoid_quotient.encode_con
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definition gloop_EMadd1 [constructor] (G : AbGroup) (n : ℕ) : Ωg (EMadd1 G n) ≃∞g gEM G n :=
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deloop_isomorphism (EM G n)
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definition gEM0_isomorphism (G : AbGroup) : gEM G 0 ≃∞g InfGroup_of_Group G :=
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!InfGroup_equiv_closed_isomorphism⁻¹ᵍ⁸ ⬝∞g gloop_EM1 G
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(gloop_EMadd1 G 0)⁻¹ᵍ⁸ ⬝∞g gloop_EM1 G
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definition gEM_functor {G H : AbGroup} (φ : G →g H) (n : ℕ) : gEM G n →∞g gEM H n :=
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inf_homomorphism.mk (EM_functor φ n) sorry
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gloop_EMadd1 H n ∘∞g Ωg→ (EMadd1_functor φ n) ∘∞g (gloop_EMadd1 G n)⁻¹ᵍ⁸
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definition EM_pmap [unfold 8] {G : AbGroup} (X : InfGroup) (n : ℕ)
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(e : AbInfGroup_of_AbGroup G →∞g Ωg'[n] X) [H : is_trunc n X] : EM G n →* X :=
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@ -366,9 +389,35 @@ namespace EM
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end
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definition EM_homomorphism_gloop [unfold 8] {G : AbGroup} (X : Type*) (n : ℕ)
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(e : AbInfGroup_of_AbGroup G →∞g Ωg[succ n] X) [H : is_trunc n X] : gEM G n →∞g Ωg X :=
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(e : AbInfGroup_of_AbGroup G →∞g Ωg[succ n] X) [H : is_trunc (n+1) X] : gEM G n →∞g Ωg X :=
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Ωg→ (EMadd1_pmap n e) ∘∞g !InfGroup_equiv_closed_isomorphism⁻¹ᵍ⁸
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definition gloopn_succ_in' (n : ℕ) (A : Type*) : Ωg[succ n] A ≃∞g Ωg'[n] (Ωg A) :=
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inf_isomorphism_of_equiv (loopn_succ_in n A)
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(by cases n with n; apply is_mul_hom_id; exact loopn_succ_in_con)
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definition EM_homomorphism_deloopable [unfold 8] {G : AbGroup} (X : Type*) (H : deloopable X) (n : ℕ)
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(e : AbInfGroup_of_AbGroup G →∞g Ωg'[n] (InfGroup_of_deloopable X)) (H : is_trunc (n+1) (deloop X)) :
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gEM G n →∞g InfGroup_of_deloopable X :=
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deloop_isomorphism X ∘∞g
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EM_homomorphism_gloop (deloop X) n
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((gloopn_succ_in' n (deloop X))⁻¹ᵍ⁸ ∘∞g Ωg'→[n] (deloop_isomorphism X)⁻¹ᵍ⁸ ∘∞g e)
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definition is_mul_hom_EM_functor [constructor] (G H : AbGroup) (n : ℕ) :
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is_mul_hom (λ(φ : G →gg H), EM_functor φ n) :=
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begin
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intro φ ψ,
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apply eq_of_phomotopy,
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exact sorry
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-- exact sorry, exact sorry, exact sorry
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-- refine idpath (φ 0 * ψ 0) ⬝ _,
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end
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/- an enriched homomorphism -/
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definition EM_ehomomorphism [constructor] (G H : AbGroup) (n : ℕ) :
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InfGroup_of_Group (G →gg H) →∞g InfGroup_of_deloopable (EM G n →** EM H n) :=
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inf_homomorphism.mk (λφ, EM_functor φ n) (is_mul_hom_EM_functor G H n)
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-- definition EM_homomorphism [unfold 8] {G : AbGroup} {X : Type*} (Y : Type*) (e : Ω Y ≃* X) (n : ℕ)
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-- (e : AbInfGroup_of_AbGroup G →∞g Ωg[succ n] X) [H : is_trunc n X] : gEM G n →∞g X :=
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-- _
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@ -2,9 +2,11 @@
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import .EM .smash_adjoint ..algebra.ring ..algebra.arrow_group
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open algebra eq EM is_equiv equiv is_trunc is_conn pointed trunc susp smash group nat
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open algebra eq EM is_equiv equiv is_trunc is_conn pointed trunc susp smash group nat function
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namespace EM
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definition EM1product_adj {R : Ring} :
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EM1 (AbGroup_of_Ring R) →* ppmap (EM1 (AbGroup_of_Ring R)) (EMadd1 (AbGroup_of_Ring R) 1) :=
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begin
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@ -25,6 +27,7 @@ definition EM0EMadd1product {A B C : AbGroup} (φ : A →g B →gg C) (n : ℕ)
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A →* EMadd1 B n →** EMadd1 C n :=
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EMadd1_pfunctor B C n ∘* pmap_of_homomorphism φ
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-- TODO: simplify
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definition EMadd1product {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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EMadd1 A n →* EMadd1 B m →** EMadd1 C (m + succ n) :=
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begin
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@ -32,10 +35,12 @@ begin
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{ refine is_trunc_pmap_of_is_conn _ (m.-1) !is_conn_EMadd1 _ _ _ _ !is_trunc_EMadd1,
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exact le_of_eq (trunc_index.of_nat_add_plus_two_of_nat m n)⁻¹ᵖ },
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apply EMadd1_pmap,
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exact sorry
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/- the underlying pointed map is: -/
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-- refine (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* ∘* ppcompose_left !loopn_EMadd1_add⁻¹ᵉ* ∘*
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-- EM0EMadd1product φ m
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refine (gloopn_pmap_isomorphism (succ n) _ _)⁻¹ᵍ⁸ ∘∞g
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gpmap_loop_homomorphism_right (EMadd1 B m) (loopn_EMadd1_add_of_eq C !succ_add)⁻¹ᵉ* ∘∞g
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gloop_pmap_isomorphism _ _ ∘∞g
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(deloop_isomorphism _)⁻¹ᵍ⁸ ∘∞g
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EM_ehomomorphism B C (succ m) ∘∞g
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inf_homomorphism_of_homomorphism φ
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end
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definition EMproduct1 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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@ -51,8 +56,9 @@ begin
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{ exact ppcompose_left (ptransport (EMadd1 C) !succ_add⁻¹) ∘* EMadd1product φ n m }}
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end
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definition EMproduct2 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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EM A n →* EM B m →** EM C (m + n) :=
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EM A n →* (EM B m →** EM C (m + n)) :=
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begin
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assert H1 : is_trunc n (gpmap_loop' (EM B m) (loop_EM C (m + n))),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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@ -75,5 +81,43 @@ begin
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exact sorry
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end
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definition EMproduct4 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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gEM A n →∞g Ωg (EM B m →** EM C (m + n + 1)) :=
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begin
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assert H1 : is_trunc (n+1) (EM B m →** EM C (m + n + 1)),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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apply EM_homomorphism_gloop,
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refine (gloopn_pmap_isomorphism _ _ _)⁻¹ᵍ⁸ ∘∞g _ ∘∞g inf_homomorphism_of_homomorphism φ,
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-- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
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-- _ ∘∞g inf_homomorphism_of_homomorphism φ
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exact sorry
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end
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definition EMproduct5 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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InfGroup_of_deloopable (EM A n) →∞g InfGroup_of_deloopable (EM B m →** EM C (m + n)) :=
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begin
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assert H1 : is_trunc (n + 1) (deloop (EM B m →** EM C (m + n))),
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{ exact is_trunc_pmap_of_is_conn_nat _ m !is_conn_EM _ _ _ !le.refl !is_trunc_EM },
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refine EM_homomorphism_deloopable _ _ _ _ _,
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-- exact _ /- (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* -/ ∘∞g _ /-ppcompose_left !loopn_EMadd1_add⁻¹ᵉ*-/ ∘∞g
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-- _ ∘∞g inf_homomorphism_of_homomorphism φ
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exact sorry
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end
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definition EMadd1product2 {A B C : AbGroup} (φ : A →g B →gg C) (n m : ℕ) :
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gEM A (n+1) →∞g Ωg[succ n] (EMadd1 B m →** EMadd1 C m) :=
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begin
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assert H1 : is_trunc (n+1) (Ω[n] (EMadd1 B m →** EMadd1 C m)),
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{ apply is_trunc_loopn, exact sorry },
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-- refine EM_homomorphism_gloop (Ω[n] (EMadd1 B m →** EMadd1 C m)) _ _,
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/- the underlying pointed map is: -/
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-- exact sorry
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-- refine (loopn_ppmap_pequiv _ _ _)⁻¹ᵉ* ∘* ppcompose_left !loopn_EMadd1_add⁻¹ᵉ* ∘*
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-- EM0EMadd1product φ m
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exact sorry
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end
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end EM
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@ -611,5 +611,29 @@ begin
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fapply phomotopy.mk, exact homotopy.rfl, refine whisker_left idp (ap_id p)⁻¹
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end
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structure deloopable.{u} [class] (A : pType.{u}) : Type.{u+1} :=
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(deloop : pType.{u})
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(deloop_pequiv : Ω deloop ≃* A)
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abbreviation deloop [unfold 2] := deloopable.deloop
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abbreviation deloop_pequiv [unfold 2] := deloopable.deloop_pequiv
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definition deloopable_loop [instance] [constructor] (A : Type*) : deloopable (Ω A) :=
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deloopable.mk A pequiv.rfl
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definition deloopable_loopn [instance] [priority 500] (n : ℕ) [H : is_succ n] (A : Type*) :
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deloopable (Ω[n] A) :=
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by induction H with n; exact deloopable.mk (Ω[n] A) pequiv.rfl
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definition inf_group_of_deloopable (A : Type*) [deloopable A] : inf_group A :=
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inf_group_equiv_closed (deloop_pequiv A) _
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definition InfGroup_of_deloopable (A : Type*) [deloopable A] : InfGroup :=
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InfGroup.mk A (inf_group_of_deloopable A)
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definition deloop_isomorphism [constructor] (A : Type*) [deloopable A] :
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Ωg (deloop A) ≃∞g InfGroup_of_deloopable A :=
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InfGroup_equiv_closed_isomorphism (Ωg (deloop A)) (deloop_pequiv A)
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end pointed
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@ -938,6 +938,15 @@ namespace pointed
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(Π*(a : A), Ω (B a)) ≃* Ω (Π*a, B a) :=
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(loop_pppi_pequiv B)⁻¹ᵉ*
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definition deloopable_pppi [instance] [constructor] {A : Type*} (B : A → Type*) [Πa, deloopable (B a)] :
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deloopable (Π*a, B a) :=
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deloopable.mk (Π*a, deloop (B a))
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(loop_pppi_pequiv (λa, deloop (B a)) ⬝e* ppi_pequiv_right (λa, deloop_pequiv (B a)))
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definition deloopable_ppmap [instance] [constructor] (A B : Type*) [deloopable B] :
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deloopable (A →** B) :=
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deloopable_pppi (λa, B)
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/- below is an alternate proof strategy for the naturality of loop_pppi_pequiv_natural,
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where we define loop_pppi_pequiv as composite of pointed equivalences, and proved the
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