279 lines
11 KiB
Text
279 lines
11 KiB
Text
/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Eilenberg MacLane spaces
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-/
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import homotopy.EM .spectrum
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open eq is_equiv equiv is_conn is_trunc unit function pointed nat group algebra trunc trunc_index
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fiber prod pointed susp EM.ops
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namespace EM
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/- Functorial action of Eilenberg-Maclane spaces -/
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definition pEM1_functor [constructor] {G H : Group} (φ : G →g H) : pEM1 G →* pEM1 H :=
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begin
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fconstructor,
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{ intro g, induction g,
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{ exact base },
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{ exact pth (φ g) },
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{ exact ap pth (respect_mul φ g h) ⬝ resp_mul (φ g) (φ h) }},
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{ reflexivity }
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end
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definition EMadd1_functor [constructor] {G H : CommGroup} (φ : G →g H) (n : ℕ) :
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EMadd1 G n →* EMadd1 H n :=
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begin
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apply ptrunc_functor,
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apply iterate_psusp_functor,
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apply pEM1_functor,
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exact φ
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end
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definition EM_functor [unfold 4] {G H : CommGroup} (φ : G →g H) (n : ℕ) :
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K G n →* K H n :=
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begin
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cases n with n,
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{ exact pmap_of_homomorphism φ },
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{ exact EMadd1_functor φ n }
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end
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-- TODO: (K G n →* K H n) ≃ (G →g H)
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/- Equivalence of Groups and pointed connected 1-truncated types -/
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definition pEM1_pequiv_ptruncconntype (X : 1-Type*[0]) : pEM1 (π₁ X) ≃* X :=
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pEM1_pequiv_type
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definition Group_equiv_ptruncconntype [constructor] : Group ≃ 1-Type*[0] :=
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equiv.MK (λG, ptruncconntype.mk (pEM1 G) _ pt !is_conn_pEM1)
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(λX, π₁ X)
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begin intro X, apply ptruncconntype_eq, esimp, exact pEM1_pequiv_type end
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begin intro G, apply eq_of_isomorphism, apply fundamental_group_pEM1 end
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/- Higher EM-spaces -/
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/- K(G, 2) is unique (see below for general case) -/
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definition loopn_EMadd1 (G : CommGroup) (n : ℕ) : Ω[succ n] (EMadd1 G n) ≃* pType_of_Group G :=
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begin
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refine _ ⬝e* loop_pEM1 G,
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cases n with n,
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{ refine !loop_ptrunc_pequiv ⬝e* _, refine ptrunc_pequiv _ _ _,
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apply is_trunc_eq, apply is_trunc_EM1},
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induction n with n IH,
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{ exact loop_pequiv_loop (loop_EM2 G)},
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refine _ ⬝e* IH,
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refine !homotopy_group_pequiv_loop_ptrunc⁻¹ᵉ* ⬝e* _ ⬝e* !homotopy_group_pequiv_loop_ptrunc,
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apply iterate_psusp_stability_pequiv,
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rexact add_mul_le_mul_add n 1 1
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end
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definition EM2_map [unfold 7] {G : CommGroup} {X : Type*} (e : Ω[2] X ≃ G)
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(r : Π(p q : Ω[2] X), e (@concat (Ω X) idp idp idp p q) = e p * e q)
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[is_conn 1 X] [is_trunc 2 X] : EMadd1 G 1 → X :=
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begin
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change trunc 2 (susp (EM1 G)) → X, intro x,
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induction x with x, induction x with x,
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{ exact pt},
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{ exact pt},
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{ change carrier (Ω X), refine EM1_map e r x}
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end
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definition pEM2_pmap [constructor] {G : CommGroup} {X : Type*} (e : Ω[2] X ≃ G)
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(r : Π(p q : Ω[2] X), e (@concat (Ω X) idp idp idp p q) = e p * e q)
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[is_conn 1 X] [is_trunc 2 X] : EMadd1 G 1 →* X :=
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pmap.mk (EM2_map e r) idp
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definition loop_pEM2_pmap {G : CommGroup} {X : Type*} (e : Ω[2] X ≃ G)
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(r : Π(p q : Ω[2] X), e (@concat (Ω X) idp idp idp p q) = e p * e q)
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[is_conn 1 X] [is_trunc 2 X] :
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Ω→[2](pEM2_pmap e r) ~ e⁻¹ᵉ ∘ loopn_EMadd1 G 1 :=
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begin
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exact sorry
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end
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-- TODO: make arguments in trivial_homotopy_group_of_is_trunc implicit
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attribute is_conn_EMadd1 is_trunc_EMadd1 [instance]
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definition pEM2_pequiv' {G : CommGroup} {X : Type*} (e : Ω[2] X ≃ G)
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(r : Π(p q : Ω[2] X), e (@concat (Ω X) idp idp idp p q) = e p * e q)
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[is_conn 1 X] [is_trunc 2 X] : EMadd1 G 1 ≃* X :=
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begin
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apply pequiv_of_pmap (pEM2_pmap e r),
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have is_conn 0 (EMadd1 G 1), from !is_conn_of_is_conn_succ,
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have is_trunc 2 (EMadd1 G 1), from !is_trunc_EMadd1,
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refine whitehead_principle_pointed 2 _ _,
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intro k, apply @nat.lt_by_cases k 2: intro H,
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{ apply @is_equiv_of_is_contr,
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do 2 exact trivial_homotopy_group_of_is_conn _ (le_of_lt_succ H)},
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{ cases H, esimp, apply is_equiv_trunc_functor, esimp,
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apply is_equiv.homotopy_closed, rotate 1,
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{ symmetry, exact loop_pEM2_pmap _ _},
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apply is_equiv_compose, apply pequiv.to_is_equiv, apply to_is_equiv},
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{ apply @is_equiv_of_is_contr,
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exact trivial_homotopy_group_of_is_trunc _ H,
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apply @trivial_homotopy_group_of_is_trunc, rotate 1, exact H, exact _inst_2}
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end
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definition pEM2_pequiv {G : CommGroup} {X : Type*} (e : πg[1+1] X ≃g G)
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[is_conn 1 X] [is_trunc 2 X] : EMadd1 G 1 ≃* X :=
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begin
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have is_set (Ω[2] X), from !is_trunc_eq,
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apply pEM2_pequiv' (!trunc_equiv⁻¹ᵉ ⬝e equiv_of_isomorphism e),
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intro p q, esimp, exact to_respect_mul e (tr p) (tr q)
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end
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-- general case
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definition EMadd1_map [unfold 8] {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃ G)
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(r : Π(p q : Ω (Ω[n] X)), e (p ⬝ q) = e p * e q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n → X :=
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begin
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revert X e r H1 H2, induction n with n f: intro X e r H1 H2,
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{ change trunc 1 (EM1 G) → X, intro x, induction x with x, exact EM1_map e r x},
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change trunc (n.+2) (susp (iterate_psusp n (pEM1 G))) → X, intro x,
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induction x with x, induction x with x,
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{ exact pt},
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{ exact pt},
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change carrier (Ω X), refine f _ _ _ _ _ (tr x),
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{ refine _⁻¹ᵉ ⬝e e, apply equiv_of_pequiv, apply loopn_succ_in},
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exact abstract begin
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intro p q, refine _ ⬝ !r, apply ap e, esimp,
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apply inv_eq_of_eq,
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refine _⁻¹ ⬝ !loopn_succ_in_con⁻¹,
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exact to_right_inv (loopn_succ_in X (succ n)) p ◾ to_right_inv (loopn_succ_in X (succ n)) q
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end end
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end
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definition pEMadd1_pmap [constructor] {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃ G)
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(r : Π(p q : Ω[succ n] X), e (@concat (Ω[n] X) pt pt pt p q) = e p * e q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n →* X :=
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pmap.mk (EMadd1_map e r) begin cases n with n: reflexivity end
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definition loop_pEMadd1_pmap {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃ G)
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(r : Π(p q : Ω[succ n] X), e (@concat (Ω[n] X) pt pt pt p q) = e p * e q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] :
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Ω→[succ n](pEMadd1_pmap e r) ~ e⁻¹ᵉ ∘ loopn_EMadd1 G n :=
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begin
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apply homotopy_of_inv_homotopy_pre (loopn_EMadd1 G n),
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intro g, esimp at *,
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revert X e r H1 H2, induction n with n IH: intro X e r H1 H2,
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{ refine !idp_con ⬝ _, refine !ap_compose'⁻¹ ⬝ _, apply elim_pth},
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{ replace (succ (succ n)) with ((succ n) + 1), rewrite [apn_succ],
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exact sorry}
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-- exact !idp_con ⬝ !elim_pth
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end
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-- definition is_conn_of_le (n : ℕ₋₂) (A : Type) [is_conn (n.+1) A] :
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-- is_conn n A :=
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-- is_trunc_trunc_of_le A -2 (trunc_index.self_le_succ n)
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-- attribute is_conn_EMadd1 is_trunc_EMadd1 [instance]
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definition pEMadd1_pequiv' {G : CommGroup} {X : Type*} {n : ℕ} (e : Ω[succ n] X ≃ G)
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(r : Π(p q : Ω[succ n] X), e (@concat (Ω[n] X) pt pt pt p q) = e p * e q)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n ≃* X :=
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begin
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apply pequiv_of_pmap (pEMadd1_pmap e r),
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have is_conn 0 (EMadd1 G n), from is_conn_of_le _ (zero_le_of_nat n),
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have is_trunc (n.+1) (EMadd1 G n), from !is_trunc_EMadd1,
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refine whitehead_principle_pointed (n.+1) _ _,
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intro k, apply @nat.lt_by_cases k (succ n): intro H,
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{ apply @is_equiv_of_is_contr,
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do 2 exact trivial_homotopy_group_of_is_conn _ (le_of_lt_succ H)},
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{ cases H, esimp, apply is_equiv_trunc_functor, esimp,
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apply is_equiv.homotopy_closed, rotate 1,
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{ symmetry, exact loop_pEMadd1_pmap _ _},
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apply is_equiv_compose, apply pequiv.to_is_equiv},
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{ apply @is_equiv_of_is_contr,
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do 2 exact trivial_homotopy_group_of_is_trunc _ H}
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end
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definition pEMadd1_pequiv {G : CommGroup} {X : Type*} {n : ℕ} (e : πg[n+1] X ≃g G)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EMadd1 G n ≃* X :=
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begin
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have is_set (Ω[succ n] X), from !is_set_loopn,
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apply pEMadd1_pequiv' (!trunc_equiv⁻¹ᵉ ⬝e equiv_of_isomorphism e),
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intro p q, esimp, exact to_respect_mul e (tr p) (tr q)
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end
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definition EM_pequiv_succ {G : CommGroup} {X : Type*} {n : ℕ} (e : πg[n+1] X ≃g G)
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[H1 : is_conn n X] [H2 : is_trunc (n.+1) X] : EM G (succ n) ≃* X :=
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pEMadd1_pequiv e
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definition EM_pequiv_zero {G : CommGroup} {X : Type*} (e : X ≃* pType_of_Group G) : EM G 0 ≃* X :=
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proof e⁻¹ᵉ* qed
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definition EM_spectrum /-[constructor]-/ (G : CommGroup) : spectrum :=
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spectrum.Mk (K G) (λn, (loop_EM G n)⁻¹ᵉ*)
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/- uniqueness of K(G,n), method 2: -/
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-- definition freudenthal_homotopy_group_pequiv (A : Type*) {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
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-- : π*[k + 1] (psusp A) ≃* π*[k] A :=
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-- calc
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-- π*[k + 1] (psusp A) ≃* π*[k] (Ω (psusp A)) : pequiv_of_eq (homotopy_group_succ_in (psusp A) k)
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-- ... ≃* Ω[k] (ptrunc k (Ω (psusp A))) : homotopy_group_pequiv_loop_ptrunc k (Ω (psusp A))
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-- ... ≃* Ω[k] (ptrunc k A) : loopn_pequiv_loopn k (freudenthal_pequiv A H)
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-- ... ≃* π*[k] A : (homotopy_group_pequiv_loop_ptrunc k A)⁻¹ᵉ*
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definition iterate_psusp_succ_pequiv (n : ℕ) (A : Type*) :
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iterate_psusp (succ n) A ≃* iterate_psusp n (psusp A) :=
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begin
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induction n with n IH,
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{ reflexivity},
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{ exact psusp_equiv IH}
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end
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definition is_conn_psusp [instance] (n : trunc_index) (A : Type*)
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[H : is_conn n A] : is_conn (n .+1) (psusp A) :=
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is_conn_susp n A
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definition iterated_freudenthal_pequiv (A : Type*) {n k m : ℕ} [HA : is_conn n A] (H : k ≤ 2 * n)
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: ptrunc k A ≃* ptrunc k (Ω[m] (iterate_psusp m A)) :=
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begin
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revert A n k HA H, induction m with m IH: intro A n k HA H,
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{ reflexivity},
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{ have H2 : succ k ≤ 2 * succ n,
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from calc
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succ k ≤ succ (2 * n) : succ_le_succ H
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... ≤ 2 * succ n : self_le_succ,
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exact calc
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ptrunc k A ≃* ptrunc k (Ω (psusp A)) : freudenthal_pequiv A H
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... ≃* Ω (ptrunc (succ k) (psusp A)) : loop_ptrunc_pequiv
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... ≃* Ω (ptrunc (succ k) (Ω[m] (iterate_psusp m (psusp A)))) :
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loop_pequiv_loop (IH (psusp A) (succ n) (succ k) _ H2)
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... ≃* ptrunc k (Ω[succ m] (iterate_psusp m (psusp A))) : loop_ptrunc_pequiv
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... ≃* ptrunc k (Ω[succ m] (iterate_psusp (succ m) A)) :
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ptrunc_pequiv_ptrunc _ (loopn_pequiv_loopn _ !iterate_psusp_succ_pequiv)}
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end
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definition pmap_eq_of_phomotopy {A B : Type*} {f g : A →* B} (p : f ~* g) : f = g :=
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pmap_eq (to_homotopy p) (to_homotopy_pt p)⁻¹
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definition pmap_equiv_pmap_right {A B : Type*} (C : Type*) (f : A ≃* B) : C →* A ≃ C →* B :=
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begin
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fapply equiv.MK,
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{ exact pcompose f},
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{ exact pcompose f⁻¹ᵉ*},
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{ intro f, apply pmap_eq_of_phomotopy,
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exact !passoc⁻¹* ⬝* pwhisker_right _ !pright_inv ⬝* !pid_pcompose},
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{ intro f, apply pmap_eq_of_phomotopy,
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exact !passoc⁻¹* ⬝* pwhisker_right _ !pleft_inv ⬝* !pid_pcompose}
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end
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definition iterate_psusp_adjoint_loopn [constructor] (X Y : Type*) (n : ℕ) :
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iterate_psusp n X →* Y ≃ X →* Ω[n] Y :=
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begin
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revert X Y, induction n with n IH: intro X Y,
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{ reflexivity},
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{ refine !susp_adjoint_loop ⬝e !IH ⬝e _, apply pmap_equiv_pmap_right,
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symmetry, apply loopn_succ_in}
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end
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end EM
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-- cohomology ∥ X → K(G,n) ∥
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-- reduced cohomology ∥ X →* K(G,n) ∥
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-- but we probably want to do this for any spectrum
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