Work on the construction of exact couples
This commit is contained in:
Floris van Doorn
parent
7ee38d255c
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cea1250ca6
8 changed files with 672 additions and 360 deletions
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@ -102,6 +102,10 @@ definition graded_hom.mk_out [constructor] (d : I ≃ I)
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(fn : Πi, M₁ (d⁻¹ i) →lm M₂ i) : M₁ →gm M₂ :=
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graded_hom.mk' d (λi j p, fn j ∘lm homomorphism_of_eq (ap M₁ (eq_inv_of_eq p)))
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definition graded_hom.mk_out' [constructor] (d : I ≃ I)
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(fn : Πi, M₁ (d i) →lm M₂ i) : M₁ →gm M₂ :=
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graded_hom.mk' d⁻¹ᵉ (λi j p, fn j ∘lm homomorphism_of_eq (ap M₁ (eq_inv_of_eq p)))
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definition graded_hom.mk_out_in [constructor] (d₁ : I ≃ I) (d₂ : I ≃ I)
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(fn : Πi, M₁ (d₁ i) →lm M₂ (d₂ i)) : M₁ →gm M₂ :=
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graded_hom.mk' (d₁⁻¹ᵉ ⬝e d₂) (λi j p, homomorphism_of_eq (ap M₂ p) ∘lm fn (d₁⁻¹ᵉ i) ∘lm
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@ -115,6 +119,16 @@ definition graded_hom_mk_refl (d : I ≃ I)
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(fn : Πi, M₁ i →lm M₂ (d i)) {i : I} (m : M₁ i) : graded_hom.mk d fn i m = fn i m :=
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by reflexivity
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definition graded_hom_mk_out'_left_inv (d : I ≃ I)
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(fn : Πi, M₁ (d i) →lm M₂ i) {i : I} (m : M₁ (d i)) :
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graded_hom.mk_out' d fn ↘ (left_inv d i) m = fn i m :=
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begin
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unfold [graded_hom.mk_out'],
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apply ap (λx, fn i (cast x m)),
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refine !ap_compose⁻¹ ⬝ ap02 _ _,
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apply is_set.elim --we can also prove this in arbitrary types
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end
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definition graded_hom_eq_zero {f : M₁ →gm M₂} {i j k : I} {q : deg f i = j} {p : deg f i = k}
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(m : M₁ i) (r : f ↘ q m = 0) : f ↘ p m = 0 :=
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have f ↘ p m = transport M₂ (q⁻¹ ⬝ p) (f ↘ q m), begin induction p, induction q, reflexivity end,
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@ -488,9 +502,6 @@ definition graded_homology_elim {g : M₂ →gm M₃} {f : M₁ →gm M₂} (h :
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(H : compose_constant h f) : graded_homology g f →gm M :=
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graded_hom.mk (deg h) (λi, homology_elim (h i) (H _ _))
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/- exact couples -/
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definition is_exact_gmod (f : M₁ →gm M₂) (f' : M₂ →gm M₃) : Type :=
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Π⦃i j k⦄ (p : deg f i = j) (q : deg f' j = k), is_exact_mod (f ↘ p) (f' ↘ q)
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@ -502,336 +513,11 @@ begin intro i j k p q; induction p; induction q; split, apply h₁, apply h₂ e
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definition gmod_im_in_ker (h : is_exact_gmod f f') : compose_constant f' f :=
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λi j k p q, is_exact.im_in_ker (h p q)
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-- structure exact_couple (M₁ M₂ : graded_module R I) : Type :=
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-- (i : M₁ →gm M₁) (j : M₁ →gm M₂) (k : M₂ →gm M₁)
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-- (exact_ij : is_exact_gmod i j)
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-- (exact_jk : is_exact_gmod j k)
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-- (exact_ki : is_exact_gmod k i)
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-- definition is_exact_gmod_mk_mk_out' {d₁ d₂ : I ≃ I} (fn₁ : Πi, M₁ i →lm M₂ (d₁ i))
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-- (fn₂ : Πi, M₂ (d₂ i) →lm M₃ i) (H : Πi, is_exact_mod (fn₁ i) _) : is_exact_gmod (graded_hom.mk d₁ fn₁) (graded_hom.mk_out' d₂ fn₂) :=
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-- begin
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end left_module
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namespace left_module
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structure exact_couple (R : Ring) (I : Set) : Type :=
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(D E : graded_module R I)
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(i : D →gm D) (j : D →gm E) (k : E →gm D)
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(ij : is_exact_gmod i j)
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(jk : is_exact_gmod j k)
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(ki : is_exact_gmod k i)
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namespace derived_couple
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section
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open exact_couple
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parameters {R : Ring} {I : Set} (X : exact_couple R I)
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local abbreviation D := D X
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local abbreviation E := E X
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local abbreviation i := i X
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local abbreviation j := j X
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local abbreviation k := k X
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local abbreviation ij := ij X
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local abbreviation jk := jk X
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local abbreviation ki := ki X
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definition d : E →gm E := j ∘gm k
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definition D' : graded_module R I := graded_image i
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definition E' : graded_module R I := graded_homology d d
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definition is_contr_E' {x : I} (H : is_contr (E x)) : is_contr (E' x) :=
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!is_contr_homology
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definition is_contr_D' {x : I} (H : is_contr (D x)) : is_contr (D' x) :=
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!is_contr_image_module
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definition i' : D' →gm D' :=
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graded_image_lift i ∘gm graded_submodule_incl _
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-- degree i + 0
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lemma is_surjective_i' {x y : I} (p : deg i' x = y)
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(H : Π⦃z⦄ (q : deg i z = x), is_surjective (i ↘ q)) : is_surjective (i' ↘ p) :=
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begin
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apply is_surjective_graded_hom_compose,
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{ intro y q, apply is_surjective_graded_image_lift },
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{ intro y q, apply is_surjective_of_is_equiv,
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induction q,
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exact to_is_equiv (equiv_of_isomorphism (image_module_isomorphism (i ← x) (H _)))
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}
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end
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lemma j_lemma1 ⦃x : I⦄ (m : D x) : d ((deg j) x) (j x m) = 0 :=
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begin
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rewrite [graded_hom_compose_fn,↑d,graded_hom_compose_fn],
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refine ap (graded_hom_fn j (deg k (deg j x))) _ ⬝
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!to_respect_zero,
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exact compose_constant.elim (gmod_im_in_ker (jk)) x m
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end
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lemma j_lemma2 : Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0),
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(graded_quotient_map _ ∘gm graded_hom_lift j j_lemma1) x m = 0 :> E' _ :=
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begin
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have Π⦃x y : I⦄ (q : deg k x = y) (r : deg d x = deg j y)
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(s : ap (deg j) q = r) ⦃m : D y⦄ (p : i y m = 0), image (d ↘ r) (j y m),
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begin
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intros, induction s, induction q,
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note m_in_im_k := is_exact.ker_in_im (ki idp _) _ p,
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induction m_in_im_k with e q,
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induction q,
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apply image.mk e idp
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end,
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have Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0), image (d ← (deg j x)) (j x m),
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begin
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intros,
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refine this _ _ _ p,
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exact to_right_inv (deg k) _ ⬝ to_left_inv (deg j) x,
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apply is_set.elim
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-- rewrite [ap_con, -adj],
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end,
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intros,
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rewrite [graded_hom_compose_fn],
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exact quotient_map_eq_zero _ (this p)
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end
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definition j' : D' →gm E' :=
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graded_image_elim (graded_homology_intro d d ∘gm graded_hom_lift j j_lemma1) j_lemma2
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-- degree deg j - deg i
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theorem k_lemma1 ⦃x : I⦄ (m : E x) : image (i ← (deg k x)) (k x m) :=
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begin
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exact sorry
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end
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theorem k_lemma2 : compose_constant (graded_hom_lift k k_lemma1 : E →gm D') d :=
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begin
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-- apply compose_constant.mk, intro x m,
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-- rewrite [graded_hom_compose_fn],
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-- refine ap (graded_hom_fn (graded_image_lift i) (deg k (deg d x))) _ ⬝ !to_respect_zero,
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-- exact compose_constant.elim (gmod_im_in_ker jk) (deg k x) (k x m)
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exact sorry
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end
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definition k' : E' →gm D' :=
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graded_homology_elim (graded_hom_lift k k_lemma1) k_lemma2
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definition deg_i' : deg i' ~ deg i := by reflexivity
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definition deg_j' : deg j' ~ deg j ∘ (deg i)⁻¹ := by reflexivity
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definition deg_k' : deg k' ~ deg k := by reflexivity
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theorem i'j' : is_exact_gmod i' j' :=
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begin
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apply is_exact_gmod.mk,
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{ intro x, refine total_image.rec _, intro m, exact sorry
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-- exact calc
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-- j' (deg i' x) (i' x ⟨(i ← x) m, image.mk m idp⟩)
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-- = j' (deg i' x) (graded_image_lift i x ((i ← x) m)) : idp
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-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
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-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
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-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
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-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
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-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
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-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
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-- ... = 0 : _
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},
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{ exact sorry }
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end
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theorem j'k' : is_exact_gmod j' k' :=
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begin
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apply is_exact_gmod.mk,
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{ exact sorry },
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{ exact sorry }
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end
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theorem k'i' : is_exact_gmod k' i' :=
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begin
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apply is_exact_gmod.mk,
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{ intro x m, exact sorry },
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{ exact sorry }
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end
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end
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end derived_couple
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section
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open derived_couple exact_couple
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definition derived_couple [constructor] {R : Ring} {I : Set}
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(X : exact_couple R I) : exact_couple R I :=
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⦃exact_couple, D := D' X, E := E' X, i := i' X, j := j' X, k := k' X,
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ij := i'j' X, jk := j'k' X, ki := k'i' X⦄
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parameters {R : Ring} {I : Set} (X : exact_couple R I) (B B' : I → ℕ)
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(Dub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))^[s] x = y → B x ≤ s → is_contr (D X y))
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(Eub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (k X))⁻¹ (iterate (deg (i X)) s ((deg (j X))⁻¹ x)) = y →
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B x ≤ s → is_contr (E X y))
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(Dlb : Π⦃x y z⦄ ⦃s : ℕ⦄ (p : deg (i X) x = y),
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iterate (deg (i X)) s y = z → B' z ≤ s → is_surjective (i X ↘ p))
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(Elb : Π⦃x y⦄ ⦃s : ℕ⦄, deg (j X) (iterate (deg (i X))⁻¹ᵉ s (deg (k X) x)) = y → B x ≤ s →
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is_contr (E X y))
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(deg_ik_commute : deg (i X) ∘ deg (k X) ~ deg (k X) ∘ deg (i X))
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definition deg_iterate_ik_commute (n : ℕ) (x : I) :
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(deg (i X))^[n] (deg (k X) x) = deg (k X) ((deg (i X))^[n] x) :=
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iterate_commute _ deg_ik_commute x
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-- we start counting pages at 0, not at 2.
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definition page (r : ℕ) : exact_couple R I :=
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iterate derived_couple r X
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definition is_contr_E (r : ℕ) (x : I) (h : is_contr (E X x)) :
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is_contr (E (page r) x) :=
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by induction r with r IH; exact h; exact is_contr_E' (page r) IH
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definition is_contr_D (r : ℕ) (x : I) (h : is_contr (D X x)) :
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is_contr (D (page r) x) :=
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by induction r with r IH; exact h; exact is_contr_D' (page r) IH
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definition deg_i (r : ℕ) : deg (i (page r)) ~ deg (i X) :=
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begin
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induction r with r IH,
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{ reflexivity },
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{ exact IH }
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end
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definition deg_k (r : ℕ) : deg (k (page r)) ~ deg (k X) :=
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begin
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induction r with r IH,
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{ reflexivity },
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{ exact IH }
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end
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definition deg_j (r : ℕ) :
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deg (j (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r :=
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begin
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induction r with r IH,
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{ reflexivity },
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{ refine hwhisker_left (deg (j (page r)))
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(to_inv_homotopy_inv (deg_i r)) ⬝hty _,
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refine hwhisker_right _ IH ⬝hty _,
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apply hwhisker_left, symmetry, apply iterate_succ }
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end
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definition deg_j_inv (r : ℕ) :
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(deg (j (page r)))⁻¹ ~ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
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have H : deg (j (page r)) ~ iterate_equiv (deg (i X))⁻¹ᵉ r ⬝e deg (j X), from deg_j r,
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λx, to_inv_homotopy_to_inv H x ⬝ iterate_inv (deg (i X))⁻¹ᵉ r ((deg (j X))⁻¹ x)
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definition deg_d (r : ℕ) :
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deg (d (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r ∘ deg (k X) :=
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compose2 (deg_j r) (deg_k r)
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definition deg_d_inv (r : ℕ) :
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(deg (d (page r)))⁻¹ ~ (deg (k X))⁻¹ ∘ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
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compose2 (to_inv_homotopy_to_inv (deg_k r)) (deg_j_inv r)
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include Elb Eub
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definition Estable {x : I} {r : ℕ} (H : B x ≤ r) :
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E (page (r + 1)) x ≃lm E (page r) x :=
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begin
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change homology (d (page r) x) (d (page r) ← x) ≃lm E (page r) x,
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apply homology_isomorphism: apply is_contr_E,
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exact Eub (deg_d_inv r x)⁻¹ H, exact Elb (deg_d r x)⁻¹ H
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end
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include Dlb
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definition is_surjective_i {x y z : I} {r s : ℕ} (H : B' z ≤ s + r)
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(p : deg (i (page r)) x = y) (q : iterate (deg (i X)) s y = z) :
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is_surjective (i (page r) ↘ p) :=
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begin
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revert x y z s H p q, induction r with r IH: intro x y z s H p q,
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{ exact Dlb p q H },
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{ change is_surjective (i' (page r) ↘ p),
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apply is_surjective_i', intro z' q',
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refine IH _ _ _ _ (le.trans H (le_of_eq (succ_add s r)⁻¹)) _ _,
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refine !iterate_succ ⬝ ap ((deg (i X))^[s]) _ ⬝ q,
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exact !deg_i⁻¹ ⬝ p }
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end
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definition Dstable {x : I} {r : ℕ} (H : B' x ≤ r) :
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D (page (r + 1)) x ≃lm D (page r) x :=
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begin
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change image_module (i (page r) ← x) ≃lm D (page r) x,
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refine image_module_isomorphism (i (page r) ← x)
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(is_surjective_i (le.trans H (le_of_eq !zero_add⁻¹)) _ _),
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reflexivity
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end
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definition Einf : graded_module R I :=
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λx, E (page (B x)) x
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definition Dinf : graded_module R I :=
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λx, D (page (B' x)) x
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definition Einfstable {x y : I} {r : ℕ} (Hr : B y ≤ r) (p : x = y) :
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Einf y ≃lm E (page r) x :=
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by symmetry; induction p; induction Hr with r Hr IH; reflexivity; exact Estable Hr ⬝lm IH
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definition Dinfstable {x y : I} {r : ℕ} (Hr : B' y ≤ r) (p : x = y) :
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Dinf y ≃lm D (page r) x :=
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by symmetry; induction p; induction Hr with r Hr IH; reflexivity; exact Dstable Hr ⬝lm IH
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parameters {x : I}
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definition r (n : ℕ) : ℕ :=
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max (max (B x + n + 1) (B ((deg (i X))^[n] x)))
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(max (B' (deg (k X) ((deg (i X))^[n] x)))
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(max (B' (deg (k X) ((deg (i X))^[n+1] x))) (B ((deg (j X))⁻¹ ((deg (i X))^[n] x)))))
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lemma rb0 (n : ℕ) : r n ≥ n + 1 :=
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ge.trans !le_max_left (ge.trans !le_max_left !le_add_left)
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lemma rb1 (n : ℕ) : B x ≤ r n - (n + 1) :=
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le_sub_of_add_le (le.trans !le_max_left !le_max_left)
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lemma rb2 (n : ℕ) : B ((deg (i X))^[n] x) ≤ r n :=
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le.trans !le_max_right !le_max_left
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lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ r n :=
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le.trans !le_max_left !le_max_right
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lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ r n :=
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le.trans (le.trans !le_max_left !le_max_right) !le_max_right
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lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ r n :=
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le.trans (le.trans !le_max_right !le_max_right) !le_max_right
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definition Einfdiag : graded_module R ℕ :=
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λn, Einf ((deg (i X))^[n] x)
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definition Dinfdiag : graded_module R ℕ :=
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λn, Dinf (deg (k X) ((deg (i X))^[n] x))
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include deg_ik_commute Dub
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definition short_exact_mod_page_r (n : ℕ) : short_exact_mod
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(E (page (r n)) ((deg (i X))^[n] x))
|
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(D (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)))
|
||||
(D (page (r n)) (deg (i (page (r n))) (deg (k (page (r n))) ((deg (i X))^[n] x)))) :=
|
||||
begin
|
||||
fapply short_exact_mod_of_is_exact,
|
||||
{ exact j (page (r n)) ← ((deg (i X))^[n] x) },
|
||||
{ exact k (page (r n)) ((deg (i X))^[n] x) },
|
||||
{ exact i (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)) },
|
||||
{ exact j (page (r n)) _ },
|
||||
{ apply is_contr_D, refine Dub !deg_j_inv⁻¹ (rb5 n) },
|
||||
{ apply is_contr_E, refine Elb _ (rb1 n),
|
||||
refine ap (deg (j X)) _ ⬝ !deg_j⁻¹,
|
||||
refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[r n]),
|
||||
exact ap (deg (i X)) (!deg_iterate_ik_commute ⬝ !deg_k⁻¹) ⬝ !deg_i⁻¹ },
|
||||
{ apply jk (page (r n)) },
|
||||
{ apply ki (page (r n)) },
|
||||
{ apply ij (page (r n)) }
|
||||
end
|
||||
|
||||
definition short_exact_mod_infpage (n : ℕ) :
|
||||
short_exact_mod (Einfdiag n) (Dinfdiag n) (Dinfdiag (n+1)) :=
|
||||
begin
|
||||
refine short_exact_mod_isomorphism _ _ _ (short_exact_mod_page_r n),
|
||||
{ exact Einfstable !rb2 idp },
|
||||
{ exact Dinfstable !rb3 !deg_k },
|
||||
{ exact Dinfstable !rb4 (!deg_i ⬝ ap (deg (i X)) !deg_k ⬝ !deg_ik_commute) }
|
||||
end
|
||||
|
||||
definition Dinfdiag0 (bound_zero : B' (deg (k X) x) = 0) : Dinfdiag 0 ≃lm D X (deg (k X) x) :=
|
||||
Dinfstable (le_of_eq bound_zero) idp
|
||||
|
||||
definition Dinfdiag_stable {s : ℕ} (h : B (deg (k X) x) ≤ s) : is_contr (Dinfdiag s) :=
|
||||
is_contr_D _ _ (Dub !deg_iterate_ik_commute h)
|
||||
|
||||
end
|
||||
-- end
|
||||
|
||||
|
||||
end left_module
|
||||
|
|
|
|||
530
algebra/module_exact_couple.hlean
Normal file
530
algebra/module_exact_couple.hlean
Normal file
|
|
@ -0,0 +1,530 @@
|
|||
/- Exact couples of graded (left-) R-modules. -/
|
||||
|
||||
-- Author: Floris van Doorn
|
||||
|
||||
import .graded ..homotopy.spectrum .product_group
|
||||
|
||||
open algebra is_trunc left_module is_equiv equiv eq function nat
|
||||
|
||||
-- move
|
||||
section
|
||||
open group int chain_complex pointed succ_str
|
||||
definition LeftModule_int_of_AbGroup [constructor] (A : AbGroup) : LeftModule rℤ :=
|
||||
LeftModule.mk A (left_module.mk sorry sorry sorry sorry 1 sorry sorry sorry sorry sorry sorry sorry sorry sorry)
|
||||
|
||||
definition lm_hom_int.mk [constructor] {A B : AbGroup} (φ : A →g B) :
|
||||
LeftModule_int_of_AbGroup A →lm LeftModule_int_of_AbGroup B :=
|
||||
homomorphism.mk φ sorry
|
||||
|
||||
definition is_exact_of_is_exact_at {N : succ_str} {A : chain_complex N} {n : N}
|
||||
(H : is_exact_at A n) : is_exact (cc_to_fn A (S n)) (cc_to_fn A n) :=
|
||||
is_exact.mk (cc_is_chain_complex A n) H
|
||||
|
||||
end
|
||||
/- exact couples -/
|
||||
|
||||
namespace left_module
|
||||
|
||||
structure exact_couple (R : Ring) (I : Set) : Type :=
|
||||
(D E : graded_module R I)
|
||||
(i : D →gm D) (j : D →gm E) (k : E →gm D)
|
||||
(ij : is_exact_gmod i j)
|
||||
(jk : is_exact_gmod j k)
|
||||
(ki : is_exact_gmod k i)
|
||||
|
||||
namespace derived_couple
|
||||
section
|
||||
open exact_couple
|
||||
|
||||
parameters {R : Ring} {I : Set} (X : exact_couple R I)
|
||||
local abbreviation D := D X
|
||||
local abbreviation E := E X
|
||||
local abbreviation i := i X
|
||||
local abbreviation j := j X
|
||||
local abbreviation k := k X
|
||||
local abbreviation ij := ij X
|
||||
local abbreviation jk := jk X
|
||||
local abbreviation ki := ki X
|
||||
|
||||
definition d : E →gm E := j ∘gm k
|
||||
definition D' : graded_module R I := graded_image i
|
||||
definition E' : graded_module R I := graded_homology d d
|
||||
|
||||
definition is_contr_E' {x : I} (H : is_contr (E x)) : is_contr (E' x) :=
|
||||
!is_contr_homology
|
||||
|
||||
definition is_contr_D' {x : I} (H : is_contr (D x)) : is_contr (D' x) :=
|
||||
!is_contr_image_module
|
||||
|
||||
definition i' : D' →gm D' :=
|
||||
graded_image_lift i ∘gm graded_submodule_incl _
|
||||
-- degree i + 0
|
||||
|
||||
lemma is_surjective_i' {x y : I} (p : deg i' x = y)
|
||||
(H : Π⦃z⦄ (q : deg i z = x), is_surjective (i ↘ q)) : is_surjective (i' ↘ p) :=
|
||||
begin
|
||||
apply is_surjective_graded_hom_compose,
|
||||
{ intro y q, apply is_surjective_graded_image_lift },
|
||||
{ intro y q, apply is_surjective_of_is_equiv,
|
||||
induction q,
|
||||
exact to_is_equiv (equiv_of_isomorphism (image_module_isomorphism (i ← x) (H _)))
|
||||
}
|
||||
end
|
||||
|
||||
lemma j_lemma1 ⦃x : I⦄ (m : D x) : d ((deg j) x) (j x m) = 0 :=
|
||||
begin
|
||||
rewrite [graded_hom_compose_fn,↑d,graded_hom_compose_fn],
|
||||
refine ap (graded_hom_fn j (deg k (deg j x))) _ ⬝
|
||||
!to_respect_zero,
|
||||
exact compose_constant.elim (gmod_im_in_ker (jk)) x m
|
||||
end
|
||||
|
||||
lemma j_lemma2 : Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0),
|
||||
(graded_quotient_map _ ∘gm graded_hom_lift j j_lemma1) x m = 0 :> E' _ :=
|
||||
begin
|
||||
have Π⦃x y : I⦄ (q : deg k x = y) (r : deg d x = deg j y)
|
||||
(s : ap (deg j) q = r) ⦃m : D y⦄ (p : i y m = 0), image (d ↘ r) (j y m),
|
||||
begin
|
||||
intros, induction s, induction q,
|
||||
note m_in_im_k := is_exact.ker_in_im (ki idp _) _ p,
|
||||
induction m_in_im_k with e q,
|
||||
induction q,
|
||||
apply image.mk e idp
|
||||
end,
|
||||
have Π⦃x : I⦄ ⦃m : D x⦄ (p : i x m = 0), image (d ← (deg j x)) (j x m),
|
||||
begin
|
||||
intros,
|
||||
refine this _ _ _ p,
|
||||
exact to_right_inv (deg k) _ ⬝ to_left_inv (deg j) x,
|
||||
apply is_set.elim
|
||||
-- rewrite [ap_con, -adj],
|
||||
end,
|
||||
intros,
|
||||
rewrite [graded_hom_compose_fn],
|
||||
exact quotient_map_eq_zero _ (this p)
|
||||
end
|
||||
|
||||
definition j' : D' →gm E' :=
|
||||
graded_image_elim (graded_homology_intro d d ∘gm graded_hom_lift j j_lemma1) j_lemma2
|
||||
-- degree deg j - deg i
|
||||
|
||||
lemma k_lemma1 ⦃x : I⦄ (m : E x) : image (i ← (deg k x)) (k x m) :=
|
||||
begin
|
||||
exact sorry
|
||||
end
|
||||
|
||||
lemma k_lemma2 : compose_constant (graded_hom_lift k k_lemma1 : E →gm D') d :=
|
||||
begin
|
||||
-- apply compose_constant.mk, intro x m,
|
||||
-- rewrite [graded_hom_compose_fn],
|
||||
-- refine ap (graded_hom_fn (graded_image_lift i) (deg k (deg d x))) _ ⬝ !to_respect_zero,
|
||||
-- exact compose_constant.elim (gmod_im_in_ker jk) (deg k x) (k x m)
|
||||
exact sorry
|
||||
end
|
||||
|
||||
definition k' : E' →gm D' :=
|
||||
graded_homology_elim (graded_hom_lift k k_lemma1) k_lemma2
|
||||
|
||||
definition deg_i' : deg i' ~ deg i := by reflexivity
|
||||
definition deg_j' : deg j' ~ deg j ∘ (deg i)⁻¹ := by reflexivity
|
||||
definition deg_k' : deg k' ~ deg k := by reflexivity
|
||||
|
||||
lemma i'j' : is_exact_gmod i' j' :=
|
||||
begin
|
||||
apply is_exact_gmod.mk,
|
||||
{ intro x, refine total_image.rec _, intro m, exact sorry
|
||||
-- exact calc
|
||||
-- j' (deg i' x) (i' x ⟨(i ← x) m, image.mk m idp⟩)
|
||||
-- = j' (deg i' x) (graded_image_lift i x ((i ← x) m)) : idp
|
||||
-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
|
||||
-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
|
||||
-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
|
||||
-- ... = graded_homology_intro d d (deg j ((deg i)⁻¹ᵉ (deg i x)))
|
||||
-- (graded_hom_lift j j_lemma1 ((deg i)⁻¹ᵉ (deg i x))
|
||||
-- (i ↘ (!to_right_inv ⬝ !to_left_inv⁻¹) m)) : _
|
||||
-- ... = 0 : _
|
||||
},
|
||||
{ exact sorry }
|
||||
end
|
||||
|
||||
lemma j'k' : is_exact_gmod j' k' :=
|
||||
begin
|
||||
apply is_exact_gmod.mk,
|
||||
{ exact sorry },
|
||||
{ exact sorry }
|
||||
end
|
||||
|
||||
lemma k'i' : is_exact_gmod k' i' :=
|
||||
begin
|
||||
apply is_exact_gmod.mk,
|
||||
{ intro x m, exact sorry },
|
||||
{ exact sorry }
|
||||
end
|
||||
|
||||
end
|
||||
end derived_couple
|
||||
|
||||
section
|
||||
open derived_couple exact_couple
|
||||
|
||||
definition derived_couple [constructor] {R : Ring} {I : Set}
|
||||
(X : exact_couple R I) : exact_couple R I :=
|
||||
⦃exact_couple, D := D' X, E := E' X, i := i' X, j := j' X, k := k' X,
|
||||
ij := i'j' X, jk := j'k' X, ki := k'i' X⦄
|
||||
|
||||
parameters {R : Ring} {I : Set} (X : exact_couple R I) (B B' : I → ℕ)
|
||||
(Dub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (i X))^[s] x = y → B x ≤ s → is_contr (D X y))
|
||||
(Eub : Π⦃x y⦄ ⦃s : ℕ⦄, (deg (k X))⁻¹ (iterate (deg (i X)) s ((deg (j X))⁻¹ x)) = y →
|
||||
B x ≤ s → is_contr (E X y))
|
||||
(Dlb : Π⦃x y z⦄ ⦃s : ℕ⦄ (p : deg (i X) x = y),
|
||||
iterate (deg (i X)) s y = z → B' z ≤ s → is_surjective (i X ↘ p))
|
||||
(Elb : Π⦃x y⦄ ⦃s : ℕ⦄, deg (j X) (iterate (deg (i X))⁻¹ᵉ s (deg (k X) x)) = y → B x ≤ s →
|
||||
is_contr (E X y))
|
||||
(deg_ik_commute : deg (i X) ∘ deg (k X) ~ deg (k X) ∘ deg (i X))
|
||||
|
||||
definition deg_iterate_ik_commute (n : ℕ) (x : I) :
|
||||
(deg (i X))^[n] (deg (k X) x) = deg (k X) ((deg (i X))^[n] x) :=
|
||||
iterate_commute _ deg_ik_commute x
|
||||
|
||||
-- we start counting pages at 0, not at 2.
|
||||
definition page (r : ℕ) : exact_couple R I :=
|
||||
iterate derived_couple r X
|
||||
|
||||
definition is_contr_E (r : ℕ) (x : I) (h : is_contr (E X x)) :
|
||||
is_contr (E (page r) x) :=
|
||||
by induction r with r IH; exact h; exact is_contr_E' (page r) IH
|
||||
|
||||
definition is_contr_D (r : ℕ) (x : I) (h : is_contr (D X x)) :
|
||||
is_contr (D (page r) x) :=
|
||||
by induction r with r IH; exact h; exact is_contr_D' (page r) IH
|
||||
|
||||
definition deg_i (r : ℕ) : deg (i (page r)) ~ deg (i X) :=
|
||||
begin
|
||||
induction r with r IH,
|
||||
{ reflexivity },
|
||||
{ exact IH }
|
||||
end
|
||||
|
||||
definition deg_k (r : ℕ) : deg (k (page r)) ~ deg (k X) :=
|
||||
begin
|
||||
induction r with r IH,
|
||||
{ reflexivity },
|
||||
{ exact IH }
|
||||
end
|
||||
|
||||
definition deg_j (r : ℕ) :
|
||||
deg (j (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r :=
|
||||
begin
|
||||
induction r with r IH,
|
||||
{ reflexivity },
|
||||
{ refine hwhisker_left (deg (j (page r)))
|
||||
(to_inv_homotopy_inv (deg_i r)) ⬝hty _,
|
||||
refine hwhisker_right _ IH ⬝hty _,
|
||||
apply hwhisker_left, symmetry, apply iterate_succ }
|
||||
end
|
||||
|
||||
definition deg_j_inv (r : ℕ) :
|
||||
(deg (j (page r)))⁻¹ ~ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
|
||||
have H : deg (j (page r)) ~ iterate_equiv (deg (i X))⁻¹ᵉ r ⬝e deg (j X), from deg_j r,
|
||||
λx, to_inv_homotopy_to_inv H x ⬝ iterate_inv (deg (i X))⁻¹ᵉ r ((deg (j X))⁻¹ x)
|
||||
|
||||
definition deg_d (r : ℕ) :
|
||||
deg (d (page r)) ~ deg (j X) ∘ iterate (deg (i X))⁻¹ r ∘ deg (k X) :=
|
||||
compose2 (deg_j r) (deg_k r)
|
||||
|
||||
definition deg_d_inv (r : ℕ) :
|
||||
(deg (d (page r)))⁻¹ ~ (deg (k X))⁻¹ ∘ iterate (deg (i X)) r ∘ (deg (j X))⁻¹ :=
|
||||
compose2 (to_inv_homotopy_to_inv (deg_k r)) (deg_j_inv r)
|
||||
|
||||
include Elb Eub
|
||||
definition Estable {x : I} {r : ℕ} (H : B x ≤ r) :
|
||||
E (page (r + 1)) x ≃lm E (page r) x :=
|
||||
begin
|
||||
change homology (d (page r) x) (d (page r) ← x) ≃lm E (page r) x,
|
||||
apply homology_isomorphism: apply is_contr_E,
|
||||
exact Eub (deg_d_inv r x)⁻¹ H, exact Elb (deg_d r x)⁻¹ H
|
||||
end
|
||||
|
||||
include Dlb
|
||||
definition is_surjective_i {x y z : I} {r s : ℕ} (H : B' z ≤ s + r)
|
||||
(p : deg (i (page r)) x = y) (q : iterate (deg (i X)) s y = z) :
|
||||
is_surjective (i (page r) ↘ p) :=
|
||||
begin
|
||||
revert x y z s H p q, induction r with r IH: intro x y z s H p q,
|
||||
{ exact Dlb p q H },
|
||||
{ change is_surjective (i' (page r) ↘ p),
|
||||
apply is_surjective_i', intro z' q',
|
||||
refine IH _ _ _ _ (le.trans H (le_of_eq (succ_add s r)⁻¹)) _ _,
|
||||
refine !iterate_succ ⬝ ap ((deg (i X))^[s]) _ ⬝ q,
|
||||
exact !deg_i⁻¹ ⬝ p }
|
||||
end
|
||||
|
||||
definition Dstable {x : I} {r : ℕ} (H : B' x ≤ r) :
|
||||
D (page (r + 1)) x ≃lm D (page r) x :=
|
||||
begin
|
||||
change image_module (i (page r) ← x) ≃lm D (page r) x,
|
||||
refine image_module_isomorphism (i (page r) ← x)
|
||||
(is_surjective_i (le.trans H (le_of_eq !zero_add⁻¹)) _ _),
|
||||
reflexivity
|
||||
end
|
||||
|
||||
definition Einf : graded_module R I :=
|
||||
λx, E (page (B x)) x
|
||||
|
||||
definition Dinf : graded_module R I :=
|
||||
λx, D (page (B' x)) x
|
||||
|
||||
definition Einfstable {x y : I} {r : ℕ} (Hr : B y ≤ r) (p : x = y) :
|
||||
Einf y ≃lm E (page r) x :=
|
||||
by symmetry; induction p; induction Hr with r Hr IH; reflexivity; exact Estable Hr ⬝lm IH
|
||||
|
||||
definition Dinfstable {x y : I} {r : ℕ} (Hr : B' y ≤ r) (p : x = y) :
|
||||
Dinf y ≃lm D (page r) x :=
|
||||
by symmetry; induction p; induction Hr with r Hr IH; reflexivity; exact Dstable Hr ⬝lm IH
|
||||
|
||||
parameters {x : I}
|
||||
|
||||
definition r (n : ℕ) : ℕ :=
|
||||
max (max (B x + n + 1) (B ((deg (i X))^[n] x)))
|
||||
(max (B' (deg (k X) ((deg (i X))^[n] x)))
|
||||
(max (B' (deg (k X) ((deg (i X))^[n+1] x))) (B ((deg (j X))⁻¹ ((deg (i X))^[n] x)))))
|
||||
|
||||
lemma rb0 (n : ℕ) : r n ≥ n + 1 :=
|
||||
ge.trans !le_max_left (ge.trans !le_max_left !le_add_left)
|
||||
lemma rb1 (n : ℕ) : B x ≤ r n - (n + 1) :=
|
||||
le_sub_of_add_le (le.trans !le_max_left !le_max_left)
|
||||
lemma rb2 (n : ℕ) : B ((deg (i X))^[n] x) ≤ r n :=
|
||||
le.trans !le_max_right !le_max_left
|
||||
lemma rb3 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n] x)) ≤ r n :=
|
||||
le.trans !le_max_left !le_max_right
|
||||
lemma rb4 (n : ℕ) : B' (deg (k X) ((deg (i X))^[n+1] x)) ≤ r n :=
|
||||
le.trans (le.trans !le_max_left !le_max_right) !le_max_right
|
||||
lemma rb5 (n : ℕ) : B ((deg (j X))⁻¹ ((deg (i X))^[n] x)) ≤ r n :=
|
||||
le.trans (le.trans !le_max_right !le_max_right) !le_max_right
|
||||
|
||||
definition Einfdiag : graded_module R ℕ :=
|
||||
λn, Einf ((deg (i X))^[n] x)
|
||||
|
||||
definition Dinfdiag : graded_module R ℕ :=
|
||||
λn, Dinf (deg (k X) ((deg (i X))^[n] x))
|
||||
|
||||
include deg_ik_commute Dub
|
||||
definition short_exact_mod_page_r (n : ℕ) : short_exact_mod
|
||||
(E (page (r n)) ((deg (i X))^[n] x))
|
||||
(D (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)))
|
||||
(D (page (r n)) (deg (i (page (r n))) (deg (k (page (r n))) ((deg (i X))^[n] x)))) :=
|
||||
begin
|
||||
fapply short_exact_mod_of_is_exact,
|
||||
{ exact j (page (r n)) ← ((deg (i X))^[n] x) },
|
||||
{ exact k (page (r n)) ((deg (i X))^[n] x) },
|
||||
{ exact i (page (r n)) (deg (k (page (r n))) ((deg (i X))^[n] x)) },
|
||||
{ exact j (page (r n)) _ },
|
||||
{ apply is_contr_D, refine Dub !deg_j_inv⁻¹ (rb5 n) },
|
||||
{ apply is_contr_E, refine Elb _ (rb1 n),
|
||||
refine ap (deg (j X)) _ ⬝ !deg_j⁻¹,
|
||||
refine iterate_sub _ !rb0 _ ⬝ _, apply ap (_^[r n]),
|
||||
exact ap (deg (i X)) (!deg_iterate_ik_commute ⬝ !deg_k⁻¹) ⬝ !deg_i⁻¹ },
|
||||
{ apply jk (page (r n)) },
|
||||
{ apply ki (page (r n)) },
|
||||
{ apply ij (page (r n)) }
|
||||
end
|
||||
|
||||
definition short_exact_mod_infpage (n : ℕ) :
|
||||
short_exact_mod (Einfdiag n) (Dinfdiag n) (Dinfdiag (n+1)) :=
|
||||
begin
|
||||
refine short_exact_mod_isomorphism _ _ _ (short_exact_mod_page_r n),
|
||||
{ exact Einfstable !rb2 idp },
|
||||
{ exact Dinfstable !rb3 !deg_k },
|
||||
{ exact Dinfstable !rb4 (!deg_i ⬝ ap (deg (i X)) !deg_k ⬝ !deg_ik_commute) }
|
||||
end
|
||||
|
||||
definition Dinfdiag0 (bound_zero : B' (deg (k X) x) = 0) : Dinfdiag 0 ≃lm D X (deg (k X) x) :=
|
||||
Dinfstable (le_of_eq bound_zero) idp
|
||||
|
||||
definition Dinfdiag_stable {s : ℕ} (h : B (deg (k X) x) ≤ s) : is_contr (Dinfdiag s) :=
|
||||
is_contr_D _ _ (Dub !deg_iterate_ik_commute h)
|
||||
|
||||
end
|
||||
|
||||
end left_module
|
||||
open left_module
|
||||
namespace pointed
|
||||
-- move
|
||||
open pointed int group is_trunc trunc is_conn
|
||||
|
||||
section
|
||||
variables {A B : Type} (f : A ≃ B) [ab_group A]
|
||||
|
||||
-- to group
|
||||
definition group_equiv_mul_comm (b b' : B) : group_equiv_mul f b b' = group_equiv_mul f b' b :=
|
||||
by rewrite [↑group_equiv_mul, mul.comm]
|
||||
|
||||
definition ab_group_equiv_closed : ab_group B :=
|
||||
⦃ab_group, group_equiv_closed f,
|
||||
mul_comm := group_equiv_mul_comm f⦄
|
||||
end
|
||||
|
||||
definition ab_group_of_is_contr (A : Type) [is_contr A] : ab_group A :=
|
||||
have ab_group unit, from ab_group_unit,
|
||||
ab_group_equiv_closed (equiv_unit_of_is_contr A)⁻¹ᵉ
|
||||
|
||||
definition group_of_is_contr (A : Type) [is_contr A] : group A :=
|
||||
have ab_group A, from ab_group_of_is_contr A, by apply _
|
||||
|
||||
definition ab_group_lift_unit : ab_group (lift unit) :=
|
||||
ab_group_of_is_contr (lift unit)
|
||||
|
||||
definition trivial_ab_group_lift : AbGroup :=
|
||||
AbGroup.mk _ ab_group_lift_unit
|
||||
|
||||
definition homomorphism_of_is_contr_right (A : Group) {B : Type} (H : is_contr B) :
|
||||
A →g Group.mk B (group_of_is_contr B) :=
|
||||
group.homomorphism.mk (λa, center _) (λa a', !is_prop.elim)
|
||||
|
||||
definition ab_group_homotopy_group_of_is_conn (n : ℕ) (A : Type*) [H : is_conn 1 A] : ab_group (π[n] A) :=
|
||||
begin
|
||||
have is_conn 0 A, from !is_conn_of_is_conn_succ,
|
||||
cases n with n,
|
||||
{ unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
|
||||
cases n with n,
|
||||
{ unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
|
||||
exact ab_group_homotopy_group n A
|
||||
end
|
||||
|
||||
definition homotopy_group_conn_nat (n : ℕ) (A : Type*[1]) : AbGroup :=
|
||||
AbGroup.mk (π[n] A) (ab_group_homotopy_group_of_is_conn n A)
|
||||
|
||||
definition homotopy_group_conn : Π(n : ℤ) (A : Type*[1]), AbGroup
|
||||
| (of_nat n) A := homotopy_group_conn_nat n A
|
||||
| (-[1+ n]) A := trivial_ab_group_lift
|
||||
|
||||
notation `πag'[`:95 n:0 `]`:0 := homotopy_group_conn n
|
||||
|
||||
definition homotopy_group_conn_nat_functor (n : ℕ) {A B : Type*[1]} (f : A →* B) :
|
||||
homotopy_group_conn_nat n A →g homotopy_group_conn_nat n B :=
|
||||
begin
|
||||
cases n with n, { apply homomorphism_of_is_contr_right },
|
||||
cases n with n, { apply homomorphism_of_is_contr_right },
|
||||
exact π→g[n+2] f
|
||||
end
|
||||
|
||||
definition homotopy_group_conn_functor : Π(n : ℤ) {A B : Type*[1]} (f : A →* B), πag'[n] A →g πag'[n] B
|
||||
| (of_nat n) A B f := homotopy_group_conn_nat_functor n f
|
||||
| (-[1+ n]) A B f := homomorphism_of_is_contr_right _ _
|
||||
|
||||
notation `π→ag'[`:95 n:0 `]`:0 := homotopy_group_conn_functor n
|
||||
|
||||
section
|
||||
open prod prod.ops fiber
|
||||
parameters {A : ℤ → Type*[1]} (f : Π(n : ℤ), A n →* A (n - 1)) [Hf : Πn, is_conn_fun 1 (f n)]
|
||||
include Hf
|
||||
definition I [constructor] : Set := trunctype.mk (ℤ × ℤ) !is_trunc_prod
|
||||
|
||||
definition D_sequence : graded_module rℤ I :=
|
||||
λv, LeftModule_int_of_AbGroup (πag'[v.2] (A (v.1)))
|
||||
|
||||
definition E_sequence : graded_module rℤ I :=
|
||||
λv, LeftModule_int_of_AbGroup (πag'[v.2] (pconntype.mk (pfiber (f (v.1))) !Hf pt))
|
||||
|
||||
definition exact_couple_sequence : exact_couple rℤ I :=
|
||||
exact_couple.mk D_sequence E_sequence sorry sorry sorry sorry sorry sorry
|
||||
|
||||
end
|
||||
|
||||
|
||||
end pointed
|
||||
|
||||
namespace spectrum
|
||||
open pointed int group is_trunc trunc is_conn prod prod.ops group fin chain_complex
|
||||
section
|
||||
-- notation `πₛ→[`:95 n:0 `]`:0 := shomotopy_group_fun n
|
||||
|
||||
definition is_equiv_mul_right [constructor] {A : Group} (a : A) : is_equiv (λb, b * a) :=
|
||||
adjointify _ (λb : A, b * a⁻¹) (λb, !inv_mul_cancel_right) (λb, !mul_inv_cancel_right)
|
||||
|
||||
definition right_action [constructor] {A : Group} (a : A) : A ≃ A :=
|
||||
equiv.mk _ (is_equiv_mul_right a)
|
||||
|
||||
definition is_equiv_add_right [constructor] {A : AddGroup} (a : A) : is_equiv (λb, b + a) :=
|
||||
adjointify _ (λb : A, b - a) (λb, !neg_add_cancel_right) (λb, !add_neg_cancel_right)
|
||||
|
||||
definition add_right_action [constructor] {A : AddGroup} (a : A) : A ≃ A :=
|
||||
equiv.mk _ (is_equiv_add_right a)
|
||||
|
||||
parameters {A : ℤ → spectrum} (f : Π(s : ℤ), A s →ₛ A (s - 1))
|
||||
|
||||
definition I [constructor] : Set := trunctype.mk (gℤ ×g gℤ) !is_trunc_prod
|
||||
|
||||
definition D_sequence : graded_module rℤ I :=
|
||||
λv, LeftModule_int_of_AbGroup (πₛ[v.1] (A (v.2)))
|
||||
|
||||
definition E_sequence : graded_module rℤ I :=
|
||||
λv, LeftModule_int_of_AbGroup (πₛ[v.1] (sfiber (f (v.2))))
|
||||
|
||||
include f
|
||||
definition i_sequence : D_sequence →gm D_sequence :=
|
||||
begin
|
||||
fapply graded_hom.mk, exact (prod_equiv_prod erfl (add_right_action (- 1))),
|
||||
intro v, induction v with n s,
|
||||
apply lm_hom_int.mk, esimp,
|
||||
-- exact homomorphism.mk _ (is_mul_hom_LES_of_shomotopy_groups (f s) (n, 0)),
|
||||
-- exact shomotopy_groups_fun (f s) (n, 0)
|
||||
exact πₛ→[n] (f s)
|
||||
end
|
||||
|
||||
definition j_sequence : D_sequence →gm E_sequence :=
|
||||
begin
|
||||
fapply graded_hom.mk_out',
|
||||
exact (prod_equiv_prod (add_right_action 1) (add_right_action (- 1))),
|
||||
intro v, induction v with n s,
|
||||
apply lm_hom_int.mk, esimp,
|
||||
rexact shomotopy_groups_fun (f s) (n, 2)
|
||||
end
|
||||
|
||||
definition k_sequence : E_sequence →gm D_sequence :=
|
||||
begin
|
||||
fapply graded_hom.mk erfl,
|
||||
intro v, induction v with n s,
|
||||
apply lm_hom_int.mk, esimp,
|
||||
-- exact homomorphism.mk _ (is_mul_hom_LES_of_shomotopy_groups (f s) (n, 1)),
|
||||
-- exact shomotopy_groups_fun (f s) (n, 1)
|
||||
exact πₛ→[n] (spoint (f s))
|
||||
end
|
||||
|
||||
lemma ij_sequence : is_exact_gmod i_sequence j_sequence :=
|
||||
begin
|
||||
intro i, induction i with n s,
|
||||
revert n, refine equiv_rect (add_right_action 1) _ _, intro n,
|
||||
esimp, intro j k p, unfold [i_sequence] at p,
|
||||
-- induction p,
|
||||
intro q, unfold [j_sequence] at q,
|
||||
note qq := left_inv (deg j_sequence) (n, s),
|
||||
unfold [j_sequence] at qq,
|
||||
revert k q,
|
||||
--refine eq.rec_to2 qq _ _
|
||||
--intro i j k p q,
|
||||
|
||||
-- revert k q,
|
||||
end
|
||||
|
||||
lemma jk_sequence : is_exact_gmod j_sequence k_sequence :=
|
||||
sorry
|
||||
|
||||
local attribute i_sequence [reducible]
|
||||
lemma ki_sequence : is_exact_gmod k_sequence i_sequence :=
|
||||
begin
|
||||
-- unfold [is_exact_gmod, is_exact_mod],
|
||||
intro i j k p q, induction p, induction q, induction i with n s,
|
||||
rexact is_exact_of_is_exact_at (is_exact_LES_of_shomotopy_groups (f s) (n, 0)),
|
||||
end
|
||||
|
||||
|
||||
definition exact_couple_sequence : exact_couple rℤ I :=
|
||||
exact_couple.mk D_sequence E_sequence i_sequence j_sequence k_sequence ij_sequence jk_sequence ki_sequence
|
||||
|
||||
end
|
||||
|
||||
|
||||
|
||||
|
||||
end spectrum
|
||||
|
|
@ -58,7 +58,19 @@ namespace group
|
|||
definition ab_product [constructor] (G G' : AbGroup) : AbGroup :=
|
||||
AbGroup.mk _ (ab_group_prod G G')
|
||||
|
||||
infix ` ×g `:30 := group.product
|
||||
infix ` ×ag `:30 := group.ab_product
|
||||
infix ` ×g `:60 := group.product
|
||||
infix ` ×ag `:60 := group.ab_product
|
||||
|
||||
definition product_functor [constructor] {G G' H H' : Group} (φ : G →g H) (ψ : G' →g H') :
|
||||
G ×g G' →g H ×g H' :=
|
||||
homomorphism.mk (λx, (φ x.1, ψ x.2)) (λx y, prod_eq !to_respect_mul !to_respect_mul)
|
||||
|
||||
infix ` ×→g `:60 := group.product_functor
|
||||
|
||||
definition product_isomorphism [constructor] {G G' H H' : Group} (φ : G ≃g H) (ψ : G' ≃g H') :
|
||||
G ×g G' ≃g H ×g H' :=
|
||||
isomorphism.mk (φ ×→g ψ) !is_equiv_prod_functor
|
||||
|
||||
infix ` ×≃g `:60 := group.product_isomorphism
|
||||
|
||||
end group
|
||||
|
|
|
|||
|
|
@ -269,17 +269,6 @@ namespace group
|
|||
definition image {G H : Group} (f : G →g H) : Group :=
|
||||
subgroup (image_subgroup f)
|
||||
|
||||
definition AbGroup_of_Group.{u} (G : Group.{u}) (H : Π (g h : G), mul g h = mul h g) : AbGroup.{u} :=
|
||||
begin
|
||||
induction G,
|
||||
induction struct,
|
||||
fapply AbGroup.mk,
|
||||
exact carrier,
|
||||
fapply ab_group.mk,
|
||||
repeat assumption,
|
||||
exact H
|
||||
end
|
||||
|
||||
definition ab_image {G : AbGroup} {H : AbGroup} (f : G →g H) : AbGroup :=
|
||||
ab_subgroup (image_subgroup f)
|
||||
|
||||
|
|
|
|||
|
|
@ -1,3 +1,7 @@
|
|||
/- submodules and quotient modules -/
|
||||
|
||||
-- Authors: Floris van Doorn
|
||||
|
||||
|
||||
import .left_module .quotient_group
|
||||
|
||||
|
|
|
|||
64
heq.hlean
Normal file
64
heq.hlean
Normal file
|
|
@ -0,0 +1,64 @@
|
|||
|
||||
open eq is_trunc
|
||||
|
||||
variables {I : Set} {P : I → Type} {i j k : I} {x x₁ x₂ : P i} {y y₁ y₂ : P j} {z : P k}
|
||||
{Q : Π⦃i⦄, P i → Type}
|
||||
|
||||
structure heq (x : P i) (y : P j) : Type :=
|
||||
(p : i = j)
|
||||
(q : x =[p] y)
|
||||
|
||||
namespace eq
|
||||
notation x ` ==[`:50 P:0 `] `:0 y:50 := @heq _ P _ _ x y
|
||||
infix ` == `:50 := heq -- mostly for printing, since it will be almost always ambiguous what P is
|
||||
|
||||
definition pathover_of_heq {p : i = j} (q : x ==[P] y) : x =[p] y :=
|
||||
change_path !is_set.elim (heq.q q)
|
||||
|
||||
definition eq_of_heq (p : x₁ ==[P] x₂) : x₁ = x₂ :=
|
||||
eq_of_pathover_idp (pathover_of_heq p)
|
||||
|
||||
definition heq.elim (p : x ==[P] y) (q : Q x) : Q y :=
|
||||
begin
|
||||
induction p with p r, induction r, exact q
|
||||
end
|
||||
|
||||
definition heq.refl [refl] (x : P i) : x ==[P] x :=
|
||||
heq.mk idp idpo
|
||||
|
||||
definition heq.rfl : x ==[P] x :=
|
||||
heq.refl x
|
||||
|
||||
definition heq.symm [symm] (p : x ==[P] y) : y ==[P] x :=
|
||||
begin
|
||||
induction p with p q, constructor, exact q⁻¹ᵒ
|
||||
end
|
||||
|
||||
definition heq_of_eq (p : x₁ = x₂) : x₁ ==[P] x₂ :=
|
||||
heq.mk idp (pathover_idp_of_eq p)
|
||||
|
||||
definition heq.trans [trans] (p : x ==[P] y) (p₂ : y ==[P] z) : x ==[P] z :=
|
||||
begin
|
||||
induction p with p q, induction p₂ with p₂ q₂, constructor, exact q ⬝o q₂
|
||||
end
|
||||
|
||||
infix ` ⬝he `:72 := heq.trans
|
||||
postfix `⁻¹ʰᵉ`:(max+10) := heq.symm
|
||||
|
||||
|
||||
definition heq_of_heq_of_eq (p : x ==[P] y) (p₂ : y = y₂) : x ==[P] y₂ :=
|
||||
p ⬝he heq_of_eq p₂
|
||||
|
||||
definition heq_of_eq_of_heq (p : x = x₂) (p₂ : x₂ ==[P] y) : x ==[P] y :=
|
||||
heq_of_eq p ⬝he p₂
|
||||
|
||||
infix ` ⬝hep `:73 := concato_eq
|
||||
infix ` ⬝phe `:74 := eq_concato
|
||||
|
||||
definition heq_tr (p : i = j) (x : P i) : x ==[P] transport P p x :=
|
||||
heq.mk p !pathover_tr
|
||||
|
||||
definition tr_heq (p : i = j) (x : P i) : transport P p x ==[P] x :=
|
||||
(heq_tr p x)⁻¹ʰᵉ
|
||||
|
||||
end eq
|
||||
|
|
@ -208,11 +208,11 @@ namespace spectrum
|
|||
-- read off the homotopy groups without any tedious case-analysis of
|
||||
-- n. We increment by 2 in order to ensure that they are all
|
||||
-- automatically abelian groups.
|
||||
definition shomotopy_group [constructor] (n : ℤ) (E : spectrum) : AbGroup := πag[2] (E (2 - n))
|
||||
definition shomotopy_group (n : ℤ) (E : spectrum) : AbGroup := πag[2] (E (2 - n))
|
||||
|
||||
notation `πₛ[`:95 n:0 `]`:0 := shomotopy_group n
|
||||
|
||||
definition shomotopy_group_fun [constructor] (n : ℤ) {E F : spectrum} (f : E →ₛ F) :
|
||||
definition shomotopy_group_fun (n : ℤ) {E F : spectrum} (f : E →ₛ F) :
|
||||
πₛ[n] E →g πₛ[n] F :=
|
||||
π→g[2] (f (2 - n))
|
||||
|
||||
|
|
@ -318,23 +318,26 @@ namespace spectrum
|
|||
(homomorphism_LES_of_homotopy_groups_fun (f (2 - n)) (1, 2) ∘g πg_glue Y n) qed
|
||||
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||||
|
||||
definition is_exact_LES_of_shomotopy_groups : is_exact LES_of_shomotopy_groups :=
|
||||
begin
|
||||
apply is_exact_splice, intro n, apply is_exact_LES_of_homotopy_groups,
|
||||
end
|
||||
|
||||
-- In the comments below is a start on an explicit description of the LES for spectra
|
||||
-- Maybe it's slightly nicer to work with than the above version
|
||||
|
||||
-- definition shomotopy_groups [reducible] : -3ℤ → AbGroup
|
||||
-- | (n, fin.mk 0 H) := πₛ[n] Y
|
||||
-- | (n, fin.mk 1 H) := πₛ[n] X
|
||||
-- | (n, fin.mk k H) := πₛ[n] (sfiber f)
|
||||
|
||||
-- definition shomotopy_groups_fun : Π(n : -3ℤ), shomotopy_groups (S n) →g shomotopy_groups n
|
||||
-- | (n, fin.mk 0 H) := proof π→g[1+1] (f (n + 2)) qed --π→[2] f (n+2)
|
||||
-- --pmap_of_homomorphism (πₛ→[n] f)
|
||||
-- | (n, fin.mk 1 H) := proof π→g[1+1] (ppoint (f (n + 2))) qed
|
||||
-- | (n, fin.mk 2 H) :=
|
||||
-- proof _ ∘g π→g[1+1] equiv_glue Y (pred n + 2) qed
|
||||
-- --π→[n] boundary_map ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y n))
|
||||
-- | (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||||
definition shomotopy_groups [reducible] : +3ℤ → AbGroup
|
||||
| (n, fin.mk 0 H) := πₛ[n] Y
|
||||
| (n, fin.mk 1 H) := πₛ[n] X
|
||||
| (n, fin.mk k H) := πₛ[n] (sfiber f)
|
||||
|
||||
definition shomotopy_groups_fun : Π(v : +3ℤ), shomotopy_groups (S v) →g shomotopy_groups v
|
||||
| (n, fin.mk 0 H) := proof πₛ→[n] f qed
|
||||
| (n, fin.mk 1 H) := proof πₛ→[n] (spoint f) qed
|
||||
| (n, fin.mk 2 H) := proof homomorphism_LES_of_homotopy_groups_fun (f (2 - n)) (nat.succ nat.zero, 2) ∘g
|
||||
πg_glue Y n ∘g (by reflexivity) qed
|
||||
| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||||
--(homomorphism_LES_of_homotopy_groups_fun (f (2 - n)) (1, 2) ∘g πg_glue Y n)
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end
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structure sp_chain_complex (N : succ_str) : Type :=
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@ -29,6 +29,9 @@ definition is_exact_g.mk {A B C : Group} {f : A →g B} {g : B →g C}
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is_exact.mk H₁ H₂
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namespace algebra
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definition ab_group_unit [constructor] : ab_group unit :=
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⦃ab_group, trivial_group, mul_comm := λx y, idp⦄
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definition inf_group_loopn (n : ℕ) (A : Type*) [H : is_succ n] : inf_group (Ω[n] A) :=
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by induction H; exact _
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@ -65,6 +68,18 @@ end algebra
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namespace eq
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definition eq.rec_to {A : Type} {a₀ : A} {P : Π⦃a₁⦄, a₀ = a₁ → Type}
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{a₁ : A} (p₀ : a₀ = a₁) (H : P p₀) {a₂ : A} (p : a₀ = a₂) : P p :=
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begin
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induction p₀, induction p, exact H
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end
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definition eq.rec_to2 {A : Type} {P : Π⦃a₀ a₁⦄, a₀ = a₁ → Type}
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{a₀ a₀' a₁' : A} (p' : a₀' = a₁') (p₀ : a₀ = a₀') (H : P p') ⦃a₁ : A⦄ (p : a₀ = a₁) : P p :=
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begin
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induction p₀, induction p', induction p, exact H
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||||
end
|
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section -- squares
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variables {A B : Type} {a a' a'' a₀₀ a₂₀ a₄₀ a₀₂ a₂₂ a₂₄ a₀₄ a₄₂ a₄₄ a₁ a₂ a₃ a₄ : A}
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/-a₀₀-/ {p₁₀ p₁₀' : a₀₀ = a₂₀} /-a₂₀-/ {p₃₀ : a₂₀ = a₄₀} /-a₄₀-/
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|
|
@ -1165,6 +1180,15 @@ structure Ring :=
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|||
attribute Ring.carrier [coercion]
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||||
attribute Ring.struct [instance]
|
||||
|
||||
namespace int
|
||||
|
||||
definition ring_int : Ring :=
|
||||
Ring.mk ℤ _
|
||||
|
||||
notation `rℤ` := ring_int
|
||||
|
||||
end int
|
||||
|
||||
namespace set_quotient
|
||||
definition is_prop_set_quotient {A : Type} (R : A → A → Prop) [is_prop A] : is_prop (set_quotient R) :=
|
||||
begin
|
||||
|
|
|
|||
Loading…
Reference in a new issue