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acyclic.lean
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acyclic.lean
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import category_theory.preadditive.additive_functor
import category_theory.limits.preserves.shapes.biproducts
import for_mathlib.derived.les2
import for_mathlib.derived.les_facts
import for_mathlib.derived.Ext_lemmas
import for_mathlib.is_quasi_iso
import for_mathlib.short_exact
import for_mathlib.homology
import for_mathlib.exact_lift_desc
.
noncomputable theory
open category_theory category_theory.limits opposite
open homotopy_category (hiding single)
open bounded_homotopy_category
-- main proof in this file is inspired by https://math.stackexchange.com/a/2118042
section
variables {𝓐 : Type*} [category 𝓐] [abelian 𝓐] {ι : Type*} {c : complex_shape ι}
def delta_to_kernel (C : homological_complex 𝓐 c) (i j k : ι) :
C.X i ⟶ kernel (C.d j k) :=
factor_thru_image _ ≫ image_to_kernel' (C.d i j) _ (C.d_comp_d _ _ _)
def delta_to_kernel_ι (C : homological_complex 𝓐 c) (i j k : ι) :
delta_to_kernel C i j k ≫ kernel.ι (C.d j k) = C.d i j :=
begin
delta delta_to_kernel image_to_kernel',
rw [category.assoc, kernel.lift_ι, image.fac],
end
def d_delta_to_kernel (C : homological_complex 𝓐 c) (h i j k : ι) :
C.d h i ≫ delta_to_kernel C i j k = 0 :=
begin
rw [← cancel_mono (kernel.ι (C.d j k)), category.assoc, delta_to_kernel_ι, C.d_comp_d, zero_comp],
end
-- move me
lemma short_exact_comp_iso {A B C D : 𝓐} (f : A ⟶ B) (g : B ⟶ C) (h : C ⟶ D) (hh : is_iso h) :
short_exact f (g ≫ h) ↔ short_exact f g :=
begin
split; intro H,
{ haveI : mono f := H.mono,
haveI : epi g,
{ haveI := H.epi, have := epi_comp (g ≫ h) (inv h), simpa only [category.assoc, is_iso.hom_inv_id, category.comp_id] },
refine ⟨_⟩, have := H.exact, rwa exact_comp_iso at this, },
{ haveI : mono f := H.mono,
haveI : epi g := H.epi,
haveI : epi (g ≫ h) := epi_comp g h,
refine ⟨_⟩, have := H.exact, rwa exact_comp_iso }
end
lemma is_acyclic_def
(C : homotopy_category 𝓐 c) :
is_acyclic C ↔ (∀ i, is_zero (C.as.homology i)) :=
begin
split,
{ apply is_acyclic.cond },
{ apply is_acyclic.mk }
end
lemma is_acyclic_iff_short_exact_to_cycles
(C : homotopy_category 𝓐 (complex_shape.up ℤ)) :
is_acyclic C ↔
(∀ i, short_exact (kernel.ι (C.as.d i (i+1))) (delta_to_kernel C.as i (i+1) (i+1+1))) :=
begin
rw is_acyclic_def,
symmetry,
apply (equiv.add_right (1 : ℤ)).forall_congr,
intro i,
let e := (homology_iso C.as i (i+1) (i+1+1) rfl rfl),
dsimp [delta_to_kernel] at e ⊢,
rw [e.is_zero_iff, homology_is_zero_iff_image_to_kernel'_is_iso],
split,
{ apply iso_of_short_exact_comp_right _ _ _, apply short_exact_kernel_factor_thru_image },
{ intro h, rw short_exact_comp_iso _ _ _ h, apply short_exact_kernel_factor_thru_image }
end
lemma is_acyclic_iff_short_exact_to_cycles'
(C : homological_complex 𝓐 (complex_shape.down ℤ)) :
(∀ i, is_zero (C.homology i)) ↔
(∀ i, short_exact (kernel.ι (C.d (i+1+1) (i+1))) (delta_to_kernel C (i+1+1) (i+1) i)) :=
begin
symmetry,
apply (equiv.add_right (1 : ℤ)).forall_congr,
intro i,
let e := (homology_iso C (i+1+1) (i+1) i rfl rfl),
dsimp [delta_to_kernel] at e ⊢,
rw [e.is_zero_iff, homology_is_zero_iff_image_to_kernel'_is_iso],
split,
{ apply iso_of_short_exact_comp_right _ _ _, apply short_exact_kernel_factor_thru_image },
{ intro h, rw short_exact_comp_iso _ _ _ h, apply short_exact_kernel_factor_thru_image }
end
end
variables {𝓐 𝓑 : Type*} [category 𝓐] [abelian 𝓐] [enough_projectives 𝓐]
variables [category 𝓑] [abelian 𝓑] [enough_projectives 𝓑]
variables (C : cochain_complex 𝓐 ℤ)
[homotopy_category.is_bounded_above ((homotopy_category.quotient _ _).obj C)]
def category_theory.functor.single (F : bounded_homotopy_category 𝓐 ⥤ 𝓑) (i : ℤ) : 𝓐 ⥤ 𝓑 :=
bounded_homotopy_category.single _ i ⋙ F
-- move me
lemma category_theory.limits.is_zero.biprod {𝓐 : Type*} [category 𝓐] [abelian 𝓐]
{X Y : 𝓐} (hX : is_zero X) (hY : is_zero Y) :
is_zero (X ⊞ Y) :=
begin
rw is_zero_iff_id_eq_zero at hX hY ⊢,
ext; simp [hX, hY],
end
instance category_theory.limits.preserves_binary_biproduct_of_additive
{𝓐 𝓑 : Type*} [category 𝓐] [category 𝓑] [abelian 𝓐] [abelian 𝓑]
(F : 𝓐 ⥤ 𝓑) [functor.additive F] (X Y : 𝓐) :
preserves_binary_biproduct X Y F :=
preserves_binary_biproduct_of_preserves_biproduct _ _ _
-- move me
@[simp] lemma category_theory.op_neg {𝓐 : Type*} [category 𝓐] [preadditive 𝓐]
{X Y : 𝓐} (f : X ⟶ Y) : (-f).op = - f.op := rfl
lemma acyclic_left_of_short_exact (B : 𝓐) {X Y Z : 𝓐} (f : X ⟶ Y) (g : Y ⟶ Z) (hfg : short_exact f g)
(hY : ∀ i > 0, is_zero (((Ext' i).obj (op $ Y)).obj B))
(hZ : ∀ i > 0, is_zero (((Ext' i).obj (op $ Z)).obj B)) :
∀ i > 0, is_zero (((Ext' i).obj (op $ X)).obj B) :=
begin
intros i hi,
have := hfg.Ext'_five_term_exact_seq B i,
refine (this.drop 1).pair.is_zero_of_is_zero_is_zero (hY _ hi) (hZ _ _),
transitivity i, { exact lt_add_one i }, { exact hi }
end
.
lemma map_is_acyclic_of_acyclic_aux
{A B C D X Y Z W : 𝓐} (f : A ⟶ B) (g : C ⟶ D) (π : B ⟶ kernel g)
{α : X ⟶ B} {β : B ⟶ Y} {γ : Y ⟶ C} {δ : C ⟶ Z} {ε : Z ⟶ D} {ζ : D ⟶ W}
(hαβ : short_exact α β) (hγδ : short_exact γ δ) (hεζ : short_exact ε ζ)
(hf : mono f) (hfπ : exact f π)
(hαπ : α ≫ π = 0) (hγg : γ ≫ g = 0) (hδε : δ ≫ ε = g)
(hπι : π ≫ kernel.ι g = β ≫ γ) :
short_exact f π :=
begin
suffices : epi π, { resetI, exact ⟨hfπ⟩ },
have hβ : epi β := hαβ.epi,
have hγ : mono γ := hγδ.mono,
have hε : mono ε := hεζ.mono,
resetI,
have hιδ : kernel.ι g ≫ δ = 0,
{ rw [← cancel_mono ε, category.assoc, hδε, kernel.condition, zero_comp], },
let e1 : Y ⟶ kernel g := hαβ.exact.epi_desc π hαπ,
let e2 : Y ⟶ kernel g := kernel.lift g γ hγg,
let e3 : kernel g ⟶ Y := hγδ.exact.mono_lift (kernel.ι g) hιδ,
have he12 : e1 = e2,
{ rw [← cancel_epi β, ← cancel_mono (kernel.ι g)],
simp only [hπι, category.assoc, kernel.lift_ι, exact.comp_epi_desc_assoc], },
have he13 : e1 ≫ e3 = 𝟙 _,
{ rw [he12, ← cancel_mono γ, category.assoc, exact.mono_lift_comp, kernel.lift_ι, category.id_comp], },
have he31 : e3 ≫ e1 = 𝟙 _,
{ rw [he12, ← cancel_mono (kernel.ι g), category.assoc, kernel.lift_ι, exact.mono_lift_comp, category.id_comp], },
let e : Y ≅ kernel g := ⟨e1, e3, he13, he31⟩,
have hπ : β ≫ e.hom = π := exact.comp_epi_desc _ _ _,
rw ← hπ, exact epi_comp _ _,
end
lemma short_exact_Ext_of_short_exact_of_acyclic {A B C : 𝓐} (Z : 𝓐) {f : A ⟶ B} {g : B ⟶ C}
(hfg : short_exact f g) (hC : ∀ i > 0, is_zero (((Ext' i).obj (op $ C)).obj Z)) :
short_exact (((Ext' 0).flip.obj Z).map g.op) (((Ext' 0).flip.obj Z).map f.op) :=
begin
have H0 := hfg.Ext'_five_term_exact_seq Z 0,
apply_with short_exact.mk {instances:=ff},
{ have H := ((hfg.Ext'_five_term_exact_seq Z (-1)).drop 2).pair,
apply H.mono_of_is_zero,
apply Ext'_is_zero_of_neg, dec_trivial },
{ apply (H0.drop 1).pair.epi_of_is_zero,
apply hC, dec_trivial },
{ exact H0.pair }
end
lemma map_is_acyclic_of_acyclic''
[is_acyclic ((homotopy_category.quotient _ _).obj C)]
(B : 𝓐)
(hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :
∀ i, is_zero (((((Ext' 0).flip.obj B).map_homological_complex _).obj C.op).homology i) :=
begin
rw is_acyclic_iff_short_exact_to_cycles',
obtain ⟨a, ha⟩ := is_bounded_above.cond ((quotient 𝓐 (complex_shape.up ℤ)).obj C),
have aux : ((quotient 𝓐 (complex_shape.up ℤ)).obj C).is_acyclic := ‹_›,
rw is_acyclic_iff_short_exact_to_cycles at aux,
intro i,
let K := λ j, kernel (C.d j (j+1)),
suffices hK : ∀ j, ∀ i > 0, is_zero (((Ext' i).obj (op $ K j)).obj B),
{ have SES1 := short_exact_Ext_of_short_exact_of_acyclic B (aux (i+1+1)) (hK _),
have SES2 := short_exact_Ext_of_short_exact_of_acyclic B (aux (i+1)) (hK _),
have SES3 := short_exact_Ext_of_short_exact_of_acyclic B (aux i) (hK _),
apply map_is_acyclic_of_acyclic_aux _ _ _ SES1 SES2 SES3 infer_instance;
clear SES1 SES2 SES3 aux,
{ delta delta_to_kernel image_to_kernel',
apply exact_comp_mono, rw exact_factor_thru_image_iff, exact exact_kernel_ι },
{ rw [← cancel_mono (kernel.ι _), zero_comp, category.assoc, delta_to_kernel_ι],
swap, apply_instance,
erw [functor.map_homological_complex_obj_d, ← functor.map_comp],
dsimp only [homological_complex.op_d, quotient_obj_as],
rw [← op_comp, d_delta_to_kernel, op_zero, functor.map_zero], },
{ erw [functor.map_homological_complex_obj_d, ← functor.map_comp],
dsimp only [homological_complex.op_d, quotient_obj_as],
rw [← op_comp, d_delta_to_kernel, op_zero, functor.map_zero], },
{ rw [← functor.map_comp, ← op_comp, delta_to_kernel_ι], refl, },
{ rw [delta_to_kernel_ι, ← functor.map_comp, ← op_comp, delta_to_kernel_ι], refl, },
{ apply_instance } },
clear i, intro j,
have : ∀ j ≥ a, ∀ i > 0, is_zero (((Ext' i).obj (op $ K j)).obj B),
{ intros j hj i hi,
apply bounded_derived_category.Ext'_zero_left_is_zero,
apply is_zero.op,
refine is_zero.of_mono (kernel.ι _) _,
exact ha j hj },
apply int.induction_on' j a,
{ exact this _ le_rfl, },
{ intros j hj aux, apply this, exact int.le_add_one hj, },
{ intros j hj IH,
obtain ⟨j, rfl⟩ : ∃ i, i + 1 = j := ⟨j - 1, sub_add_cancel _ _⟩,
rw add_sub_cancel,
apply acyclic_left_of_short_exact B (kernel.ι _) (delta_to_kernel _ _ _ _) _ (hC _) IH,
exact aux j, }
end
lemma map_is_acyclic_of_acyclic'
[is_acyclic ((homotopy_category.quotient _ _).obj C)]
(B : 𝓐)
(hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :
is_acyclic ((((Ext' 0).flip.obj B).right_op.map_homotopy_category _).obj ((homotopy_category.quotient _ _).obj C)) :=
begin
rw is_acyclic_def,
intro i,
have h1 : (complex_shape.up ℤ).rel (i - 1) i, { dsimp, apply sub_add_cancel },
refine is_zero.of_iso _ (homology_iso' _ (i-1) i (i+1) h1 rfl),
dsimp only [functor.map_homotopy_category_obj, quotient_obj_as,
functor.right_op_map, functor.map_homological_complex_obj_d],
apply exact.homology_is_zero,
apply exact.op,
refine exact_of_homology_is_zero _,
{ rw [← category_theory.functor.map_comp, ← op_comp, homological_complex.d_comp_d, op_zero, functor.map_zero], },
have := map_is_acyclic_of_acyclic'' C B hC i,
apply this.of_iso _, clear this,
let C' := (((Ext' 0).flip.obj B).map_homological_complex (complex_shape.up ℤ).symm).obj (homological_complex.op C),
have h1 : (complex_shape.down ℤ).rel i (i - 1), { dsimp, apply sub_add_cancel },
exact (homology_iso' C' (i+1) i (i-1) rfl h1).symm,
end
lemma map_is_acyclic_of_acyclic
[is_acyclic ((homotopy_category.quotient _ _).obj C)]
(B : 𝓐)
(hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B)) :
is_acyclic (((preadditive_yoneda.obj B).right_op.map_homotopy_category _).obj ((homotopy_category.quotient _ _).obj C)) :=
begin
have := map_is_acyclic_of_acyclic' C B hC,
rw is_acyclic_def at this ⊢,
intro i, specialize this i,
apply this.of_iso _, clear this,
have h1 : (complex_shape.up ℤ).rel (i - 1) i, { dsimp, apply sub_add_cancel },
refine (homology_iso' _ (i-1) i (i+1) h1 rfl) ≪≫ _ ≪≫ (homology_iso' _ (i-1) i (i+1) h1 rfl).symm,
dsimp only [functor.map_homotopy_category_obj, quotient_obj_as,
functor.right_op_map, functor.map_homological_complex_obj_d],
let e := λ i, ((bounded_derived_category.Ext'_zero_flip_iso _ B).app (op $ C.X i)).op,
refine homology.map_iso _ _ (arrow.iso_mk (e _) (e _) _) (arrow.iso_mk (e _) (e _) _) rfl,
{ simp only [iso.op_hom, iso.app_hom, arrow.mk_hom, functor.flip_obj_map, ← op_comp, ← nat_trans.naturality], },
{ simp only [iso.op_hom, iso.app_hom, arrow.mk_hom, functor.flip_obj_map, ← op_comp, ← nat_trans.naturality], },
end
lemma acyclic_of_projective (P B : 𝓐) [projective P] (i : ℤ) (hi : 0 < i) :
is_zero (((Ext' i).obj (op P)).obj B) :=
begin
rw (Ext'_iso (op P) B i _ (𝟙 _) _).is_zero_iff,
{ rcases i with ((_|i)|i),
{ exfalso, revert hi, dec_trivial },
swap, { exfalso, revert hi, dec_trivial },
refine is_zero.homology_is_zero _ _ _ _,
apply AddCommGroup.is_zero_of_eq,
intros,
apply is_zero.eq_of_src,
apply is_zero_zero, },
{ refine ⟨_, _, _⟩,
{ rintro (_|n), { assumption }, { dsimp, apply_instance } },
{ exact exact_zero_mono (𝟙 P) },
{ rintro (_|n); exact exact_of_zero 0 0 } }
end
def Ext_compute_with_acyclic
(B : 𝓐)
(hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))
(i : ℤ) :
((Ext i).obj (op $ of' C)).obj ((single _ 0).obj B) ≅
(((preadditive_yoneda.obj B).right_op.map_homological_complex _).obj C).unop.homology (-i) :=
begin
let P := (of' C).replace,
refine (preadditive_yoneda.map_iso _).app (op P) ≪≫ _,
{ exact (single 𝓐 (-i)).obj B },
{ exact (shift_single_iso 0 i).app B ≪≫ eq_to_iso (by rw zero_sub) },
refine hom_single_iso _ _ _ ≪≫ _,
let π : P ⟶ of' C := (of' C).π,
let HomB := (preadditive_yoneda.obj B).right_op.map_homological_complex (complex_shape.up ℤ) ⋙ homological_complex.unop_functor.right_op,
let fq := (homotopy_category.quotient _ _).map (HomB.map π.out).unop,
suffices hf : is_quasi_iso fq,
{ have := @is_quasi_iso.cond _ _ _ _ _ _ _ _ hf (-i),
resetI,
let e := as_iso ((homotopy_category.homology_functor Ab _ (-i)).map fq),
exact e.symm, },
-- that was the data,
-- now the proof obligation ...
/-
The proof strategy is roughly the following (https://math.stackexchange.com/a/2118042):
the map is a quasi-iso iff its cone is acyclic
the cone commutes with the additive functor
so you end up with this functor applied to the cone of `π`
the cone of `π` is acyclic, since `π` is a quasi-iso
by induction, the other cone is also acyclic
-/
apply is_quasi_iso_of_op,
let f := homological_complex.op_functor.map (HomB.map (quot.out π)),
have := cone_triangleₕ_mem_distinguished_triangles _ _ f,
replace := is_quasi_iso_iff_is_acyclic _ this,
dsimp [homological_complex.cone.triangleₕ] at this,
erw this, clear this i,
constructor,
intro i, obtain ⟨i, rfl⟩ : ∃ j, j + 1 = i := ⟨i - 1, sub_add_cancel _ _⟩,
refine is_zero.of_iso _ (homology_iso _ i (i+1) (i+1+1) _ _),
rotate, { dsimp, refl }, { dsimp, refl },
apply exact.homology_is_zero _,
dsimp only [homotopy_category.quotient, quotient.functor_obj_as, homological_complex.cone_d],
have hπ : is_quasi_iso π, { dsimp [π], apply_instance },
have := cone_triangleₕ_mem_distinguished_triangles _ _ π.out,
replace := is_quasi_iso_iff_is_acyclic _ this,
dsimp [homological_complex.cone.triangleₕ] at this,
simp only [quotient_map_out] at this,
replace := this.mp _,
swap, { convert hπ using 1, generalize : P.val = X, cases X, refl, },
haveI preaux : ((quotient 𝓐 (complex_shape.up ℤ)).obj (homological_complex.cone (quot.out π))).is_bounded_above,
{ constructor,
obtain ⟨a, ha⟩ := is_bounded_above.cond ((quotient 𝓐 (complex_shape.up ℤ)).obj C),
obtain ⟨b, hb⟩ := is_bounded_above.cond P.val,
refine ⟨max a b, _⟩,
intros k hk,
refine category_theory.limits.is_zero.biprod _ _,
{ apply hb, refine (le_max_right _ _).trans (hk.trans (lt_add_one _).le) },
{ apply ha, exact (le_max_left _ _).trans hk, } },
have aux := @map_is_acyclic_of_acyclic _ _ _ _ _ _ this B _,
{ replace := (@is_acyclic.cond _ _ _ _ _ _ aux (i+1)).of_iso (homology_iso _ i (i+1) (i+1+1) _ _).symm,
rotate, { dsimp, refl }, { dsimp, refl },
dsimp only [homotopy_category.quotient, quotient.functor_obj_as, homological_complex.cone_d,
functor.map_homotopy_category_obj, functor.map_homological_complex_obj_d] at this,
replace := exact_of_homology_is_zero this,
let e := functor.map_biprod (preadditive_yoneda.obj B).right_op,
refine preadditive.exact_of_iso_of_exact' _ _ _ _ (e _ _) (e _ _) (e _ _) _ _ this;
dsimp only [e, functor.map_biprod_hom],
all_goals
{ ext,
{ simp only [category.assoc, functor.right_op_map, homological_complex.cone.d, biprod.lift_fst,
eq_self_iff_true, functor.map_homological_complex_obj_d, functor.right_op_map,
homological_complex.X_eq_to_iso_refl, category.comp_id, dite_eq_ite, if_true,
biprod.lift_fst, biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero],
simp only [functor.map_homological_complex_obj_d, functor.right_op_map, functor.comp_map,
biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero,
← op_comp, ← category_theory.functor.map_comp, biprod.lift_fst],
simp only [biprod.desc_eq, comp_zero, add_zero, preadditive.comp_neg,
category_theory.op_neg, functor.map_neg, op_comp, category_theory.functor.map_comp],
refl },
{ simp only [category.assoc, functor.right_op_map, homological_complex.cone.d, biprod.lift_snd,
eq_self_iff_true, functor.map_homological_complex_obj_d, functor.right_op_map,
functor.map_homological_complex_map_f, homological_complex.X_eq_to_iso_refl,
category.comp_id, dite_eq_ite, if_true, biprod.lift_snd, biprod.lift_desc],
simp only [functor.map_homological_complex_obj_d, functor.right_op_map, functor.comp_map,
biprod.lift_desc, preadditive.comp_neg, comp_zero, add_zero,
← op_comp, ← category_theory.functor.map_comp, biprod.lift_snd],
simp only [biprod.desc_eq, op_add, functor.map_neg, functor.map_add, op_comp,
category_theory.functor.map_comp],
refl } } },
{ clear i, intros k i hi,
let e := functor.map_biprod ((Ext' i).flip.obj B).right_op
(P.val.as.X (k + 1)) ((of' C).val.as.X k),
refine is_zero.of_iso (is_zero.unop _) e.symm.unop,
refine category_theory.limits.is_zero.biprod _ _,
{ simp only [functor.right_op_obj, functor.flip_obj_obj, is_zero_op],
exact acyclic_of_projective (P.val.as.X (k + 1)) B i hi, },
{ exact (hC k _ hi).op, }, },
end
.
def homological_complex.single_iso (B : 𝓐) {i j : ℤ} (h : j = i) :
((homological_complex.single _ (complex_shape.up ℤ) i).obj B).X j ≅ B :=
eq_to_iso (if_pos h)
def cochain_complex.hom_to_single_of_hom
(C : cochain_complex 𝓐 ℤ) (B : 𝓐) (i : ℤ) (f : C.X i ⟶ B) :
C ⟶ (homological_complex.single _ _ i).obj B :=
{ f := λ j, if h : j = i then eq_to_hom (by rw h) ≫ f ≫ (homological_complex.single_iso _ h).inv
else 0,
comm' := sorry }
def Ext_compute_with_acyclic_inv_eq_aux (B) (i) :
AddCommGroup.of (C.X (-i) ⟶ B) ⟶ ((Ext i).obj (op (of' C))).obj ((single 𝓐 0).obj B) :=
{ to_fun := λ f, (of' C).π ≫ begin
dsimp at f,
refine (homotopy_category.quotient _ _).map _,
refine _ ≫ (homological_complex.single_shift _ _).inv.app _,
refine cochain_complex.hom_to_single_of_hom _ _ _ _,
refine _ ≫ f,
refine eq_to_hom _,
apply congr_arg,
exact zero_sub _,
end,
map_zero' := sorry,
map_add' := sorry }
lemma Ext_compute_with_acylic_inv_eq (B : 𝓐)
(hC : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C.X k)).obj B))
(i : ℤ) :
(Ext_compute_with_acyclic _ B hC i).inv =
homology.desc' _ _ _
(kernel.ι _ ≫ Ext_compute_with_acyclic_inv_eq_aux _ _ _)
sorry := sorry
lemma homology.lift_desc (X Y Z : 𝓐) (f : X ⟶ Y) (g : Y ⟶ Z) (w)
(U : 𝓐) (e : _ ⟶ U) (he : f ≫ e = 0) (V : 𝓐) (t : V ⟶ _) (ht : t ≫ g = 0) :
homology.lift f g w (t ≫ cokernel.π _) (by { simp [ht] } ) ≫
homology.desc' _ _ _ (kernel.ι _ ≫ e) (by { simp [he] }) =
t ≫ e :=
begin
let s := _, change s ≫ _ = _,
have hs : s = kernel.lift _ t ht ≫ homology.π' _ _ _,
{ apply homology.hom_to_ext,
simp only [homology.lift_ι, category.assoc, projective.homology.π'_ι, kernel.lift_ι_assoc] },
simp [hs],
end
lemma homology.lift_desc' (X Y Z : 𝓐) (f : X ⟶ Y) (g : Y ⟶ Z) (w)
(U : 𝓐) (e : _ ⟶ U) (he : f ≫ e = 0) (V : 𝓐) (t : V ⟶ _) (ht : t ≫ g = 0)
(u v) (hu : u = t ≫ cokernel.π _) (hv : v = kernel.ι _ ≫ e) :
homology.lift f g w u (by simpa [hu] ) ≫ homology.desc' _ _ _ v (by simpa [hv]) = t ≫ e :=
begin
subst hu,
subst hv,
apply homology.lift_desc,
assumption'
end
lemma Ext_compute_with_acyclic_naturality (C₁ C₂ : cochain_complex 𝓐 ℤ)
[((quotient 𝓐 (complex_shape.up ℤ)).obj C₁).is_bounded_above]
[((quotient 𝓐 (complex_shape.up ℤ)).obj C₂).is_bounded_above]
(B : 𝓐)
(hC₁ : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C₁.X k)).obj B))
(hC₂ : ∀ k, ∀ i > 0, is_zero (((Ext' i).obj (op $ C₂.X k)).obj B))
(f : C₁ ⟶ C₂)
(i : ℤ) :
((Ext i).flip.obj ((single _ 0).obj B)).map (quiver.hom.op $
show (of' C₁).val ⟶ (of' C₂).val, from (homotopy_category.quotient _ _).map f) ≫
(Ext_compute_with_acyclic C₁ B hC₁ i).hom =
(Ext_compute_with_acyclic C₂ B hC₂ i).hom ≫
(((preadditive_yoneda.obj B).right_op.map_homological_complex _ ⋙
homological_complex.unop_functor.right_op ⋙ (_root_.homology_functor _ _ (-i)).op).map f).unop :=
begin
rw [← iso.inv_comp_eq, ← category.assoc, ← iso.eq_comp_inv],
rw Ext_compute_with_acylic_inv_eq,
rw Ext_compute_with_acylic_inv_eq,
apply homology.hom_from_ext,
simp only [category.assoc, homology.π'_desc'_assoc],
dsimp only [functor.comp_map, functor.op_map, homology_functor_map],
erw homology.map_eq_desc'_lift_left,
simp only [category.assoc],
erw [homology.π'_desc'_assoc],
dsimp,
rw (homology.lift_desc' _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ rfl),
rotate 3,
{ exact kernel.ι _ ≫ (preadditive_yoneda.obj _).map (f.f _).op },
swap,
{ simpa },
{ dsimp, sorry },
{ dsimp, simp only [category.assoc],
congr' 1,
sorry },
{ apply_instance },
{ sorry }
end