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feat: the mapping cone of a monomorphism, up to a quasi-isomorphism (#13675)
In this PR, given a short exact sequence `S` in the category of cochain complexes in an abelian category, we compare the homology sequence of `S`, and the homology sequence of the homological functor on the homotopy category applied to the distinguished triangle of the mapping cone of the monomorphism `S.f`. In particular, up to a quasi-isomorphism, the mapping cone of `S.f` identifies to `S.X₃`. In future PRs, this shall be used in order to attach a distinguished triangle in the derived category to `S` (#13742), and compare the homology sequence of `S` and the homology sequence attached to this distinguished triangle. Co-authored-by: Joël Riou <[email protected]>
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‎Mathlib.lean

+2
Original file line numberDiff line numberDiff line change
@@ -320,6 +320,7 @@ import Mathlib.Algebra.Homology.HomotopyCategory.MappingCone
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import Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
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import Mathlib.Algebra.Homology.HomotopyCategory.Shift
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import Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
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import Mathlib.Algebra.Homology.HomotopyCategory.ShortExact
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import Mathlib.Algebra.Homology.HomotopyCategory.SingleFunctors
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import Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
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import Mathlib.Algebra.Homology.HomotopyCofiber
@@ -330,6 +331,7 @@ import Mathlib.Algebra.Homology.Localization
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import Mathlib.Algebra.Homology.ModuleCat
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import Mathlib.Algebra.Homology.Opposite
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import Mathlib.Algebra.Homology.QuasiIso
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import Mathlib.Algebra.Homology.Refinements
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import Mathlib.Algebra.Homology.ShortComplex.Ab
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import Mathlib.Algebra.Homology.ShortComplex.Abelian
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import Mathlib.Algebra.Homology.ShortComplex.Basic

‎Mathlib/Algebra/Homology/HomologySequenceLemmas.lean

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@@ -72,7 +72,7 @@ noncomputable def composableArrows₅ (i j : ι) (hij : c.Rel i j) : ComposableA
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mk₅ (homologyMap S₁.f i) (homologyMap S₁.g i) (hS₁.δ i j hij)
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(homologyMap S₁.f j) (homologyMap S₁.g j)
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lemma composableArrows₅_exact (i j : ι) (hij : c.Rel i j):
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lemma composableArrows₅_exact (i j : ι) (hij : c.Rel i j) :
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(composableArrows₅ hS₁ i j hij).Exact :=
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exact_of_δ₀ (hS₁.homology_exact₂ i).exact_toComposableArrows
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(exact_of_δ₀ (hS₁.homology_exact₃ i j hij).exact_toComposableArrows

‎Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean

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@@ -174,6 +174,17 @@ lemma ext_from_iff (i j : ℤ) (hij : j + 1 = i) {A : C} (f g : (mappingCone φ)
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· rintro ⟨h₁, h₂⟩
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exact ext_from φ i j hij h₁ h₂
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lemma decomp_to {i : ℤ} {A : C} (f : A ⟶ (mappingCone φ).X i) (j : ℤ) (hij : i + 1 = j) :
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∃ (a : A ⟶ F.X j) (b : A ⟶ G.X i), f = a ≫ (inl φ).v j i (by omega) + b ≫ (inr φ).f i :=
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⟨f ≫ (fst φ).1.v i j hij, f ≫ (snd φ).v i i (add_zero i),
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by apply ext_to φ i j hij <;> simp⟩
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lemma decomp_from {j : ℤ} {A : C} (f : (mappingCone φ).X j ⟶ A) (i : ℤ) (hij : j + 1 = i) :
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∃ (a : F.X i ⟶ A) (b : G.X j ⟶ A),
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f = (fst φ).1.v j i hij ≫ a + (snd φ).v j j (add_zero j) ≫ b :=
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⟨(inl φ).v i j (by omega) ≫ f, (inr φ).f j ≫ f,
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by apply ext_from φ i j hij <;> simp⟩
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lemma ext_cochain_to_iff (i j : ℤ) (hij : i + 1 = j)
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{K : CochainComplex C ℤ} {γ₁ γ₂ : Cochain K (mappingCone φ) i} :
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γ₁ = γ₂ ↔ γ₁.comp (fst φ).1 hij = γ₂.comp (fst φ).1 hij ∧

‎Mathlib/Algebra/Homology/HomotopyCategory/ShiftSequence.lean

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@@ -18,7 +18,7 @@ and `HomotopyCategory` namespaces.
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-/
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open CategoryTheory Category
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open CategoryTheory Category ComplexShape Limits
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variable (C : Type*) [Category C] [Preadditive C]
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@@ -124,6 +124,34 @@ instance {K L : CochainComplex C ℤ} (φ : K ⟶ L) (n : ℤ) [QuasiIso φ] :
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change IsIso (homologyMap φ _)
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infer_instance
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variable (C) in
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lemma homologyFunctor_shift (n : ℤ) :
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(homologyFunctor C (ComplexShape.up ℤ) 0).shift n =
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homologyFunctor C (ComplexShape.up ℤ) n := rfl
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@[reassoc]
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lemma liftCycles_shift_homologyπ
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(K : CochainComplex C ℤ) {A : C} {n i : ℤ} (f : A ⟶ (K⟦n⟧).X i) (j : ℤ)
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(hj : (up ℤ).next i = j) (hf : f ≫ (K⟦n⟧).d i j = 0) (i' : ℤ) (hi' : n + i = i') (j' : ℤ)
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(hj' : (up ℤ).next i' = j') :
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(K⟦n⟧).liftCycles f j hj hf ≫ (K⟦n⟧).homologyπ i =
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K.liftCycles (f ≫ (K.shiftFunctorObjXIso n i i' (by omega)).hom) j' hj' (by
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simp only [next] at hj hj'
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obtain rfl : i' = i + n := by omega
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obtain rfl : j' = j + n := by omega
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dsimp at hf ⊢
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simp only [Linear.comp_units_smul] at hf
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apply (one_smul (M := ℤˣ) _).symm.trans _
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rw [← Int.units_mul_self n.negOnePow, mul_smul, comp_id, hf, smul_zero]) ≫
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K.homologyπ i' ≫
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((HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftIso n i i' hi').inv.app K := by
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simp only [liftCycles, homologyπ,
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shiftFunctorObjXIso, Functor.shiftIso, Functor.ShiftSequence.shiftIso,
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ShiftSequence.shiftIso_inv_app, ShortComplex.homologyπ_naturality,
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ShortComplex.liftCycles_comp_cyclesMap_assoc, shiftShortComplexFunctorIso_inv_app_τ₂,
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assoc, Iso.hom_inv_id, comp_id]
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rfl
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end CochainComplex
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namespace HomotopyCategory
@@ -147,4 +175,14 @@ lemma homologyShiftIso_hom_app (n a a' : ℤ) (ha' : n + a = a') (K : CochainCom
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(homologyFunctorFactors _ _ a').inv.app K := by
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apply Functor.ShiftSequence.induced_shiftIso_hom_app_obj
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@[reassoc]
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lemma homologyFunctor_shiftMap
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{K L : CochainComplex C ℤ} {n : ℤ} (f : K ⟶ L⟦n⟧) (a a' : ℤ) (h : n + a = a') :
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(homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap
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((quotient _ _).map f ≫ ((quotient _ _).commShiftIso n).hom.app _) a a' h =
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(homologyFunctorFactors _ _ a).hom.app K ≫
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(HomologicalComplex.homologyFunctor C (ComplexShape.up ℤ) 0).shiftMap f a a' h ≫
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(homologyFunctorFactors _ _ a').inv.app L := by
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apply Functor.ShiftSequence.induced_shiftMap
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end HomotopyCategory

‎Mathlib/Algebra/Homology/HomotopyCategory/ShortExact.lean

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@@ -0,0 +1,146 @@
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/-
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Copyright (c) 2024 Joël Riou. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joël Riou
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-/
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import Mathlib.Algebra.Homology.HomotopyCategory.HomologicalFunctor
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import Mathlib.Algebra.Homology.HomotopyCategory.ShiftSequence
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import Mathlib.Algebra.Homology.HomologySequenceLemmas
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import Mathlib.Algebra.Homology.Refinements
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/-!
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# The mapping cone of a monomorphism, up to a quasi-isomophism
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If `S` is a short exact short complex of cochain complexes in an abelian category,
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we construct a quasi-isomorphism `descShortComplex S : mappingCone S.f ⟶ S.X₃`.
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We obtain this by comparing the homology sequence of `S` and the homology
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sequence of the homology functor on the homotopy category, applied to the
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distinguished triangle attached to the mapping cone of `S.f`.
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-/
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open CategoryTheory Category ComplexShape HomotopyCategory Limits
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HomologicalComplex.HomologySequence Pretriangulated Preadditive
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variable {C : Type*} [Category C] [Abelian C]
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namespace CochainComplex
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@[reassoc]
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lemma homologySequenceδ_quotient_mapTriangle_obj
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(T : Triangle (CochainComplex C ℤ)) (n₀ n₁ : ℤ) (h : n₀ + 1 = n₁) :
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(homologyFunctor C (up ℤ) 0).homologySequenceδ
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((quotient C (up ℤ)).mapTriangle.obj T) n₀ n₁ h =
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(homologyFunctorFactors C (up ℤ) n₀).hom.app _ ≫
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(HomologicalComplex.homologyFunctor C (up ℤ) 0).shiftMap T.mor₃ n₀ n₁ (by omega) ≫
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(homologyFunctorFactors C (up ℤ) n₁).inv.app _ := by
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apply homologyFunctor_shiftMap
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namespace mappingCone
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variable (S : ShortComplex (CochainComplex C ℤ)) (hS : S.ShortExact)
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/-- The canonical morphism `mappingCone S.f ⟶ S.X₃` when `S` is a short complex
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of cochain complexes. -/
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noncomputable def descShortComplex : mappingCone S.f ⟶ S.X₃ := desc S.f 0 S.g (by simp)
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@[reassoc (attr := simp)]
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lemma inr_descShortComplex : inr S.f ≫ descShortComplex S = S.g := by
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simp [descShortComplex]
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@[reassoc (attr := simp)]
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lemma inr_f_descShortComplex_f (n : ℤ) : (inr S.f).f n ≫ (descShortComplex S).f n = S.g.f n := by
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simp [descShortComplex]
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@[reassoc (attr := simp)]
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lemma inl_v_descShortComplex_f (i j : ℤ) (h : i + (-1) = j) :
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(inl S.f).v i j h ≫ (descShortComplex S).f j = 0 := by
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simp [descShortComplex]
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variable {S}
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lemma homologySequenceδ_triangleh (n₀ : ℤ) (n₁ : ℤ) (h : n₀ + 1 = n₁) :
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(homologyFunctor C (up ℤ) 0).homologySequenceδ (triangleh S.f) n₀ n₁ h =
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(homologyFunctorFactors C (up ℤ) n₀).hom.app _ ≫
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HomologicalComplex.homologyMap (descShortComplex S) n₀ ≫ hS.δ n₀ n₁ h ≫
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(homologyFunctorFactors C (up ℤ) n₁).inv.app _ := by
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/- We proceed by diagram chase. We test the identity on
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cocycles `x' : A' ⟶ (mappingCone S.f).X n₀` -/
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dsimp
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rw [← cancel_mono ((homologyFunctorFactors C (up ℤ) n₁).hom.app _),
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assoc, assoc, assoc, Iso.inv_hom_id_app,
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← cancel_epi ((homologyFunctorFactors C (up ℤ) n₀).inv.app _), Iso.inv_hom_id_app_assoc]
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apply yoneda.map_injective
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ext ⟨A⟩ (x : A ⟶ _)
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obtain ⟨A', π, _, x', w, hx'⟩ :=
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(mappingCone S.f).eq_liftCycles_homologyπ_up_to_refinements x n₁ (by simpa using h)
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erw [homologySequenceδ_quotient_mapTriangle_obj_assoc _ _ _ h]
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dsimp
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rw [comp_id, Iso.inv_hom_id_app_assoc, Iso.inv_hom_id_app]
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erw [comp_id]
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rw [← cancel_epi π, reassoc_of% hx', reassoc_of% hx',
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HomologicalComplex.homologyπ_naturality_assoc,
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HomologicalComplex.liftCycles_comp_cyclesMap_assoc]
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/- We decompose the cocycle `x'` into two morphisms `a : A' ⟶ S.X₁.X n₁`
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and `b : A' ⟶ S.X₂.X n₀` satisfying certain relations. -/
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obtain ⟨a, b, hab⟩ := decomp_to _ x' n₁ h
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rw [hab, ext_to_iff _ n₁ (n₁ + 1) rfl, add_comp, assoc, assoc, inr_f_d, add_comp, assoc,
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assoc, assoc, assoc, inr_f_fst_v, comp_zero, comp_zero, add_zero, zero_comp,
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d_fst_v _ _ _ _ h, comp_neg, inl_v_fst_v_assoc, comp_neg, neg_eq_zero,
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add_comp, assoc, assoc, assoc, assoc, inr_f_snd_v, comp_id, zero_comp,
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d_snd_v _ _ _ h, comp_add, inl_v_fst_v_assoc, inl_v_snd_v_assoc, zero_comp, add_zero] at w
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/- We simplify the RHS. -/
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conv_rhs => simp only [hab, add_comp, assoc, inr_f_descShortComplex_f,
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inl_v_descShortComplex_f, comp_zero, zero_add]
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rw [hS.δ_eq n₀ n₁ (by simpa using h) (b ≫ S.g.f n₀) _ b rfl (-a)
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(by simp only [neg_comp, neg_eq_iff_add_eq_zero, w.2]) (n₁ + 1) (by simp)]
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/- We simplify the LHS. -/
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dsimp [Functor.shiftMap, homologyFunctor_shift]
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rw [HomologicalComplex.homologyπ_naturality_assoc,
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HomologicalComplex.liftCycles_comp_cyclesMap_assoc,
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S.X₁.liftCycles_shift_homologyπ_assoc _ _ _ _ n₁ (by omega) (n₁ + 1) (by simp),
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Iso.inv_hom_id_app]
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dsimp [homologyFunctor_shift]
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simp only [hab, add_comp, assoc, inl_v_triangle_mor₃_f_assoc,
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shiftFunctorObjXIso, neg_comp, Iso.inv_hom_id, comp_neg, comp_id,
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inr_f_triangle_mor₃_f_assoc, zero_comp, comp_zero, add_zero]
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open ComposableArrows
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set_option simprocs false
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lemma quasiIso_descShortComplex : QuasiIso (descShortComplex S) where
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quasiIsoAt n := by
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rw [quasiIsoAt_iff_isIso_homologyMap]
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let φ : ((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅
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(triangleh S.f) n _ rfl).δlast ⟶ (composableArrows₅ hS n _ rfl).δlast :=
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homMk₄ ((homologyFunctorFactors C (up ℤ) _).hom.app _)
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((homologyFunctorFactors C (up ℤ) _).hom.app _)
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((homologyFunctorFactors C (up ℤ) _).hom.app _ ≫
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HomologicalComplex.homologyMap (descShortComplex S) n)
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((homologyFunctorFactors C (up ℤ) _).hom.app _)
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((homologyFunctorFactors C (up ℤ) _).hom.app _)
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((homologyFunctorFactors C (up ℤ) _).hom.naturality S.f)
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(by
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erw [(homologyFunctorFactors C (up ℤ) n).hom.naturality_assoc]
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dsimp
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rw [← HomologicalComplex.homologyMap_comp, inr_descShortComplex])
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(by
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dsimp
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erw [homologySequenceδ_triangleh hS]
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simp only [Functor.comp_obj, HomologicalComplex.homologyFunctor_obj, assoc,
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Iso.inv_hom_id_app, comp_id])
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((homologyFunctorFactors C (up ℤ) _).hom.naturality S.f)
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have : IsIso ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f) ≫
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HomologicalComplex.homologyMap (descShortComplex S) n) := by
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apply Abelian.isIso_of_epi_of_isIso_of_isIso_of_mono
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((homologyFunctor C (up ℤ) 0).homologySequenceComposableArrows₅_exact _
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(mappingCone_triangleh_distinguished S.f) n _ rfl).δlast
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(composableArrows₅_exact hS n _ rfl).δlast φ
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all_goals dsimp [φ]; infer_instance
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apply IsIso.of_isIso_comp_left ((homologyFunctorFactors C (up ℤ) n).hom.app (mappingCone S.f))
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end mappingCone
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end CochainComplex

‎Mathlib/Algebra/Homology/Refinements.lean

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@@ -0,0 +1,34 @@
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/-
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Copyright (c) 2024 Joël Riou. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joël Riou
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-/
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import Mathlib.CategoryTheory.Abelian.Refinements
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import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
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/-!
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# Refinements
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This file contains lemmas about "refinements" that are specific to
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the study of the homology of `HomologicalComplex`. General
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lemmas about refinements and the case of `ShortComplex` appear
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in the file `CategoryTheory.Abelian.Refinements`.
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-/
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open CategoryTheory
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variable {C ι : Type*} [Category C] [Abelian C] {c : ComplexShape ι}
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(K : HomologicalComplex C c)
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namespace HomologicalComplex
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lemma eq_liftCycles_homologyπ_up_to_refinements {A : C} {i : ι} (γ : A ⟶ K.homology i)
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(j : ι) (hj : c.next i = j) :
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∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (z : A' ⟶ K.X i) (hz : z ≫ K.d i j = 0),
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π ≫ γ = K.liftCycles z j hj hz ≫ K.homologyπ i := by
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subst hj
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exact (K.sc i).eq_liftCycles_homologyπ_up_to_refinements γ
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end HomologicalComplex

‎Mathlib/CategoryTheory/Abelian/Refinements.lean

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@@ -110,4 +110,13 @@ lemma ShortComplex.Exact.exact_up_to_refinements
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rw [ShortComplex.exact_iff_exact_up_to_refinements] at hS
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exact hS x₂ hx₂
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lemma ShortComplex.eq_liftCycles_homologyπ_up_to_refinements {A : C} (γ : A ⟶ S.homology) :
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∃ (A' : C) (π : A' ⟶ A) (_ : Epi π) (z : A' ⟶ S.X₂) (hz : z ≫ S.g = 0),
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π ≫ γ = S.liftCycles z hz ≫ S.homologyπ := by
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obtain ⟨A', π, hπ, z, hz⟩ := surjective_up_to_refinements_of_epi S.homologyπ γ
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refine ⟨A', π, hπ, z ≫ S.iCycles, by simp, ?_⟩
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rw [hz]
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congr 1
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rw [← cancel_mono S.iCycles, liftCycles_i]
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end CategoryTheory

‎Mathlib/CategoryTheory/Shift/InducedShiftSequence.lean

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@@ -119,6 +119,18 @@ lemma induced_shiftIso_hom_app_obj (n a a' : M) (ha' : n + a = a') (X : C) :
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(G.shiftIso n a a' ha').hom.app X ≫ (e' a').inv.app X := by
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apply induced.shiftIso_hom_app_obj
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@[reassoc]
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lemma induced_shiftMap {n : M} {X Y : C} (f : X ⟶ Y⟦n⟧) (a a' : M) (h : n + a = a') :
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letI := induced e M F' e'
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F.shiftMap (L.map f ≫ (L.commShiftIso n).hom.app _) a a' h =
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(e' a).hom.app X ≫ G.shiftMap f a a' h ≫ (e' a').inv.app Y := by
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dsimp [shiftMap]
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rw [Functor.map_comp, induced_shiftIso_hom_app_obj, assoc, assoc]
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nth_rw 2 [← Functor.map_comp_assoc]
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simp only [comp_obj, Iso.hom_inv_id_app, map_id, id_comp]
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rw [← NatTrans.naturality_assoc]
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rfl
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end ShiftSequence
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end Functor

‎Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean

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@@ -7,6 +7,7 @@ import Mathlib.Algebra.Homology.ShortComplex.Exact
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import Mathlib.CategoryTheory.Shift.ShiftSequence
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import Mathlib.CategoryTheory.Triangulated.Functor
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import Mathlib.CategoryTheory.Triangulated.Subcategory
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import Mathlib.Algebra.Homology.ExactSequence
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/-! # Homological functors
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@@ -249,6 +250,21 @@ lemma mem_homologicalKernel_W_iff {X Y : C} (f : X ⟶ Y) :
249250
· intros
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constructor <;> infer_instance
251252

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open ComposableArrows
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/-- The exact sequence with six terms starting from `(F.shift n₀).obj T.obj₁` until
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`(F.shift n₁).obj T.obj₃` when `T` is a distinguished triangle and `F` a homological functor. -/
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@[simp] noncomputable def homologySequenceComposableArrows₅ : ComposableArrows A 5 :=
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mk₅ ((F.shift n₀).map T.mor₁) ((F.shift n₀).map T.mor₂)
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(F.homologySequenceδ T n₀ n₁ h) ((F.shift n₁).map T.mor₁) ((F.shift n₁).map T.mor₂)
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lemma homologySequenceComposableArrows₅_exact :
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(F.homologySequenceComposableArrows₅ T n₀ n₁ h).Exact :=
263+
exact_of_δ₀ (F.homologySequence_exact₂ T hT n₀).exact_toComposableArrows
264+
(exact_of_δ₀ (F.homologySequence_exact₃ T hT n₀ n₁ h).exact_toComposableArrows
265+
(exact_of_δ₀ (F.homologySequence_exact₁ T hT n₀ n₁ h).exact_toComposableArrows
266+
(F.homologySequence_exact₂ T hT n₁).exact_toComposableArrows))
267+
252268
end
253269

254270
end Functor

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