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Found 108 declarations mentioning CategoryTheory.Abelian.SpectralObject.opcycles.

• CategoryTheory.Abelian.SpectralObject.opcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n : ℤ) : C
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcycles_X₃ 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).X₃ = X.opcycles f g n₁
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcycles_X₁ 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : (X.kernelSequenceOpcycles f g fg h n).X₁ = X.opcycles f g n
• CategoryTheory.Abelian.SpectralObject.isZero_opcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n : ℤ) (h : CategoryTheory.Limits.IsZero ((X.H n).obj (CategoryTheory.ComposableArrows.mk₁ f))) : CategoryTheory.Limits.IsZero (X.opcycles f g n)
• CategoryTheory.Abelian.SpectralObject.pOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n : ℤ) : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ f) ⟶ X.opcycles f g n
• CategoryTheory.Abelian.SpectralObject.instEpiPOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n : ℤ) : CategoryTheory.Epi (X.pOpcycles f g n)
• CategoryTheory.Abelian.SpectralObject.δFromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : X.opcycles f₂ f₃ n₀ ⟶ (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₁)
• CategoryTheory.Abelian.SpectralObject.fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : X.opcycles f g n ⟶ (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg)
• CategoryTheory.Abelian.SpectralObject.instMonoFromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : CategoryTheory.Mono (X.fromOpcycles f g fg h n)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_id 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n : ℤ) : X.opcyclesMap f g f g (CategoryTheory.CategoryStruct.id (CategoryTheory.ComposableArrows.mk₂ f g)) n = CategoryTheory.CategoryStruct.id (X.opcycles f g n)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) {i' j' k' : ι} (f' : i' ⟶ j') (g' : j' ⟶ k') (α : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f' g') (n : ℤ) : X.opcycles f g n ⟶ X.opcycles f' g' n
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcycles_g 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : (X.cokernelSequenceOpcycles f g n₀ n₁ hn₁).g = X.pOpcycles f g n₁
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcycles_f 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : (X.kernelSequenceOpcycles f g fg h n).f = X.fromOpcycles f g fg h n
• CategoryTheory.Abelian.SpectralObject.isIso_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) (hg : CategoryTheory.Limits.IsZero ((X.H n).obj (CategoryTheory.ComposableArrows.mk₁ g))) : CategoryTheory.IsIso (X.fromOpcycles f g fg h n)
• CategoryTheory.Abelian.SpectralObject.fromOpcyles_δ 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₀) (X.δ f₁ f₂₃ n₀ n₁ hn₁) = X.δFromOpcycles f₁ f₂ f₃ n₀ n₁ hn₁
• CategoryTheory.Abelian.SpectralObject.pOpcycles_δFromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f₂ f₃ n₀) (X.δFromOpcycles f₁ f₂ f₃ n₀ n₁ hn₁) = X.δ f₁ f₂ n₀ n₁ hn₁
• CategoryTheory.Abelian.SpectralObject.p_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n) (X.fromOpcycles f g fg h n) = (X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)
• CategoryTheory.Abelian.SpectralObject.fromOpcyles_δ_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₁) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₀) (CategoryTheory.CategoryStruct.comp (X.δ f₁ f₂₃ n₀ n₁ hn₁) h) = CategoryTheory.CategoryStruct.comp (X.δFromOpcycles f₁ f₂ f₃ n₀ n₁ hn₁) h
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoKernel 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : X.opcycles f g n ≅ CategoryTheory.Limits.kernel ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h))
• CategoryTheory.Abelian.SpectralObject.pOpcycles_δFromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁) {Z : C} (h : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₁) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f₂ f₃ n₀) (CategoryTheory.CategoryStruct.comp (X.δFromOpcycles f₁ f₂ f₃ n₀ n₁ hn₁) h) = CategoryTheory.CategoryStruct.comp (X.δ f₁ f₂ n₀ n₁ hn₁) h
• CategoryTheory.Abelian.SpectralObject.δ_pOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.δ f g n₀ n₁ hn₁) (X.pOpcycles f g n₁) = 0
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_comp 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) {i' j' k' : ι} (f' : i' ⟶ j') (g' : j' ⟶ k') {i'' j'' k'' : ι} (f'' : i'' ⟶ j'') (g'' : j'' ⟶ k'') (α : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f' g') (α' : CategoryTheory.ComposableArrows.mk₂ f' g' ⟶ CategoryTheory.ComposableArrows.mk₂ f'' g'') (α'' : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f'' g'') (n : ℤ) (h : CategoryTheory.CategoryStruct.comp α α' = α'' := by cat_disch) : CategoryTheory.CategoryStruct.comp (X.opcyclesMap f g f' g' α n) (X.opcyclesMap f' g' f'' g'' α' n) = X.opcyclesMap f g f'' g'' α'' n
• CategoryTheory.Abelian.SpectralObject.descOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁) {A : C} (x : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f) ⟶ A) (hx : CategoryTheory.CategoryStruct.comp (X.δ f g n₀ n₁ hn₁) x = 0) : X.opcycles f g n₁ ⟶ A
• CategoryTheory.Abelian.SpectralObject.p_fromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) {Z : C} (h✝ : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f g fg h n) h✝) = CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) h✝
• CategoryTheory.Abelian.SpectralObject.fromOpcycles_H_map_twoδ₁Toδ₀ 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : CategoryTheory.CategoryStruct.comp (X.fromOpcycles f g fg h n) ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) = 0
• CategoryTheory.Abelian.SpectralObject.liftOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) {A : C} {n : ℤ} (x : A ⟶ (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg)) (hx : CategoryTheory.CategoryStruct.comp x ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) = 0) : A ⟶ X.opcycles f g n
• CategoryTheory.Abelian.SpectralObject.δ_pOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : X.opcycles f g n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.δ f g n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n₁) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Abelian.SpectralObject.p_descOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁) {A : C} (x : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f) ⟶ A) (hx : CategoryTheory.CategoryStruct.comp (X.δ f g n₀ n₁ hn₁) x = 0) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n₁) (X.descOpcycles f g n₀ n₁ hn₁ x hx) = x
• CategoryTheory.Abelian.SpectralObject.fromOpcycles_H_map_twoδ₁Toδ₀_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) {Z : C} (h✝ : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ g) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.fromOpcycles f g fg h n) (CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) h✝) = CategoryTheory.CategoryStruct.comp 0 h✝
• CategoryTheory.Abelian.SpectralObject.liftOpcycles_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) {A : C} {n : ℤ} (x : A ⟶ (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg)) (hx : CategoryTheory.CategoryStruct.comp x ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) = 0) : CategoryTheory.CategoryStruct.comp (X.liftOpcycles f g fg h x hx) (X.fromOpcycles f g fg h n) = x
• CategoryTheory.Abelian.SpectralObject.p_descOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁) {A : C} (x : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f) ⟶ A) (hx : CategoryTheory.CategoryStruct.comp (X.δ f g n₀ n₁ hn₁) x = 0) {Z : C} (h : A ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n₁) (CategoryTheory.CategoryStruct.comp (X.descOpcycles f g n₀ n₁ hn₁ x hx) h) = CategoryTheory.CategoryStruct.comp x h
• CategoryTheory.Abelian.SpectralObject.liftOpcycles_fromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) {A : C} {n : ℤ} (x : A ⟶ (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg)) (hx : CategoryTheory.CategoryStruct.comp x ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) = 0) {Z : C} (h✝ : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.liftOpcycles f g fg h x hx) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f g fg h n) h✝) = CategoryTheory.CategoryStruct.comp x h✝
• CategoryTheory.Abelian.SpectralObject.p_opcyclesMap 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) {i' j' k' : ι} (f' : i' ⟶ j') (g' : j' ⟶ k') (α : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f' g') (β : CategoryTheory.ComposableArrows.mk₁ f ⟶ CategoryTheory.ComposableArrows.mk₁ f') (n : ℤ) (hβ : β = CategoryTheory.ComposableArrows.homMk₁ (α.app 0) (α.app 1) ⋯ := by cat_disch) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n) (X.opcyclesMap f g f' g' α n) = CategoryTheory.CategoryStruct.comp ((X.H n).map β) (X.pOpcycles f' g' n)
• CategoryTheory.Abelian.SpectralObject.p_opcyclesMap_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) {i' j' k' : ι} (f' : i' ⟶ j') (g' : j' ⟶ k') (α : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f' g') (β : CategoryTheory.ComposableArrows.mk₁ f ⟶ CategoryTheory.ComposableArrows.mk₁ f') (n : ℤ) (hβ : β = CategoryTheory.ComposableArrows.homMk₁ (α.app 0) (α.app 1) ⋯ := by cat_disch) {Z : C} (h : X.opcycles f' g' n ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n) (CategoryTheory.CategoryStruct.comp (X.opcyclesMap f g f' g' α n) h) = CategoryTheory.CategoryStruct.comp ((X.H n).map β) (CategoryTheory.CategoryStruct.comp (X.pOpcycles f' g' n) h)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) {i' j' k' : ι} (f' : i' ⟶ j') (g' : j' ⟶ k') (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (fg' : i' ⟶ k') (h' : CategoryTheory.CategoryStruct.comp f' g' = fg') (α : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f' g') (β : CategoryTheory.ComposableArrows.mk₁ fg ⟶ CategoryTheory.ComposableArrows.mk₁ fg') (n : ℤ) (hβ₀ : β.app 0 = α.app 0 := by cat_disch) (hβ₁ : β.app 1 = α.app 2 := by cat_disch) : CategoryTheory.CategoryStruct.comp (X.opcyclesMap f g f' g' α n) (X.fromOpcycles f' g' fg' h' n) = CategoryTheory.CategoryStruct.comp (X.fromOpcycles f g fg h n) ((X.H n).map β)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_fromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) {i' j' k' : ι} (f' : i' ⟶ j') (g' : j' ⟶ k') (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (fg' : i' ⟶ k') (h' : CategoryTheory.CategoryStruct.comp f' g' = fg') (α : CategoryTheory.ComposableArrows.mk₂ f g ⟶ CategoryTheory.ComposableArrows.mk₂ f' g') (β : CategoryTheory.ComposableArrows.mk₁ fg ⟶ CategoryTheory.ComposableArrows.mk₁ fg') (n : ℤ) (hβ₀ : β.app 0 = α.app 0 := by cat_disch) (hβ₁ : β.app 1 = α.app 2 := by cat_disch) {Z : C} (h✝ : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg') ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.opcyclesMap f g f' g' α n) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f' g' fg' h' n) h✝) = CategoryTheory.CategoryStruct.comp (X.fromOpcycles f g fg h n) (CategoryTheory.CategoryStruct.comp ((X.H n).map β) h✝)
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoKernel_hom_fac 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n) (CategoryTheory.CategoryStruct.comp (X.opcyclesIsoKernel f g fg h n).hom (CategoryTheory.Limits.kernel.ι ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)))) = (X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoKernel_hom_fac_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Cycles
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k : ι} (f : i ⟶ j) (g : j ⟶ k) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) {Z : C} (h✝ : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f g n) (CategoryTheory.CategoryStruct.comp (X.opcyclesIsoKernel f g fg h n).hom (CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.kernel.ι ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h))) h✝)) = CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) h✝
• CategoryTheory.Abelian.SpectralObject.ιE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ⟶ X.opcycles f₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesE_X₂ 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.kernelSequenceOpcyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₂ = X.opcycles f₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.instMonoιE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_2} {ι : Type u_4} [CategoryTheory.Category.{u_1, u_2} C] [CategoryTheory.Category.{u_3, u_4} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁) (hn₂ : n₁ + 1 = n₂) : CategoryTheory.Mono (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂)
• CategoryTheory.Abelian.SpectralObject.rightHomologyDataShortComplex_Q 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.rightHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).Q = X.opcycles f₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.opcyclesToE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : X.opcycles f₁₂ f₃ n₁ ⟶ X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcyclesE_X₂ 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).X₂ = X.opcycles f₁₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.instEpiOpcyclesToE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁) (hn₂ : n₁ + 1 = n₂) : CategoryTheory.Epi (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂)
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoH 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : X.opcycles f (CategoryTheory.CategoryStruct.id i₁) n₁ ≅ (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f)
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesE_f 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.kernelSequenceOpcyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f = X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.opcyclesIso 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).opcycles ≅ X.opcycles f₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.shortComplexOpcyclesThreeδ₂Toδ₁_X₁ 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.shortComplexOpcyclesThreeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₁ = X.opcycles f₁ f₂₃ n₁
• CategoryTheory.Abelian.SpectralObject.shortComplexOpcyclesThreeδ₂Toδ₁_X₂ 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.shortComplexOpcyclesThreeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₂ = X.opcycles f₁₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcyclesE_g 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).g = X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.shortComplexOpcyclesThreeδ₂Toδ₁_g 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.shortComplexOpcyclesThreeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃ n₀ n₁ n₂ hn₁ hn₂).g = X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.shortComplexOpcyclesThreeδ₂Toδ₁_f 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.shortComplexOpcyclesThreeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f = X.opcyclesMap f₁ f₂₃ f₁₂ f₃ (CategoryTheory.ComposableArrows.threeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃) n₁
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesE_g 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.kernelSequenceOpcyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g = X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂
• CategoryTheory.Abelian.SpectralObject.opcyclesToE_ιE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) = X.opcyclesMap f₁₂ f₃ f₂ f₃ (CategoryTheory.ComposableArrows.threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₁
• CategoryTheory.Abelian.SpectralObject.rightHomologyDataShortComplex_p 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.rightHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).p = X.pOpcycles f₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.rightHomologyDataShortComplex_H 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.rightHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).H = CategoryTheory.Limits.kernel (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoH_hom 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : (X.opcyclesIsoH f n₀ n₁ hn₁).hom = X.fromOpcycles f (CategoryTheory.CategoryStruct.id i₁) f ⋯ n₁
• CategoryTheory.Abelian.SpectralObject.opcyclesToE_ιE_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.opcycles f₂ f₃ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₁₂ f₃ f₂ f₃ (CategoryTheory.ComposableArrows.threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₁) h
• CategoryTheory.Abelian.SpectralObject.πE_ιE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) = CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂ n₁) (X.pOpcycles f₂ f₃ n₁)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_threeδ₂Toδ₁_opcyclesToE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₁ f₂₃ f₁₂ f₃ (CategoryTheory.ComposableArrows.threeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃) n₁) (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) = 0
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoH_hom_inv_id 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.fromOpcycles f (CategoryTheory.CategoryStruct.id i₁) f ⋯ n₁) (X.opcyclesIsoH f n₀ n₁ hn₁).inv = CategoryTheory.CategoryStruct.id (X.opcycles f (CategoryTheory.CategoryStruct.id i₁) n₁)
• CategoryTheory.Abelian.SpectralObject.kernelSequenceE_f 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.kernelSequenceE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f = CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯) (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₁)
• CategoryTheory.Abelian.SpectralObject.rightHomologyDataShortComplex_ι 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.rightHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).ι = CategoryTheory.Limits.kernel.ι (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ ⋯)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_threeδ₂Toδ₁_opcyclesToE_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (f₂₃ : i₁ ⟶ i₃) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₁ f₂₃ f₁₂ f₃ (CategoryTheory.ComposableArrows.threeδ₂Toδ₁ f₁ f₂ f₃ f₁₂ f₂₃ h₁₂ h₂₃) n₁) (CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Abelian.SpectralObject.p_opcyclesIso_hom 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).pOpcycles (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom = X.pOpcycles f₂ f₃ n₁
• CategoryTheory.Abelian.SpectralObject.πE_ιE_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.opcycles f₂ f₃ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂ n₁) (CategoryTheory.CategoryStruct.comp (X.pOpcycles f₂ f₃ n₁) h)
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoH_hom_inv_id_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : X.opcycles f (CategoryTheory.CategoryStruct.id i₁) n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.fromOpcycles f (CategoryTheory.CategoryStruct.id i₁) f ⋯ n₁) (CategoryTheory.CategoryStruct.comp (X.opcyclesIsoH f n₀ n₁ hn₁).inv h) = h
• CategoryTheory.Abelian.SpectralObject.EIsoH_hom_opcyclesIsoH_inv 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁) (hn₂ : n₁ + 1 = n₂) {i j : ι} (f : i ⟶ j) : CategoryTheory.CategoryStruct.comp (X.EIsoH f n₀ n₁ n₂ hn₁ hn₂).hom (X.opcyclesIsoH f n₀ n₁ hn₁).inv = X.ιE (CategoryTheory.CategoryStruct.id i) f (CategoryTheory.CategoryStruct.id j) n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.EToCycles_i 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₂₃ : i₁ ⟶ i₃) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.EToCycles f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂) (X.iCycles f₁ f₂₃ n₁) = CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₁)
• CategoryTheory.Abelian.SpectralObject.p_opcyclesToE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f₁₂ f₃ n₁) (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) = CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂)
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcyclesE_f 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.cokernelSequenceOpcyclesE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).f = CategoryTheory.CategoryStruct.comp ((X.H n₁).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂)) (X.pOpcycles f₁₂ f₃ n₁)
• CategoryTheory.Abelian.SpectralObject.EToCycles_i_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₂₃ : i₁ ⟶ i₃) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₂₃) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.EToCycles f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂₃ n₁) h) = CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₁) h)
• CategoryTheory.Abelian.SpectralObject.p_opcyclesToE_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f₁₂ f₃ n₁) (CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) (CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) h)
• CategoryTheory.Abelian.SpectralObject.p_opcyclesIso_hom_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.opcycles f₂ f₃ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).pOpcycles (CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom h) = CategoryTheory.CategoryStruct.comp (X.pOpcycles f₂ f₃ n₁) h
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoH_inv_hom_id_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.opcyclesIsoH f n₀ n₁ hn₁).inv (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f (CategoryTheory.CategoryStruct.id i₁) f ⋯ n₁) h) = h
• CategoryTheory.Abelian.SpectralObject.p_opcyclesIso_inv 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f₂ f₃ n₁) (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv = (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).pOpcycles
• CategoryTheory.Abelian.SpectralObject.opcyclesIso_hom_δFromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂) = (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).fromOpcycles
• CategoryTheory.Abelian.SpectralObject.opcyclesIsoH_inv_hom_id 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.opcyclesIsoH f n₀ n₁ hn₁).inv (X.fromOpcycles f (CategoryTheory.CategoryStruct.id i₁) f ⋯ n₁) = CategoryTheory.CategoryStruct.id ((X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f))
• CategoryTheory.Abelian.SpectralObject.EIsoH_hom_opcyclesIsoH_inv_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁) (hn₂ : n₁ + 1 = n₂) {i j : ι} (f : i ⟶ j) {Z : C} (h : X.opcycles f (CategoryTheory.CategoryStruct.id j) n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.EIsoH f n₀ n₁ n₂ hn₁ hn₂).hom (CategoryTheory.CategoryStruct.comp (X.opcyclesIsoH f n₀ n₁ hn₁).inv h) = CategoryTheory.CategoryStruct.comp (X.ιE (CategoryTheory.CategoryStruct.id i) f (CategoryTheory.CategoryStruct.id j) n₀ n₁ n₂ hn₁ hn₂) h
• CategoryTheory.Abelian.SpectralObject.ιE_δFromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂) = 0
• CategoryTheory.Abelian.SpectralObject.ιE_δFromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : (X.H n₂).obj (CategoryTheory.ComposableArrows.mk₁ f₁) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Abelian.SpectralObject.p_opcyclesIso_inv_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).opcycles ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.pOpcycles f₂ f₃ n₁) (CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv h) = CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).pOpcycles h
• CategoryTheory.Abelian.SpectralObject.opcyclesIso_hom_δFromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : (X.H n₂).obj (CategoryTheory.ComposableArrows.mk₁ f₁) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom (CategoryTheory.CategoryStruct.comp (X.δFromOpcycles f₁ f₂ f₃ n₁ n₂ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).fromOpcycles h
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesEIso_hom_τ₂ 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.kernelSequenceOpcyclesEIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom.τ₂ = (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv
• CategoryTheory.Abelian.SpectralObject.kernelSequenceOpcyclesEIso_inv_τ₂ 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.kernelSequenceOpcyclesEIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv.τ₂ = (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom
• CategoryTheory.Abelian.SpectralObject.map_ιE 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃') (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (γ : CategoryTheory.ComposableArrows.mk₂ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₂' f₃') (n₀ n₁ n₂ : ℤ) (hγ : γ = CategoryTheory.ComposableArrows.homMk₂ (α.app 1) (α.app 2) (α.app 3) ⋯ ⋯ := by cat_disch) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂) (X.ιE f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂) = CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.opcyclesMap f₂ f₃ f₂' f₃' γ n₁)
• CategoryTheory.Abelian.SpectralObject.map_ιE_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃') (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (γ : CategoryTheory.ComposableArrows.mk₂ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₂' f₃') (n₀ n₁ n₂ : ℤ) (hγ : γ = CategoryTheory.ComposableArrows.homMk₂ (α.app 1) (α.app 2) (α.app 3) ⋯ ⋯ := by cat_disch) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.opcycles f₂' f₃' n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.ιE f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₂ f₃ f₂' f₃' γ n₁) h)
• CategoryTheory.Abelian.SpectralObject.opcyclesToE_map 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃') (f₁₂' : i₀' ⟶ i₂') (h₁₂' : CategoryTheory.CategoryStruct.comp f₁' f₂' = f₁₂') (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (β : CategoryTheory.ComposableArrows.mk₂ f₁₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁₂' f₃') (n₀ n₁ n₂ : ℤ) (h₀ : β.app 0 = α.app 0 := by cat_disch) (h₁ : β.app 1 = α.app 2 := by cat_disch) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯) = CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₁₂ f₃ f₁₂' f₃' β n₁) (X.opcyclesToE f₁' f₂' f₃' f₁₂' h₁₂' n₀ n₁ n₂ hn₁ hn₂)
• CategoryTheory.Abelian.SpectralObject.opcyclesToE_map_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃') (f₁₂' : i₀' ⟶ i₂') (h₁₂' : CategoryTheory.CategoryStruct.comp f₁' f₂' = f₁₂') (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (β : CategoryTheory.ComposableArrows.mk₂ f₁₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁₂' f₃') (n₀ n₁ n₂ : ℤ) (h₀ : β.app 0 = α.app 0 := by cat_disch) (h₁ : β.app 1 = α.app 2 := by cat_disch) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.E f₁' f₂' f₃' n₀ n₁ n₂ ⋯ ⋯ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ ⋯ ⋯) h) = CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₁₂ f₃ f₁₂' f₃' β n₁) (CategoryTheory.CategoryStruct.comp (X.opcyclesToE f₁' f₂' f₃' f₁₂' h₁₂' n₀ n₁ n₂ hn₁ hn₂) h)
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_opcyclesIso_hom 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃') (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (γ : CategoryTheory.ComposableArrows.mk₂ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₂' f₃') (hγ : γ = CategoryTheory.ComposableArrows.homMk₂ (α.app 1) (α.app 2) (α.app 3) ⋯ ⋯ := by cat_disch) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)) (X.opcyclesIso f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂).hom = CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom (X.opcyclesMap f₂ f₃ f₂' f₃' γ n₁)
• CategoryTheory.Abelian.SpectralObject.liftE_ιE_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) {A : C} (x : A ⟶ (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₂₃)) (h : CategoryTheory.CategoryStruct.comp x ((X.H n₁).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃)) = 0) (hn₂ : n₁ + 1 = n₂) (h' : CategoryTheory.CategoryStruct.comp x (X.δ f₁ f₂₃ n₁ n₂ hn₂) = 0) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.liftE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ x h hn₂ h' hn₁) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₁)) = x
• CategoryTheory.Abelian.SpectralObject.liftE_ιE_fromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ n₂ : ℤ) {A : C} (x : A ⟶ (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₂₃)) (h : CategoryTheory.CategoryStruct.comp x ((X.H n₁).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃)) = 0) (hn₂ : n₁ + 1 = n₂) (h' : CategoryTheory.CategoryStruct.comp x (X.δ f₁ f₂₃ n₁ n₂ hn₂) = 0) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h✝ : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₂₃) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.liftE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ x h hn₂ h' hn₁) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₁) h✝)) = CategoryTheory.CategoryStruct.comp x h✝
• CategoryTheory.Abelian.SpectralObject.opcyclesMap_opcyclesIso_hom_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Page
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃') (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (γ : CategoryTheory.ComposableArrows.mk₂ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₂' f₃') (hγ : γ = CategoryTheory.ComposableArrows.homMk₂ (α.app 1) (α.app 2) (α.app 3) ⋯ ⋯ := by cat_disch) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h : X.opcycles f₂' f₃' n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.opcyclesMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)) (CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂).hom h) = CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom (CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₂ f₃ f₂' f₃' γ n₁) h)
• CategoryTheory.Abelian.SpectralObject.Ψ 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : X.cycles f₂ f₃ n₀ ⟶ X.opcycles f₁ f₂ n₁
• CategoryTheory.Abelian.SpectralObject.cyclesMap_Ψ_exact 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : { X₁ := X.cycles f₁₂ f₃ n₀, X₂ := X.cycles f₂ f₃ n₀, X₃ := X.opcycles f₁ f₂ n₁, f := X.cyclesMap f₁₂ f₃ f₂ f₃ (CategoryTheory.ComposableArrows.threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀, g := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, zero := ⋯ }.Exact
• CategoryTheory.Abelian.SpectralObject.Ψ_opcyclesMap_exact 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : { X₁ := X.cycles f₂ f₃ n₀, X₂ := X.opcycles f₁ f₂ n₁, X₃ := X.opcycles f₁ f₂₃ n₁, f := X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁, g := X.opcyclesMap f₁ f₂ f₁ f₂₃ (CategoryTheory.ComposableArrows.threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁, zero := ⋯ }.Exact
• CategoryTheory.Abelian.SpectralObject.πE_d_ιE 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ i₄ i₅ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₄ : i₃ ⟶ i₄) (f₅ : i₄ ⟶ i₅) (n₀ n₁ n₂ n₃ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) (hn₃ : n₂ + 1 = n₃ := by lia) : CategoryTheory.CategoryStruct.comp (X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃) (X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃)) = X.Ψ f₂ f₃ f₄ n₁ n₂ hn₂
• CategoryTheory.Abelian.SpectralObject.cyclesMap_Ψ 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁₂ f₃ f₂ f₃ (CategoryTheory.ComposableArrows.threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀) (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) = 0
• CategoryTheory.Abelian.SpectralObject.Ψ_opcyclesMap 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) (X.opcyclesMap f₁ f₂ f₁ f₂₃ (CategoryTheory.ComposableArrows.threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁) = 0
• CategoryTheory.Abelian.SpectralObject.πE_d_ιE_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ i₄ i₅ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₄ : i₃ ⟶ i₄) (f₅ : i₄ ⟶ i₅) (n₀ n₁ n₂ n₃ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) (hn₃ : n₂ + 1 = n₃ := by lia) {Z : C} (h : X.opcycles f₂ f₃ n₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃) h)) = CategoryTheory.CategoryStruct.comp (X.Ψ f₂ f₃ f₄ n₁ n₂ hn₂) h
• CategoryTheory.Abelian.SpectralObject.cyclesMap_Ψ_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : X.opcycles f₁ f₂ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁₂ f₃ f₂ f₃ (CategoryTheory.ComposableArrows.threeδ₁Toδ₀ f₁ f₂ f₃ f₁₂ h₁₂) n₀) (CategoryTheory.CategoryStruct.comp (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Abelian.SpectralObject.toCycles_Ψ 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.toCycles f₂ f₃ f₂₃ h₂₃ n₀) (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) = CategoryTheory.CategoryStruct.comp (X.δ f₁ f₂₃ n₀ n₁ hn₁) (X.pOpcycles f₁ f₂ n₁)
• CategoryTheory.Abelian.SpectralObject.Ψ_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) (X.fromOpcycles f₁ f₂ f₁₂ h₁₂ n₁) = CategoryTheory.CategoryStruct.comp (X.iCycles f₂ f₃ n₀) (X.δ f₁₂ f₃ n₀ n₁ hn₁)
• CategoryTheory.Abelian.SpectralObject.toCycles_Ψ_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₂₃ : j ⟶ l) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : X.opcycles f₁ f₂ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.toCycles f₂ f₃ f₂₃ h₂₃ n₀) (CategoryTheory.CategoryStruct.comp (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) h) = CategoryTheory.CategoryStruct.comp (X.δ f₁ f₂₃ n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.pOpcycles f₁ f₂ n₁) h)
• CategoryTheory.Abelian.SpectralObject.Ψ_fromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₁₂) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.Ψ f₁ f₂ f₃ n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₁ f₂ f₁₂ h₁₂ n₁) h) = CategoryTheory.CategoryStruct.comp (X.iCycles f₂ f₃ n₀) (CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃ n₀ n₁ hn₁) h)
• CategoryTheory.Abelian.SpectralObject.d_ιE_fromOpcycles_assoc 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ i₄ i₅ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₄ : i₃ ⟶ i₄) (f₅ : i₄ ⟶ i₅) (f₂₃ : i₁ ⟶ i₃) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (f₃₄ : i₂ ⟶ i₄) (h₃₄ : CategoryTheory.CategoryStruct.comp f₃ f₄ = f₃₄) (f₄₅ : i₃ ⟶ i₅) (h₄₅ : CategoryTheory.CategoryStruct.comp f₄ f₅ = f₄₅) (n₀ n₁ n₂ n₃ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) (hn₃ : n₂ + 1 = n₃ := by lia) {Z : C} (h : (X.H n₂).obj (CategoryTheory.ComposableArrows.mk₁ f₂₃) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₂) h)) = CategoryTheory.CategoryStruct.comp (X.ιE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₄ f₅ f₄₅ h₄₅ n₁) (CategoryTheory.CategoryStruct.comp (X.δ f₂₃ f₄₅ n₁ n₂ hn₂) h))
• CategoryTheory.Abelian.SpectralObject.d_ιE_fromOpcycles 📋 Mathlib.Algebra.Homology.SpectralObject.Differentials
  {C : Type u_1} {ι : Type u_2} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Category.{v_2, u_2} ι] [CategoryTheory.Abelian C] (X : CategoryTheory.Abelian.SpectralObject C ι) {i₀ i₁ i₂ i₃ i₄ i₅ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃) (f₄ : i₃ ⟶ i₄) (f₅ : i₄ ⟶ i₅) (f₂₃ : i₁ ⟶ i₃) (h₂₃ : CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃) (f₃₄ : i₂ ⟶ i₄) (h₃₄ : CategoryTheory.CategoryStruct.comp f₃ f₄ = f₃₄) (f₄₅ : i₃ ⟶ i₅) (h₄₅ : CategoryTheory.CategoryStruct.comp f₄ f₅ = f₄₅) (n₀ n₁ n₂ n₃ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) (hn₃ : n₂ + 1 = n₃ := by lia) : CategoryTheory.CategoryStruct.comp (X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃) (CategoryTheory.CategoryStruct.comp (X.ιE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃) (X.fromOpcycles f₂ f₃ f₂₃ h₂₃ n₂)) = CategoryTheory.CategoryStruct.comp (X.ιE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.fromOpcycles f₄ f₅ f₄₅ h₄₅ n₁) (X.δ f₂₃ f₄₅ n₁ n₂ hn₂))

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59