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Found 103 declarations mentioning CategoryTheory.Abelian.SpectralObject.cycles.

• CategoryTheory.Abelian.SpectralObject.cycles 📋 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.kernelSequenceCycles_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) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : (X.kernelSequenceCycles f g n₀ n₁ hn₁).X₁ = X.cycles f g n₀
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCycles_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) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : (X.cokernelSequenceCycles f g fg h n).X₃ = X.cycles f g n
• CategoryTheory.Abelian.SpectralObject.isZero_cycles 📋 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 g))) : CategoryTheory.Limits.IsZero (X.cycles f g n)
• CategoryTheory.Abelian.SpectralObject.iCycles 📋 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.cycles f g n ⟶ (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ g)
• CategoryTheory.Abelian.SpectralObject.instMonoICycles 📋 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.Mono (X.iCycles f g n)
• CategoryTheory.Abelian.SpectralObject.δToCycles 📋 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.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ f₃) ⟶ X.cycles f₁ f₂ n₁
• CategoryTheory.Abelian.SpectralObject.toCycles 📋 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.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ X.cycles f g n
• CategoryTheory.Abelian.SpectralObject.instEpiToCycles 📋 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.Epi (X.toCycles f g fg h n)
• CategoryTheory.Abelian.SpectralObject.cyclesMap_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.cyclesMap f g f g (CategoryTheory.CategoryStruct.id (CategoryTheory.ComposableArrows.mk₂ f g)) n = CategoryTheory.CategoryStruct.id (X.cycles f g n)
• CategoryTheory.Abelian.SpectralObject.cyclesMap 📋 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.cycles f g n ⟶ X.cycles f' g' n
• CategoryTheory.Abelian.SpectralObject.kernelSequenceCycles_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) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : (X.kernelSequenceCycles f g n₀ n₁ hn₁).f = X.iCycles f g n₀
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCycles_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) (fg : i ⟶ k) (h : CategoryTheory.CategoryStruct.comp f g = fg) (n : ℤ) : (X.cokernelSequenceCycles f g fg h n).g = X.toCycles f g fg h n
• CategoryTheory.Abelian.SpectralObject.isIso_toCycles 📋 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) (hf : CategoryTheory.Limits.IsZero ((X.H n).obj (CategoryTheory.ComposableArrows.mk₁ f))) : CategoryTheory.IsIso (X.toCycles f g fg h n)
• CategoryTheory.Abelian.SpectralObject.δ_toCycles 📋 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) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃ n₀ n₁ hn₁) (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) = X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁
• CategoryTheory.Abelian.SpectralObject.δToCycles_iCycles 📋 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.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) (X.iCycles f₁ f₂ n₁) = X.δ f₂ f₃ n₀ n₁ hn₁
• CategoryTheory.Abelian.SpectralObject.toCycles_i 📋 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.toCycles f g fg h n) (X.iCycles f g n) = (X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)
• CategoryTheory.Abelian.SpectralObject.δ_toCycles_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) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) {Z : C} (h : X.cycles f₁ f₂ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃ n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) h) = CategoryTheory.CategoryStruct.comp (X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) h
• CategoryTheory.Abelian.SpectralObject.cokernelIsoCycles 📋 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.Limits.cokernel ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) ≅ X.cycles f g n
• CategoryTheory.Abelian.SpectralObject.δToCycles_iCycles_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 f₂) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂ n₁) h) = CategoryTheory.CategoryStruct.comp (X.δ f₂ f₃ n₀ n₁ hn₁) h
• CategoryTheory.Abelian.SpectralObject.iCycles_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 : ι} (f : i ⟶ j) (g : j ⟶ k) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : CategoryTheory.CategoryStruct.comp (X.iCycles f g n₀) (X.δ f g n₀ n₁ hn₁) = 0
• CategoryTheory.Abelian.SpectralObject.cyclesMap_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.cyclesMap f g f' g' α n) (X.cyclesMap f' g' f'' g'' α' n) = X.cyclesMap f g f'' g'' α'' n
• CategoryTheory.Abelian.SpectralObject.liftCycles 📋 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 : A ⟶ (X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ g)) (hx : CategoryTheory.CategoryStruct.comp x (X.δ f g n₀ n₁ hn₁) = 0) : A ⟶ X.cycles f g n₀
• CategoryTheory.Abelian.SpectralObject.cyclesMap_comp_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') {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) {Z : C} (h✝ : X.cycles f'' g'' n ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesMap f g f' g' α n) (CategoryTheory.CategoryStruct.comp (X.cyclesMap f' g' f'' g'' α' n) h✝) = CategoryTheory.CategoryStruct.comp (X.cyclesMap f g f'' g'' α'' n) h✝
• CategoryTheory.Abelian.SpectralObject.toCycles_i_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.toCycles f g fg h n) (CategoryTheory.CategoryStruct.comp (X.iCycles f g n) h✝) = CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) h✝
• CategoryTheory.Abelian.SpectralObject.H_map_twoδ₂Toδ₁_toCycles 📋 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.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) (X.toCycles f g fg h n) = 0
• CategoryTheory.Abelian.SpectralObject.descCycles 📋 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 : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ A) (hx : CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) x = 0) : X.cycles f g n ⟶ A
• CategoryTheory.Abelian.SpectralObject.iCycles_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 : ι} (f : i ⟶ j) (g : j ⟶ k) (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.iCycles f g n₀) (CategoryTheory.CategoryStruct.comp (X.δ f g n₀ n₁ hn₁) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Abelian.SpectralObject.liftCycles_i 📋 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 : A ⟶ (X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ g)) (hx : CategoryTheory.CategoryStruct.comp x (X.δ f g n₀ n₁ hn₁) = 0) : CategoryTheory.CategoryStruct.comp (X.liftCycles f g n₀ n₁ hn₁ x hx) (X.iCycles f g n₀) = x
• CategoryTheory.Abelian.SpectralObject.H_map_twoδ₂Toδ₁_toCycles_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.cycles f g n ⟶ Z) : CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) (CategoryTheory.CategoryStruct.comp (X.toCycles f g fg h n) h✝) = CategoryTheory.CategoryStruct.comp 0 h✝
• CategoryTheory.Abelian.SpectralObject.toCycles_descCycles 📋 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 : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ A) (hx : CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) x = 0) : CategoryTheory.CategoryStruct.comp (X.toCycles f g fg h n) (X.descCycles f g fg h x hx) = x
• CategoryTheory.Abelian.SpectralObject.liftCycles_i_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 : A ⟶ (X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ g)) (hx : CategoryTheory.CategoryStruct.comp x (X.δ f g n₀ n₁ hn₁) = 0) {Z : C} (h : (X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ g) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.liftCycles f g n₀ n₁ hn₁ x hx) (CategoryTheory.CategoryStruct.comp (X.iCycles f g n₀) h) = CategoryTheory.CategoryStruct.comp x h
• CategoryTheory.Abelian.SpectralObject.toCycles_descCycles_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 : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ fg) ⟶ A) (hx : CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h)) x = 0) {Z : C} (h✝ : A ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.toCycles f g fg h n) (CategoryTheory.CategoryStruct.comp (X.descCycles f g fg h x hx) h✝) = CategoryTheory.CategoryStruct.comp x h✝
• CategoryTheory.Abelian.SpectralObject.cyclesMap_i 📋 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 g ⟶ CategoryTheory.ComposableArrows.mk₁ g') (n : ℤ) (hβ : β = CategoryTheory.ComposableArrows.homMk₁ (α.app 1) (α.app 2) ⋯ := by cat_disch) : CategoryTheory.CategoryStruct.comp (X.cyclesMap f g f' g' α n) (X.iCycles f' g' n) = CategoryTheory.CategoryStruct.comp (X.iCycles f g n) ((X.H n).map β)
• CategoryTheory.Abelian.SpectralObject.cyclesMap_i_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 g ⟶ CategoryTheory.ComposableArrows.mk₁ g') (n : ℤ) (hβ : β = CategoryTheory.ComposableArrows.homMk₁ (α.app 1) (α.app 2) ⋯ := by cat_disch) {Z : C} (h : (X.H n).obj (CategoryTheory.ComposableArrows.mk₁ g') ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesMap f g f' g' α n) (CategoryTheory.CategoryStruct.comp (X.iCycles f' g' n) h) = CategoryTheory.CategoryStruct.comp (X.iCycles f g n) (CategoryTheory.CategoryStruct.comp ((X.H n).map β) h)
• CategoryTheory.Abelian.SpectralObject.toCycles_cyclesMap 📋 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.toCycles f g fg h n) (X.cyclesMap f g f' g' α n) = CategoryTheory.CategoryStruct.comp ((X.H n).map β) (X.toCycles f' g' fg' h' n)
• CategoryTheory.Abelian.SpectralObject.toCycles_cyclesMap_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.cycles f' g' n ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.toCycles f g fg h n) (CategoryTheory.CategoryStruct.comp (X.cyclesMap f g f' g' α n) h✝) = CategoryTheory.CategoryStruct.comp ((X.H n).map β) (CategoryTheory.CategoryStruct.comp (X.toCycles f' g' fg' h' n) h✝)
• CategoryTheory.Abelian.SpectralObject.cokernelIsoCycles_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 (CategoryTheory.Limits.cokernel.π ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h))) (CategoryTheory.CategoryStruct.comp (X.cokernelIsoCycles f g fg h n).hom (X.iCycles f g n)) = (X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)
• CategoryTheory.Abelian.SpectralObject.cokernelIsoCycles_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 g) ⟶ Z) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.cokernel.π ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f g fg h))) (CategoryTheory.CategoryStruct.comp (X.cokernelIsoCycles f g fg h n).hom (CategoryTheory.CategoryStruct.comp (X.iCycles f g n) h✝)) = CategoryTheory.CategoryStruct.comp ((X.H n).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f g fg h)) h✝
• CategoryTheory.Abelian.SpectralObject.πE 📋 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.cycles f₁ f₂ n₁ ⟶ X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesE_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.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).X₂ = X.cycles f₁ f₂ n₁
• CategoryTheory.Abelian.SpectralObject.instEpiπE 📋 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.Epi (X✝.πE f₁✝ f₂✝ f₃✝ n₀✝ n₁✝ n₂✝ hn₁✝ hn₂✝)
• CategoryTheory.Abelian.SpectralObject.leftHomologyDataShortComplex_K 📋 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.leftHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).K = X.cycles f₁ f₂ n₁
• CategoryTheory.Abelian.SpectralObject.EToCycles 📋 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) : X.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ⟶ X.cycles f₁ f₂₃ n₁
• CategoryTheory.Abelian.SpectralObject.kernelSequenceCyclesE_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) (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) : (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).X₂ = X.cycles f₁ f₂₃ n₁
• CategoryTheory.Abelian.SpectralObject.instMonoEToCycles 📋 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) (hn₂ : n₁ + 1 = n₂) : CategoryTheory.Mono (X.EToCycles f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂)
• CategoryTheory.Abelian.SpectralObject.cyclesIsoH 📋 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.cycles (CategoryTheory.CategoryStruct.id i₀) f n₀ ≅ (X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ f)
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesE_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.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).g = X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.cyclesIso 📋 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).cycles ≅ X.cycles f₁ f₂ n₁
• CategoryTheory.Abelian.SpectralObject.kernelSequenceCyclesE_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) (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) : (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).f = X.EToCycles f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesE_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.cokernelSequenceCyclesE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).f = X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁
• CategoryTheory.Abelian.SpectralObject.leftHomologyDataShortComplex_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) {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.leftHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).i = X.iCycles f₁ f₂ n₁
• CategoryTheory.Abelian.SpectralObject.πE_EToCycles 📋 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.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.EToCycles f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂) = X.cyclesMap f₁ f₂ f₁ f₂₃ (CategoryTheory.ComposableArrows.threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) n₁
• CategoryTheory.Abelian.SpectralObject.leftHomologyDataShortComplex_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.leftHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).H = CategoryTheory.Limits.cokernel (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)
• CategoryTheory.Abelian.SpectralObject.cyclesIsoH_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) {i₀ i₁ : ι} (f : i₀ ⟶ i₁) (n₀ n₁ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) : (X.cyclesIsoH f n₀ n₁ hn₁).inv = X.toCycles (CategoryTheory.CategoryStruct.id i₀) f f ⋯ n₀
• CategoryTheory.Abelian.SpectralObject.πE_EToCycles_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.cycles f₁ f₂₃ n₁ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.EToCycles f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁ f₂ f₁ f₂₃ (CategoryTheory.ComposableArrows.threeδ₃Toδ₂ f₁ f₂ f₃ f₂₃ h₂₃) 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.cyclesIsoH_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.cyclesIsoH f n₀ n₁ hn₁).hom (X.toCycles (CategoryTheory.CategoryStruct.id i₀) f f ⋯ n₀) = CategoryTheory.CategoryStruct.id (X.cycles (CategoryTheory.CategoryStruct.id i₀) f n₀)
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceE_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) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.cokernelSequenceE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ hn₁ hn₂).g = CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) (X.πE f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯)
• CategoryTheory.Abelian.SpectralObject.leftHomologyDataShortComplex_π 📋 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.leftHomologyDataShortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).π = CategoryTheory.Limits.cokernel.π (X.δToCycles f₁ f₂ f₃ n₀ n₁ ⋯)
• CategoryTheory.Abelian.SpectralObject.cyclesIso_inv_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) {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.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).iCycles = X.iCycles f₁ f₂ 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.cyclesIsoH_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.cycles (CategoryTheory.CategoryStruct.id i₀) f n₀ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesIsoH f n₀ n₁ hn₁).hom (CategoryTheory.CategoryStruct.comp (X.toCycles (CategoryTheory.CategoryStruct.id i₀) f f ⋯ n₀) h) = h
• CategoryTheory.Abelian.SpectralObject.cyclesIsoH_hom_EIsoH_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.cyclesIsoH f n₁ n₂ hn₂).hom (X.EIsoH f n₀ n₁ n₂ hn₁ hn₂).inv = X.πE (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.kernelSequenceCyclesE_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) (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) : (X.kernelSequenceCyclesE f₁ f₂ f₃ f₂₃ h₂₃ n₀ n₁ n₂ hn₁ hn₂).g = CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂₃ n₁) ((X.H n₁).map (CategoryTheory.ComposableArrows.twoδ₁Toδ₀ f₂ f₃ f₂₃ h₂₃))
• 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.cyclesIso_inv_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) {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).X₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv (CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).iCycles h) = CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂ n₁) h
• CategoryTheory.Abelian.SpectralObject.cyclesIsoH_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 n₀).obj (CategoryTheory.ComposableArrows.mk₁ f) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.toCycles (CategoryTheory.CategoryStruct.id i₀) f f ⋯ n₀) (CategoryTheory.CategoryStruct.comp (X.cyclesIsoH f n₀ n₁ hn₁).hom h) = h
• CategoryTheory.Abelian.SpectralObject.cyclesIso_hom_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) {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.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom (X.iCycles f₁ f₂ n₁) = (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).iCycles
• CategoryTheory.Abelian.SpectralObject.δToCycles_cyclesIso_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.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv = (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).toCycles
• CategoryTheory.Abelian.SpectralObject.cyclesIsoH_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.toCycles (CategoryTheory.CategoryStruct.id i₀) f f ⋯ n₀) (X.cyclesIsoH f n₀ n₁ hn₁).hom = CategoryTheory.CategoryStruct.id ((X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ f))
• CategoryTheory.Abelian.SpectralObject.cyclesIsoH_hom_EIsoH_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.E (CategoryTheory.CategoryStruct.id i) f (CategoryTheory.CategoryStruct.id j) n₀ n₁ n₂ hn₁ hn₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesIsoH f n₁ n₂ hn₂).hom (CategoryTheory.CategoryStruct.comp (X.EIsoH f n₀ n₁ n₂ hn₁ hn₂).inv h) = CategoryTheory.CategoryStruct.comp (X.πE (CategoryTheory.CategoryStruct.id i) f (CategoryTheory.CategoryStruct.id j) n₀ n₁ n₂ hn₁ hn₂) h
• CategoryTheory.Abelian.SpectralObject.δToCycles_πE 📋 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.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) = 0
• CategoryTheory.Abelian.SpectralObject.δToCycles_πE_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.E f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp 0 h
• CategoryTheory.Abelian.SpectralObject.cyclesIso_hom_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) {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 n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₂) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom (CategoryTheory.CategoryStruct.comp (X.iCycles f₁ f₂ n₁) h) = CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).iCycles h
• CategoryTheory.Abelian.SpectralObject.δToCycles_cyclesIso_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).cycles ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.δToCycles f₁ f₂ f₃ n₀ n₁ hn₁) (CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv h) = CategoryTheory.CategoryStruct.comp (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).toCycles h
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesEIso_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.cokernelSequenceCyclesEIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom.τ₂ = (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).inv
• CategoryTheory.Abelian.SpectralObject.cokernelSequenceCyclesEIso_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.cokernelSequenceCyclesEIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv.τ₂ = (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ ⋯ ⋯).hom
• CategoryTheory.Abelian.SpectralObject.πE_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) {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') (β : CategoryTheory.ComposableArrows.mk₂ f₁ f₂ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁' f₂') (n₀ n₁ n₂ : ℤ) (hβ : β = CategoryTheory.ComposableArrows.homMk₂ (α.app 0) (α.app 1) (α.app 2) ⋯ ⋯ := by cat_disch) (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.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂) = CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁ f₂ f₁' f₂' β n₁) (X.πE f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂)
• CategoryTheory.Abelian.SpectralObject.πE_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) {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') (β : CategoryTheory.ComposableArrows.mk₂ f₁ f₂ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁' f₂') (n₀ n₁ n₂ : ℤ) (hβ : β = CategoryTheory.ComposableArrows.homMk₂ (α.app 0) (α.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 hn₁ hn₂ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂) h) = CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁ f₂ f₁' f₂' β n₁) (CategoryTheory.CategoryStruct.comp (X.πE f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂) h)
• CategoryTheory.Abelian.SpectralObject.cyclesIso_inv_cyclesMap 📋 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') (β : CategoryTheory.ComposableArrows.mk₂ f₁ f₂ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁' f₂') (hβ : β = CategoryTheory.ComposableArrows.homMk₂ (α.app 0) (α.app 1) (α.app 2) ⋯ ⋯) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv (CategoryTheory.ShortComplex.cyclesMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)) = CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁ f₂ f₁' f₂' β n₁) (X.cyclesIso f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂).inv
• CategoryTheory.Abelian.SpectralObject.toCycles_πE_descE 📋 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) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) {A : C} (x : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₁₂) ⟶ A) (h : CategoryTheory.CategoryStruct.comp ((X.H n₁).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂)) x = 0) (hn₁ : n₀ + 1 = n₁) (h' : CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃ n₀ n₁ hn₁) x = 0) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) (CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (X.descE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ x h hn₁ h' hn₂)) = x
• CategoryTheory.Abelian.SpectralObject.toCycles_πE_descE_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) (f₁₂ : i ⟶ k) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (n₀ n₁ n₂ : ℤ) {A : C} (x : (X.H n₁).obj (CategoryTheory.ComposableArrows.mk₁ f₁₂) ⟶ A) (h : CategoryTheory.CategoryStruct.comp ((X.H n₁).map (CategoryTheory.ComposableArrows.twoδ₂Toδ₁ f₁ f₂ f₁₂ h₁₂)) x = 0) (hn₁ : n₀ + 1 = n₁) (h' : CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃ n₀ n₁ hn₁) x = 0) (hn₂ : n₁ + 1 = n₂ := by lia) {Z : C} (h✝ : A ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₁) (CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.descE f₁ f₂ f₃ f₁₂ h₁₂ n₀ n₁ n₂ x h hn₁ h' hn₂) h✝)) = CategoryTheory.CategoryStruct.comp x h✝
• CategoryTheory.Abelian.SpectralObject.cyclesIso_inv_cyclesMap_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') (β : CategoryTheory.ComposableArrows.mk₂ f₁ f₂ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁' f₂') (hβ : β = CategoryTheory.ComposableArrows.homMk₂ (α.app 0) (α.app 1) (α.app 2) ⋯ ⋯) (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 ⋯ ⋯).cycles ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv (CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.cyclesMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)) h) = CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁ f₂ f₁' f₂' β n₁) (CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂).inv 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.πE_EIsoH_hom 📋 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 : ι} (f₁ : i₀ ⟶ i₁) (n₀ n₁ n₂ : ℤ) (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (X.πE (CategoryTheory.CategoryStruct.id i₀) f₁ (CategoryTheory.CategoryStruct.id i₁) n₀ n₁ n₂ hn₁ hn₂) (X.EIsoH f₁ n₀ n₁ n₂ hn₁ hn₂).hom = (X.cyclesIsoH f₁ n₁ n₂ hn₂).hom
• 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.πE_EIsoH_hom_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 : ι} (f₁ : i₀ ⟶ i₁) (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 f₁) ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.πE (CategoryTheory.CategoryStruct.id i₀) f₁ (CategoryTheory.CategoryStruct.id i₁) n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.CategoryStruct.comp (X.EIsoH f₁ n₀ n₁ n₂ hn₁ hn₂).hom h) = CategoryTheory.CategoryStruct.comp (X.cyclesIsoH f₁ n₁ n₂ hn₂).hom h
• CategoryTheory.Abelian.SpectralObject.toCycles_πE_d 📋 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) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₃₄ : i₂ ⟶ i₄) (h₃₄ : CategoryTheory.CategoryStruct.comp f₃ f₄ = f₃₄) (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.toCycles f₃ f₄ f₃₄ h₃₄ n₁) (CategoryTheory.CategoryStruct.comp (X.πE f₃ f₄ f₅ n₀ n₁ n₂ hn₁ hn₂) (X.d f₁ f₂ f₃ f₄ f₅ n₀ n₁ n₂ n₃ hn₁ hn₂ hn₃)) = CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃₄ n₁ n₂ hn₂) (CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₂) (X.πE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃))
• CategoryTheory.Abelian.SpectralObject.toCycles_πE_d_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) (f₁₂ : i₀ ⟶ i₂) (h₁₂ : CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂) (f₃₄ : i₂ ⟶ i₄) (h₃₄ : CategoryTheory.CategoryStruct.comp f₃ f₄ = f₃₄) (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.E f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃ ⟶ Z) : CategoryTheory.CategoryStruct.comp (X.toCycles f₃ f₄ f₃₄ h₃₄ n₁) (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) h)) = CategoryTheory.CategoryStruct.comp (X.δ f₁₂ f₃₄ n₁ n₂ hn₂) (CategoryTheory.CategoryStruct.comp (X.toCycles f₁ f₂ f₁₂ h₁₂ n₂) (CategoryTheory.CategoryStruct.comp (X.πE f₁ f₂ f₃ n₁ n₂ n₃ hn₂ hn₃) h))

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