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Found 27 declarations mentioning CategoryTheory.Abelian.SpectralObject.iCycles.

• 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.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.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.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.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.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.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.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.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.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_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.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.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.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.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.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.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.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.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)

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