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

• 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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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.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)

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