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Found 35 declarations mentioning CategoryTheory.Abelian.SpectralObject.SpectralSequence.page.
- CategoryTheory.Abelian.SpectralObject.SpectralSequence.page 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r : ℤ) (hr : r₀ ≤ r) : HomologicalComplex C (c r) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.page_X 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r : ℤ) (hr : r₀ ≤ r) (pq : κ) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).X pq = CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageX X data r pq ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.page_d 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r : ℤ) (hr : r₀ ≤ r) (pq pq' : κ) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).d pq pq' = CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageD X data r pq pq' ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyIso 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq' : κ) [X.HasSpectralSequence data] : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).homology pq' ≅ (CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r' ⋯).X pq' - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).sc' pq pq' pq'').HomologyData - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.cc 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' : κ) (i₀ i₁ i₂ i₃ i₃' : ι) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.Limits.CokernelCofork ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).d pq pq') - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kf 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq' pq'' : κ) (i₀' i₀ i₁ i₂ i₃ : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.Limits.KernelFork ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).d pq' pq'') - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.ccSc_f 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' : κ) (i₀ i₁ i₂ i₃ i₃' : ι) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.ccSc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).f = CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageD X data r pq pq' ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kfSc_g 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq' pq'' : κ) (i₀' i₀ i₁ i₂ i₃ : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kfSc X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).g = CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageD X data r pq' pq'' ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyIso' 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).sc' pq pq' pq'').homology ≅ (CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r' ⋯).X pq' - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_left_H 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.H = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_right_H 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.H = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_right_Q 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.Q = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_left_H 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.H = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_right_H 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.H = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_left_K 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.K = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_right_Q 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.Q = X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isLimitKf 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq' pq'' : κ) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.Limits.IsLimit (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kf X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isColimitCc 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' : κ) (hpq : (c r).prev pq' = pq) (i₀ i₁ i₂ i₃ i₃' : ι) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.Limits.IsColimit (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.cc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.shortComplexIso 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r : ℤ) (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).Rel pq pq') (hpq' : (c r).Rel pq' pq'') (n₀ n₁ n₂ n₃ n₄ : ℤ) (hn₁ : n₀ + 1 = n₁) (hn₂ : n₁ + 1 = n₂) (hn₃ : n₂ + 1 = n₃) (hn₄ : n₃ + 1 = n₄) (hn₂' : n₂ = data.deg pq') : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).sc' pq pq' pq'' ≅ X.dShortComplex (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ n₃ n₄ hn₁ hn₂ hn₃ hn₄ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_left_π 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.π = X.mapFourδ₄Toδ₃' i₀' i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_right_ι 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.ι = X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_left_π 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.π = X.mapFourδ₄Toδ₃' i₀' i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_right_ι 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.ι = X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯ - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_left_K 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.K = (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kf X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯).pt - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.ccSc_g 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' : κ) (i₀ i₁ i₂ i₃ i₃' : ι) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.ccSc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).g = CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯).hom (X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_left_i 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).left.i = CategoryTheory.Limits.Fork.ι (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kf X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_right_p 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).right.p = CategoryTheory.Limits.Cofork.π (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.cc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kf_w 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq' pq'' : κ) (i₀' i₀ i₁ i₂ i₃ : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (CategoryTheory.CategoryStruct.comp (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃ ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ hn₁ hn₂) (CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' hn₁ hn₂).inv) ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).d pq' pq'') = 0 - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.cc_w 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' : κ) (i₀ i₁ i₂ i₃ i₃' : ι) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).d pq pq') (CategoryTheory.CategoryStruct.comp (CategoryTheory.Abelian.SpectralObject.SpectralSequence.pageXIso X data r hr pq' i₀ i₁ i₂ i₃ hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯).hom (X.mapFourδ₄Toδ₃' i₀ i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯)) = 0 - CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.fac 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : CategoryTheory.CategoryStruct.comp (CategoryTheory.Limits.Fork.ι (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.kf X data r r' hrr' hr pq' pq'' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯)) (CategoryTheory.Limits.Cofork.π (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.cc X data r r' hrr' hr pq pq' i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯)) = CategoryTheory.CategoryStruct.comp (X.mapFourδ₄Toδ₃' i₀' i₁ i₂ i₃ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯) (X.mapFourδ₁Toδ₀' i₀' i₀ i₁ i₂ i₃' ⋯ ⋯ ⋯ ⋯ n₀ n₁ n₂ ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_iso_hom 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).iso.hom = CategoryTheory.CategoryStruct.id (X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData_iso_inv 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) [X.HasSpectralSequence data] (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (X.spectralSequenceHomologyData data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).iso.inv = CategoryTheory.CategoryStruct.id (X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯) - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_iso_hom 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).iso.hom = CategoryTheory.CategoryStruct.id (CategoryTheory.ShortComplex.HomologyData.ofEpiMonoFactorisation.leftHomologyData ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).sc' pq pq' pq'') (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isLimitKf X data r r' hrr' hr pq' pq'' hpq' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isColimitCc X data r r' hrr' hr pq pq' hpq i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' ⋯ ⋯) ⋯).H - CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData_iso_inv 📋 Mathlib.Algebra.Homology.SpectralObject.SpectralSequence
{C : Type u_1} {ι : Type u_2} {κ : Type u_3} [CategoryTheory.Category.{v_1, u_1} C] [CategoryTheory.Abelian C] [Preorder ι] (X : CategoryTheory.Abelian.SpectralObject C ι) {c : ℤ → ComplexShape κ} {r₀ : ℤ} (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) (r r' : ℤ) (hrr' : r + 1 = r') (hr : r₀ ≤ r) (pq pq' pq'' : κ) (hpq : (c r).prev pq' = pq) (hpq' : (c r).next pq' = pq'') (i₀' i₀ i₁ i₂ i₃ i₃' : ι) (hi₀' : i₀' = data.i₀ r' pq' ⋯) (hi₀ : i₀ = data.i₀ r pq' ⋯) (hi₁ : i₁ = data.i₁ pq') (hi₂ : i₂ = data.i₂ pq') (hi₃ : i₃ = data.i₃ r pq' ⋯) (hi₃' : i₃' = data.i₃ r' pq' ⋯) (n₀ n₁ n₂ : ℤ) (hn₁' : n₁ = data.deg pq') [X.HasSpectralSequence data] (hn₁ : n₀ + 1 = n₁ := by lia) (hn₂ : n₁ + 1 = n₂ := by lia) : (CategoryTheory.Abelian.SpectralObject.SpectralSequence.homologyData X data r r' hrr' hr pq pq' pq'' hpq hpq' i₀' i₀ i₁ i₂ i₃ i₃' hi₀' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ hn₁' hn₁ hn₂).iso.inv = CategoryTheory.CategoryStruct.id (CategoryTheory.ShortComplex.HomologyData.ofEpiMonoFactorisation.leftHomologyData ((CategoryTheory.Abelian.SpectralObject.SpectralSequence.page X data r hr).sc' pq pq' pq'') (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isLimitKf X data r r' hrr' hr pq' pq'' hpq' i₀' i₀ i₁ i₂ i₃ hi₀' hi₀ hi₁ hi₂ hi₃ n₀ n₁ n₂ hn₁' ⋯ ⋯) (CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isColimitCc X data r r' hrr' hr pq pq' hpq i₀ i₁ i₂ i₃ i₃' hi₀ hi₁ hi₂ hi₃ hi₃' n₀ n₁ n₂ 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:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"somewhere in their lemma name.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 ?aIf 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 ?bBy 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 findtsum_lt_tsumeven though the hypothesisf i < g iis not the last.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