Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.spectralSequencePageXIso

{C : Type u_1} →
  {ι : Type u_2} →
    {κ : Type u_3} →
      [inst : CategoryTheory.Category.{v_1, u_1} C] →
        [inst_1 : CategoryTheory.Abelian C] →
          [inst_2 : Preorder ι] →
            (X : CategoryTheory.Abelian.SpectralObject C ι) →
              {c : ℤ → ComplexShape κ} →
                {r₀ : ℤ} →
                  (data : CategoryTheory.Abelian.SpectralObject.SpectralSequenceDataCore ι c r₀) →
                    [inst_3 : X.HasSpectralSequence data] →
                      (r : ℤ) →
                        (hr : r₀ ≤ r) →
                          (pq : κ) →
                            (i₀ i₁ i₂ i₃ : ι) →
                              (h₀ : i₀ = data.i₀ r pq ⋯) →
                                (h₁ : i₁ = data.i₁ pq) →
                                  (h₂ : i₂ = data.i₂ pq) →
                                    (h₃ : i₃ = data.i₃ r pq ⋯) →
                                      (n₀ n₁ n₂ : ℤ) →
                                        n₁ = data.deg pq →
                                          (hn₁ :
                                              autoParam (n₀ + 1 = n₁)
                                                CategoryTheory.Abelian.SpectralObject.spectralSequencePageXIso._auto_1) →
                                            (hn₂ :
                                                autoParam (n₁ + 1 = n₂)
                                                  CategoryTheory.Abelian.SpectralObject.spectralSequencePageXIso._auto_3) →
                                              ((X.spectralSequence data).page r ⋯).X pq ≅
                                                X.E (CategoryTheory.homOfLE ⋯) (CategoryTheory.homOfLE ⋯)
                                                  (CategoryTheory.homOfLE ⋯) n₀ n₁ n₂ ⋯ ⋯

The objects on the pages of a spectral sequence attached to a spectral object X identifies an object X.E.

Defined in

Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Cited by

11 results in Mathlib

Foundations

Depth 130 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianPreorderCategoryTheory.Abelian.SpectralObject.HasSpectralSequence