Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData

{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 r' : ℤ) →
                        (hrr' : r + 1 = r') →
                          (hr : r₀ ≤ r) →
                            (pq pq' pq'' : κ) →
                              (c r).prev pq' = pq →
                                (c r).next pq' = pq'' →
                                  (i₀' i₀ i₁ i₂ i₃ i₃' : ι) →
                                    i₀' = data.i₀ r' pq' ⋯ →
                                      i₀ = data.i₀ r pq' ⋯ →
                                        i₁ = data.i₁ pq' →
                                          i₂ = data.i₂ pq' →
                                            i₃ = data.i₃ r pq' ⋯ →
                                              i₃' = data.i₃ r' pq' ⋯ →
                                                (n₀ n₁ n₂ : ℤ) →
                                                  n₁ = data.deg pq' →
                                                    autoParam (n₀ + 1 = n₁)
                                                        CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData._auto_1 →
                                                      autoParam (n₁ + 1 = n₂)
                                                          CategoryTheory.Abelian.SpectralObject.spectralSequenceHomologyData._auto_3 →
                                                        (((X.spectralSequence data).page r hr).sc' pq pq'
                                                            pq'').HomologyData

The homology data for the short complexes given by the differentials of a spectral sequence attached to a spectral object in an abelian category.

Defined in

Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Cited by

12 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianPreorderCategoryTheory.Abelian.SpectralObject.HasSpectralSequence