Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isLimitKf

{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) →
                    (r r' : ℤ) →
                      (hrr' : r + 1 = r') →
                        (hr : r₀ ≤ r) →
                          (pq' pq'' : κ) →
                            (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 :
                                                    autoParam (n₀ + 1 = n₁)
                                                      CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isLimitKf._auto_1) →
                                                  (hn₂ :
                                                      autoParam (n₁ + 1 = n₂)
                                                        CategoryTheory.Abelian.SpectralObject.SpectralSequence.HomologyData.isLimitKf._auto_3) →
                                                    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)

The kernel fork kf is a limit.

Defined in

Mathlib.Algebra.Homology.SpectralObject.SpectralSequence

Cited by

4 results in Mathlib

Foundations

Depth 124 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianPreorderCategoryTheory.Abelian.SpectralObject.HasSpectralSequence