Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcyclesE

{C : Type u_1} →
  {ι : Type u_2} →
    [inst : CategoryTheory.Category.{v_1, u_1} C] →
      [inst_1 : CategoryTheory.Category.{v_2, u_2} ι] →
        [inst_2 : CategoryTheory.Abelian C] →
          CategoryTheory.Abelian.SpectralObject C ι →
            {i₀ i₁ i₂ i₃ : ι} →
              (f₁ : i₀ ⟶ i₁) →
                (f₂ : i₁ ⟶ i₂) →
                  (i₂ ⟶ i₃) →
                    (f₁₂ : i₀ ⟶ i₂) →
                      CategoryTheory.CategoryStruct.comp f₁ f₂ = f₁₂ →
                        (n₀ n₁ n₂ : ℤ) →
                          autoParam (n₀ + 1 = n₁)
                              CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcyclesE._auto_1 →
                            autoParam (n₁ + 1 = n₂)
                                CategoryTheory.Abelian.SpectralObject.cokernelSequenceOpcyclesE._auto_3 →
                              CategoryTheory.ShortComplex C

The (exact) sequence H^n(f₁) ⟶ opZ^n(f₁ ≫ f₂, f₃) ⟶ E^n(f₁, f₂, f₃) ⟶ 0.

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

7 results in Mathlib

Foundations

Depth 108 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian