Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.kernelSequenceE

{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 →
            {i j k l : ι} →
              (i ⟶ j) →
                (f₂ : j ⟶ k) →
                  (f₃ : k ⟶ l) →
                    (f₂₃ : j ⟶ l) →
                      CategoryTheory.CategoryStruct.comp f₂ f₃ = f₂₃ →
                        (n₀ n₁ n₂ : ℤ) →
                          autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.kernelSequenceE._auto_1 →
                            autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.kernelSequenceE._auto_3 →
                              CategoryTheory.ShortComplex C

The (exact) sequence 0 ⟶ E^n(f₁, f₂, f₃) ⟶ H^n(f₂ ≫ f₃) ⟶ H^n(f₃) ⊞ H^{n+1}(f₁).

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

7 results in Mathlib

Foundations

Depth 105 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian