Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.cyclesIso

{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] →
          (X : CategoryTheory.Abelian.SpectralObject C ι) →
            {i j k l : ι} →
              (f₁ : i ⟶ j) →
                (f₂ : j ⟶ k) →
                  (f₃ : k ⟶ l) →
                    (n₀ n₁ n₂ : ℤ) →
                      (hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.cyclesIso._auto_1) →
                        (hn₂ : autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.cyclesIso._auto_3) →
                          (X.shortComplex f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).cycles ≅ X.cycles f₁ f₂ n₁

The cycles of the short complex shortComplex at E^{n₁}(f₁, f₂, f₃) identifies to Z^{n₁}(f₁, f₂).

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

15 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian