Theorems · Definition · category theory

CategoryTheory.Abelian.SpectralObject.cyclesIsoH

{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) →
            {i₀ i₁ : ι} →
              (f : i₀ ⟶ i₁) →
                (n₀ n₁ : ℤ) →
                  autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.cyclesIsoH._auto_1 →
                    (X.cycles (CategoryTheory.CategoryStruct.id i₀) f n₀ ≅
                      (X.H n₀).obj (CategoryTheory.ComposableArrows.mk₁ f))

The isomorphism Z^n(𝟙 _, f) ≅ H^n(f).

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

11 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian