Theorems · Theorem · category theory

CategoryTheory.Abelian.SpectralObject.cyclesIso_inv_cyclesMap_assoc

∀ {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 i₂ i₃ : ι} (f₁ : i₀ ⟶ i₁) (f₂ : i₁ ⟶ i₂) (f₃ : i₂ ⟶ i₃)
  {i₀' i₁' i₂' i₃' : ι} (f₁' : i₀' ⟶ i₁') (f₂' : i₁' ⟶ i₂') (f₃' : i₂' ⟶ i₃')
  (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃')
  (β : CategoryTheory.ComposableArrows.mk₂ f₁ f₂ ⟶ CategoryTheory.ComposableArrows.mk₂ f₁' f₂'),
  β = CategoryTheory.ComposableArrows.homMk₂ (α.app 0) (α.app 1) (α.app 2) ⋯ ⋯ →
    ∀ (n₀ n₁ n₂ : ℤ)
      (hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.cyclesIso_inv_cyclesMap._auto_1)
      (hn₂ : autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.cyclesIso_inv_cyclesMap._auto_3) {Z : C}
      (h : (X.shortComplex f₁' f₂' f₃' n₀ n₁ n₂ ⋯ ⋯).cycles ⟶ Z),
      CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).inv
          (CategoryTheory.CategoryStruct.comp
            (CategoryTheory.ShortComplex.cyclesMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)) h) =
        CategoryTheory.CategoryStruct.comp (X.cyclesMap f₁ f₂ f₁' f₂' β n₁)
          (CategoryTheory.CategoryStruct.comp (X.cyclesIso f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂).inv h)

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

1 results in Mathlib

Foundations

Depth 103 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian