Theorems · Theorem · category theory

CategoryTheory.Abelian.SpectralObject.opcyclesMap_opcyclesIso_hom_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')
  (γ : CategoryTheory.ComposableArrows.mk₂ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₂ f₂' f₃'),
  autoParam (γ = CategoryTheory.ComposableArrows.homMk₂ (α.app 1) (α.app 2) (α.app 3) ⋯ ⋯)
      CategoryTheory.Abelian.SpectralObject.opcyclesMap_opcyclesIso_hom._auto_1 →
    ∀ (n₀ n₁ n₂ : ℤ)
      (hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.opcyclesMap_opcyclesIso_hom._auto_3)
      (hn₂ : autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.opcyclesMap_opcyclesIso_hom._auto_5) {Z : C}
      (h : X.opcycles f₂' f₃' n₁ ⟶ Z),
      CategoryTheory.CategoryStruct.comp
          (CategoryTheory.ShortComplex.opcyclesMap (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂))
          (CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁' f₂' f₃' n₀ n₁ n₂ hn₁ hn₂).hom h) =
        CategoryTheory.CategoryStruct.comp (X.opcyclesIso f₁ f₂ f₃ n₀ n₁ n₂ hn₁ hn₂).hom
          (CategoryTheory.CategoryStruct.comp (X.opcyclesMap f₂ f₃ f₂' f₃' γ n₁) h)

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian