Theorems · Theorem · category theory

CategoryTheory.Abelian.SpectralObject.shortComplexMap_comp_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) {i j k l : ι} (f₁ : i ⟶ j) (f₂ : j ⟶ k) (f₃ : k ⟶ l) {i' j' k' l' : ι}
  (f₁' : i' ⟶ j') (f₂' : j' ⟶ k') (f₃' : k' ⟶ l') {i'' j'' k'' l'' : ι} (f₁'' : i'' ⟶ j'') (f₂'' : j'' ⟶ k'')
  (f₃'' : k'' ⟶ l'')
  (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃')
  (β : CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃' ⟶ CategoryTheory.ComposableArrows.mk₃ f₁'' f₂'' f₃'')
  (n₀ n₁ n₂ : ℤ) (hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.shortComplexMap_comp._auto_1)
  (hn₂ : autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.shortComplexMap_comp._auto_3)
  {Z : CategoryTheory.ShortComplex C} (h : X.shortComplex f₁'' f₂'' f₃'' n₀ n₁ n₂ ⋯ ⋯ ⟶ Z),
  CategoryTheory.CategoryStruct.comp
      (X.shortComplexMap f₁ f₂ f₃ f₁'' f₂'' f₃'' (CategoryTheory.CategoryStruct.comp α β) n₀ n₁ n₂ hn₁ hn₂) h =
    CategoryTheory.CategoryStruct.comp (X.shortComplexMap f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)
      (CategoryTheory.CategoryStruct.comp (X.shortComplexMap f₁' f₂' f₃' f₁'' f₂'' f₃'' β n₀ n₁ n₂ hn₁ hn₂) h)

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

0 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian