Theorems · Theorem · category theory

CategoryTheory.Abelian.SpectralObject.isIso_map

∀ {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')
  (α : CategoryTheory.ComposableArrows.mk₃ f₁ f₂ f₃ ⟶ CategoryTheory.ComposableArrows.mk₃ f₁' f₂' f₃') (n₀ n₁ n₂ : ℤ),
  CategoryTheory.IsIso
      ((X.H n₀).map
        ((CategoryTheory.ComposableArrows.functorArrows ι 2 3 3 CategoryTheory.Abelian.SpectralObject.isIso_map._proof_6
              CategoryTheory.Abelian.SpectralObject.shortComplexMap._proof_3).map
          α)) →
    CategoryTheory.IsIso
        ((X.H n₁).map
          ((CategoryTheory.ComposableArrows.functorArrows ι 1 2 3
                CategoryTheory.Abelian.SpectralObject.isIso_map._proof_8
                CategoryTheory.Abelian.SpectralObject.isIso_map._proof_6).map
            α)) →
      CategoryTheory.IsIso
          ((X.H n₂).map
            ((CategoryTheory.ComposableArrows.functorArrows ι 0 1 3
                  CategoryTheory.Abelian.SpectralObject.isIso_map._proof_10
                  CategoryTheory.Abelian.SpectralObject.isIso_map._proof_12).map
              α)) →
        ∀ (hn₁ : autoParam (n₀ + 1 = n₁) CategoryTheory.Abelian.SpectralObject.isIso_map._auto_1)
          (hn₂ : autoParam (n₁ + 1 = n₂) CategoryTheory.Abelian.SpectralObject.isIso_map._auto_3),
          CategoryTheory.IsIso (X.map f₁ f₂ f₃ f₁' f₂' f₃' α n₀ n₁ n₂ hn₁ hn₂)

Defined in

Mathlib.Algebra.Homology.SpectralObject.Page

Cited by

1 results in Mathlib

Foundations

Depth 97 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Abelian