Theorems · Theorem · category theory

CategoryTheory.InjectiveResolution.iso_inv_naturality_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
  [inst_2 : CategoryTheory.HasInjectiveResolutions C] {X Y : C} (f : X ⟶ Y) (I : CategoryTheory.InjectiveResolution X)
  (J : CategoryTheory.InjectiveResolution Y) (φ : I.cocomplex ⟶ J.cocomplex),
  CategoryTheory.CategoryStruct.comp (I.ι.f 0) (φ.f 0) = CategoryTheory.CategoryStruct.comp f (J.ι.f 0) →
    ∀ {Z : HomotopyCategory C (ComplexShape.up ℕ)} (h : (CategoryTheory.injectiveResolutions C).obj Y ⟶ Z),
      CategoryTheory.CategoryStruct.comp I.iso.inv
          (CategoryTheory.CategoryStruct.comp ((CategoryTheory.injectiveResolutions C).map f) h) =
        CategoryTheory.CategoryStruct.comp ((HomotopyCategory.quotient C (ComplexShape.up ℕ)).map φ)
          (CategoryTheory.CategoryStruct.comp J.iso.inv h)

Defined in

Mathlib.CategoryTheory.Abelian.Injective.Resolution

Cited by

0 results in Mathlib

Foundations

Depth 112 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasInjectiveResolutions