Theorems · Theorem · category theory

CategoryTheory.ProjectiveResolution.iso_hom_naturality_assoc

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C]
  [inst_2 : CategoryTheory.HasProjectiveResolutions C] {X Y : C} (f : X ⟶ Y) (P : CategoryTheory.ProjectiveResolution X)
  (Q : CategoryTheory.ProjectiveResolution Y) (φ : P.complex ⟶ Q.complex),
  CategoryTheory.CategoryStruct.comp (φ.f 0) (Q.π.f 0) = CategoryTheory.CategoryStruct.comp (P.π.f 0) f →
    ∀ {Z : HomotopyCategory C (ComplexShape.down ℕ)}
      (h : (HomotopyCategory.quotient C (ComplexShape.down ℕ)).obj Q.complex ⟶ Z),
      CategoryTheory.CategoryStruct.comp ((CategoryTheory.projectiveResolutions C).map f)
          (CategoryTheory.CategoryStruct.comp Q.iso.hom h) =
        CategoryTheory.CategoryStruct.comp P.iso.hom
          (CategoryTheory.CategoryStruct.comp ((HomotopyCategory.quotient C (ComplexShape.down ℕ)).map φ) h)

Defined in

Mathlib.CategoryTheory.Abelian.Projective.Resolution

Cited by

0 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.HasProjectiveResolutions