Theorems · Theorem · category theory

CochainComplex.mappingCone.ext_to_iff

∀ {C : Type u_1} [inst : CategoryTheory.Category.{v, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  {F G : CochainComplex C ℤ} (φ : F ⟶ G) [inst_2 : HomologicalComplex.HasHomotopyCofiber φ] (i j : ℤ) (hij : i + 1 = j)
  {A : C} (f g : A ⟶ (CochainComplex.mappingCone φ).X i),
  f = g ↔
    CategoryTheory.CategoryStruct.comp f ((↑(CochainComplex.mappingCone.fst φ)).v i j hij) =
        CategoryTheory.CategoryStruct.comp g ((↑(CochainComplex.mappingCone.fst φ)).v i j hij) ∧
      CategoryTheory.CategoryStruct.comp f ((CochainComplex.mappingCone.snd φ).v i i ⋯) =
        CategoryTheory.CategoryStruct.comp g ((CochainComplex.mappingCone.snd φ).v i i ⋯)

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Cited by

4 results in Mathlib

Foundations

Depth 59 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveHomologicalComplex.HasHomotopyCofiber