Theorems · Definition · category theory

CochainComplex.mappingConeCompHomotopyEquiv

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] →
        {X₁ X₂ X₃ : CochainComplex C ℤ} →
          (f : X₁ ⟶ X₂) →
            (g : X₂ ⟶ X₃) →
              HomotopyEquiv (CochainComplex.mappingCone g)
                (CochainComplex.mappingCone (CochainComplex.mappingConeCompTriangle f g).mor₁)

Given two composable morphisms f and g in the category of cochain complexes, this is the homotopy equivalence mappingConeCompHomotopyEquiv f g between mappingCone g and the mapping cone of the morphism mappingCone f ⟶ mappingCone (f ≫ g).

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Cited by

9 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasBinaryBiproducts