Theorems · Definition · category theory

CochainComplex.mappingConeCompTriangleh

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [CategoryTheory.Limits.HasBinaryBiproducts C] →
        {X₁ X₂ X₃ : CochainComplex C ℤ} →
          (X₁ ⟶ X₂) → (X₂ ⟶ X₃) → CategoryTheory.Pretriangulated.Triangle (HomotopyCategory C (ComplexShape.up ℤ))

Given two composable morphisms f : X₁ ⟶ X₂ and g : X₂ ⟶ X₃ in the category of cochain complexes, this is the canonical triangle mappingCone f ⟶ mappingCone (f ≫ g) ⟶ mappingCone g ⟶ (mappingCone f)⟦1⟧ in the homotopy category. It is a distinguished triangle, see HomotopyCategory.mappingConeCompTriangleh_distinguished.

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.Triangulated

Cited by

3 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasBinaryBiproducts