Theorems · Definition · category theory

CochainComplex.mappingCone.triangle

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [CategoryTheory.Limits.HasBinaryBiproducts C] →
        {K L : CochainComplex C ℤ} → (K ⟶ L) → CategoryTheory.Pretriangulated.Triangle (CochainComplex C ℤ)

The standard triangle K ⟶ L ⟶ mappingCone φ ⟶ K⟦(1 : ℤ)⟧ in CochainComplex C ℤ attached to a morphism φ : K ⟶ L. It involves φ, inr φ : L ⟶ mappingCone φ and the morphism induced by the 1-cocycle -mappingCone.fst φ.

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated

Cited by

48 results in Mathlib

Foundations

Depth 69 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.Limits.HasBinaryBiproducts