Theorems · Definition · category theory

CochainComplex.mappingCone.shiftIso

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v_1, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      [inst_2 : CategoryTheory.Limits.HasBinaryBiproducts C] →
        {K L : CochainComplex C ℤ} →
          (φ : K ⟶ L) →
            (n : ℤ) →
              (CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).obj (CochainComplex.mappingCone φ) ≅
                CochainComplex.mappingCone ((CategoryTheory.shiftFunctor (CochainComplex C ℤ) n).map φ)

The canonical isomorphism (mappingCone φ)⟦n⟧ ≅ mappingCone (φ⟦n⟧').

Defined in: Mathlib.Algebra.Homology.HomotopyCategory.Pretriangulated
Cited by: 0 results in Mathlib
Foundations: Depth 72 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Limits.HasBinaryBiproducts

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

CategoryTheory.Categorystatement and proof · cited by 32,673
Quiver.Homstatement and proof · cited by 32,603
CategoryTheory.Functor.objstatement · cited by 19,642
CategoryTheory.Functor.mapstatement and proof · cited by 8,698
CategoryTheory.Isostatement · cited by 3,963
CategoryTheory.Preadditivestatement and proof · cited by 3,309
HomologicalComplex.Xstatement · cited by 1,839
CategoryTheory.shiftFunctorstatement and proof · cited by 1,553
ComplexShape.upstatement · cited by 1,123
CochainComplexstatement and proof · cited by 1,016
CochainComplex.mappingConestatement · cited by 181
CategoryTheory.Limits.HasBinaryBiproductsstatement and proof · cited by 165

Cited by1

Results whose statement or proof uses this declaration.

CochainComplex.mappingCone.shiftTriangleIsoproof · cited by 0