Theorems · Theorem · category theory

CochainComplex.MappingConeCompHomotopyEquiv.hom_inv_id

∀ {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),
  CategoryTheory.CategoryStruct.comp (CochainComplex.MappingConeCompHomotopyEquiv.hom f g)
      (CochainComplex.MappingConeCompHomotopyEquiv.inv f g) =
    CategoryTheory.CategoryStruct.id (CochainComplex.mappingCone g)

Defined in: Mathlib.Algebra.Homology.HomotopyCategory.Triangulated
Cited by: 2 results in Mathlib
Foundations: Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Preadditive CategoryTheory.Limits.HasBinaryBiproducts

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Cites46

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.CategoryStruct.compstatement and proof · cited by 17,999
CategoryTheory.CategoryStruct.idstatement and proof · cited by 6,235
CategoryTheory.Preadditivestatement and proof · cited by 3,309
add_zeroproof · cited by 2,707
zero_addproof · cited by 2,366
CategoryTheory.Category.comp_idproof · cited by 2,119
HomologicalComplex.Xstatement and proof · cited by 1,839
ComplexShape.upstatement · cited by 1,123
CochainComplexstatement and proof · cited by 1,016
HomologicalComplex.Hom.fproof · cited by 845

Cited by2

Results whose statement or proof uses this declaration.

CochainComplex.mappingConeCompHomotopyEquiv_hom_inv_idproof · cited by 2
CochainComplex.MappingConeCompHomotopyEquiv.hom_inv_id_assocproof · cited by 0