Theorems · Definition · category theory

CochainComplex.mappingCone.liftHomotopy

{C : Type u_1} →
  [inst : CategoryTheory.Category.{v, u_1} C] →
    [inst_1 : CategoryTheory.Preadditive C] →
      {F G : CochainComplex C ℤ} →
        (φ : F ⟶ G) →
          [inst_2 : HomologicalComplex.HasHomotopyCofiber φ] →
            {K : CochainComplex C ℤ} →
              (f₁ f₂ : K ⟶ CochainComplex.mappingCone φ) →
                (α : CochainComplex.HomComplex.Cochain K F 0) →
                  (β : CochainComplex.HomComplex.Cochain K G (-1)) →
                    (CochainComplex.HomComplex.Cochain.ofHom f₁).comp ↑(CochainComplex.mappingCone.fst φ) ⋯ =
                        -CochainComplex.HomComplex.δ 0 1 α +
                          (CochainComplex.HomComplex.Cochain.ofHom f₂).comp ↑(CochainComplex.mappingCone.fst φ) ⋯ →
                      (CochainComplex.HomComplex.Cochain.ofHom f₁).comp (CochainComplex.mappingCone.snd φ) ⋯ =
                          CochainComplex.HomComplex.δ (-1) 0 β + α.comp (CochainComplex.HomComplex.Cochain.ofHom φ) ⋯ +
                            (CochainComplex.HomComplex.Cochain.ofHom f₂).comp (CochainComplex.mappingCone.snd φ) ⋯ →
                        Homotopy f₁ f₂

Constructor for homotopies between morphisms to a mapping cone.

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Cited by

0 results in Mathlib

Foundations

Depth 62 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveHomologicalComplex.HasHomotopyCofiber