Theorems · Definition · category theory

CochainComplex.mappingCone.descCocycle

{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} →
              {n m : ℤ} →
                (α : CochainComplex.HomComplex.Cochain F K m) →
                  (β : CochainComplex.HomComplex.Cocycle G K n) →
                    m + 1 = n →
                      CochainComplex.HomComplex.δ m n α =
                          n.negOnePow • (CochainComplex.HomComplex.Cochain.ofHom φ).comp ↑β ⋯ →
                        CochainComplex.HomComplex.Cocycle (CochainComplex.mappingCone φ) K n

Given φ : F ⟶ G, this is the cocycle in Cocycle (mappingCone φ) K n that is constructed from α : Cochain F K m (with m + 1 = n) and β : Cocycle F K n, when a suitable cocycle relation is satisfied.

Defined in

Mathlib.Algebra.Homology.HomotopyCategory.MappingCone

Cited by

8 results in Mathlib

Foundations

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

Assumes

CategoryTheory.CategoryCategoryTheory.PreadditiveHomologicalComplex.HasHomotopyCofiber