Theorems · Definition · category theory

CochainComplex.IsKProjective.homotopyZero

{C : Type u_2} →
  [inst : CategoryTheory.Category.{u_3, u_2} C] →
    [inst_1 : CategoryTheory.Abelian C] →
      {K L : CochainComplex C ℤ} → (f : K ⟶ L) → HomologicalComplex.Acyclic L → [K.IsKProjective] → Homotopy f 0

A choice of homotopy to zero for a morphism from a K-projective cochain complex to an acyclic cochain complex.

Defined in: Mathlib.Algebra.Homology.HomotopyCategory.KProjective
Cited by: 3 results in Mathlib
Foundations: Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CategoryTheory.Category CategoryTheory.Abelian CochainComplex.IsKProjective

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Cites9

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

CategoryTheory.Categorystatement · cited by 32,673
Quiver.Homstatement · cited by 32,603
CategoryTheory.Abelianstatement · cited by 1,753
HomologicalComplexstatement · cited by 1,691
ComplexShape.upstatement · cited by 1,123
CochainComplexstatement · cited by 1,016
Homotopystatement · cited by 106
HomologicalComplex.Acyclicstatement · cited by 28
CochainComplex.IsKProjectivestatement · cited by 15

Cited by3

Results whose statement or proof uses this declaration.

HomotopyEquiv.isKProjectiveproof · cited by 1
CochainComplex.isKProjective_iff_leftOrthogonalproof · cited by 1
CochainComplex.IsKProjective.homotopyZero_defstatement · cited by 0