Theorems · Theorem · category theory
CategoryTheory.ShortComplex.quasiIso_iff_isIso_liftCycles
∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Limits.HasZeroMorphisms C]
{S₁ S₂ : CategoryTheory.ShortComplex C} [inst_2 : S₁.HasHomology] [inst_3 : S₂.HasHomology] (φ : S₁ ⟶ S₂),
S₁.f = 0 →
∀ (hg₁ : S₁.g = 0),
S₂.f = 0 → (CategoryTheory.ShortComplex.QuasiIso φ ↔ CategoryTheory.IsIso (S₂.liftCycles φ.τ₂ ⋯))- Cited by
- 2 results in Mathlib
- Foundations
- Depth 42 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites34
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.compproof · cited by 17,999
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.Category.comp_idproof · cited by 2,119
- CategoryTheory.Category.id_compproof · cited by 1,998
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.IsIsostatement · cited by 1,156
- CategoryTheory.ShortComplex.X₂statement and proof · cited by 1,115
- CategoryTheory.ShortComplex.X₁statement · cited by 889
- CategoryTheory.ShortComplex.X₃statement and proof · cited by 876
- CategoryTheory.ShortComplex.gstatement and proof · cited by 658
Cited by2
Results whose statement or proof uses this declaration.
- CategoryTheory.ShortComplex.quasiIso_iff_of_zerosproof · cited by 2
- CochainComplex.isIso_liftCycles_iffproof · cited by 0