Theorems · Theorem · category theory
CategoryTheory.ShortComplex.quasiIso_map_iff_of_preservesLeftHomology
∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
[inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Limits.HasZeroMorphisms C]
[inst_3 : CategoryTheory.Limits.HasZeroMorphisms D] (F : CategoryTheory.Functor C D)
[inst_4 : F.PreservesZeroMorphisms] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂) [inst_5 : S₁.HasHomology]
[inst_6 : S₂.HasHomology] [inst_7 : (F.mapShortComplex.obj S₁).HasHomology]
[inst_8 : (F.mapShortComplex.obj S₂).HasHomology] [F.PreservesLeftHomologyOf S₁] [F.PreservesLeftHomologyOf S₂]
[F.ReflectsIsomorphisms],
CategoryTheory.ShortComplex.QuasiIso (F.mapShortComplex.map φ) ↔ CategoryTheory.ShortComplex.QuasiIso φ- Cited by
- 1 results in Mathlib
- Foundations
- Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.ShortComplex.HasHomologyCategoryTheory.ShortComplex.HasHomologyCategoryTheory.ShortComplex.HasHomologyCategoryTheory.ShortComplex.HasHomologyCategoryTheory.Functor.PreservesLeftHomologyOfCategoryTheory.Functor.PreservesLeftHomologyOfCategoryTheory.Functor.ReflectsIsomorphisms
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
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.Functor.objstatement and proof · cited by 19,642
- CategoryTheory.Functorstatement and proof · cited by 16,252
- CategoryTheory.Functor.mapstatement and proof · cited by 8,698
- CategoryTheory.Limits.HasZeroMorphismsstatement and proof · cited by 3,275
- CategoryTheory.ShortComplexstatement and proof · cited by 1,850
- CategoryTheory.IsIsoproof · cited by 1,156
- CategoryTheory.Functor.PreservesZeroMorphismsstatement and proof · cited by 458
- CategoryTheory.ShortComplex.HasHomologystatement and proof · cited by 253
- CategoryTheory.ShortComplex.LeftHomologyData.Hproof · cited by 236
- CategoryTheory.ShortComplex.leftHomologyDataproof · cited by 83
Cited by1
Results whose statement or proof uses this declaration.
- HomologicalComplex.quasiIsoAt_map_iff_of_preservesHomologyproof · cited by 1