Theorems · Theorem · category theory

CategoryTheory.ShortComplex.RightHomologyMapData.quasiIso_map_iff

∀ {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} {hr₁ : S₁.RightHomologyData}
  {hr₂ : S₂.RightHomologyData} (ψr : CategoryTheory.ShortComplex.RightHomologyMapData φ hr₁ hr₂)
  [inst_5 : (F.mapShortComplex.obj S₁).HasHomology] [inst_6 : (F.mapShortComplex.obj S₂).HasHomology]
  [hr₁.IsPreservedBy F] [hr₂.IsPreservedBy F],
  CategoryTheory.ShortComplex.QuasiIso (F.mapShortComplex.map φ) ↔ CategoryTheory.IsIso (F.map ψr.φH)

Defined in

Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Cited by

0 results in Mathlib

Foundations

Depth 43 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.ShortComplex.HasHomologyCategoryTheory.ShortComplex.HasHomologyCategoryTheory.ShortComplex.RightHomologyData.IsPreservedByCategoryTheory.ShortComplex.RightHomologyData.IsPreservedBy