Theorems · Theorem · category theory

CategoryTheory.ShortComplex.mapHomologyIso_inv_naturality_assoc

∀ {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] {S₁ S₂ : CategoryTheory.ShortComplex C} (φ : S₁ ⟶ S₂)
  (F : CategoryTheory.Functor C D) [inst_4 : F.PreservesZeroMorphisms] [inst_5 : S₁.HasHomology]
  [inst_6 : S₂.HasHomology] [inst_7 : (S₁.map F).HasHomology] [inst_8 : (S₂.map F).HasHomology]
  [inst_9 : F.PreservesLeftHomologyOf S₁] [inst_10 : F.PreservesLeftHomologyOf S₂] {Z : D}
  (h : (S₂.map F).homology ⟶ Z),
  CategoryTheory.CategoryStruct.comp (F.map (CategoryTheory.ShortComplex.homologyMap φ))
      (CategoryTheory.CategoryStruct.comp (S₂.mapHomologyIso F).inv h) =
    CategoryTheory.CategoryStruct.comp (S₁.mapHomologyIso F).inv
      (CategoryTheory.CategoryStruct.comp (CategoryTheory.ShortComplex.homologyMap (F.mapShortComplex.map φ)) h)

Defined in

Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Cited by

0 results in Mathlib

Foundations

Depth 48 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.PreservesLeftHomologyOf