Theorems · Theorem · category theory

CategoryTheory.ShortComplex.homologyMap_mapNatTrans

∀ {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] (S : CategoryTheory.ShortComplex C)
  {F G : CategoryTheory.Functor C D} [inst_4 : F.PreservesZeroMorphisms] [inst_5 : G.PreservesZeroMorphisms]
  [inst_6 : F.PreservesLeftHomologyOf S] [inst_7 : G.PreservesLeftHomologyOf S] [inst_8 : F.PreservesRightHomologyOf S]
  [inst_9 : G.PreservesRightHomologyOf S] [inst_10 : S.HasHomology] (τ : F ⟶ G),
  CategoryTheory.ShortComplex.homologyMap (S.mapNatTrans τ) =
    CategoryTheory.CategoryStruct.comp (S.mapHomologyIso F).hom
      (CategoryTheory.CategoryStruct.comp (τ.app S.homology) (S.mapHomologyIso G).inv)

Defined in

Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Cited by

1 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.Functor.PreservesZeroMorphismsCategoryTheory.Functor.PreservesLeftHomologyOfCategoryTheory.Functor.PreservesLeftHomologyOfCategoryTheory.Functor.PreservesRightHomologyOfCategoryTheory.Functor.PreservesRightHomologyOfCategoryTheory.ShortComplex.HasHomology