Theorems · Theorem · category theory

CategoryTheory.NatTrans.app_homology

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

Defined in

Mathlib.Algebra.Homology.ShortComplex.PreservesHomology

Cited by

0 results in Mathlib

Foundations

Depth 44 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.Limits.HasZeroMorphismsCategoryTheory.ShortComplex.HasHomologyCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Functor.PreservesZeroMorphismsCategoryTheory.Functor.PreservesLeftHomologyOfCategoryTheory.Functor.PreservesLeftHomologyOfCategoryTheory.Functor.PreservesRightHomologyOfCategoryTheory.Functor.PreservesRightHomologyOf