Theorems · Theorem · category theory

CategoryTheory.Functor.mapHomologicalComplexUpToQuasiIsoFactorsh_hom_app_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] (F : CategoryTheory.Functor C D) {ι : Type u_3} {c : ComplexShape ι}
  [inst_2 : CategoryTheory.Preadditive C] [inst_3 : CategoryTheory.Preadditive D]
  [inst_4 : CategoryTheory.CategoryWithHomology C] [inst_5 : CategoryTheory.CategoryWithHomology D]
  [inst_6 : (HomologicalComplex.quasiIso D c).HasLocalization] [inst_7 : F.Additive] [inst_8 : F.PreservesHomology]
  [inst_9 : (HomologicalComplex.quasiIso C c).HasLocalization] [inst_10 : c.QFactorsThroughHomotopy C]
  [inst_11 : c.QFactorsThroughHomotopy D]
  [inst_12 : (HomotopyCategory.quotient C c).IsLocalization (HomologicalComplex.homotopyEquivalences C c)]
  (K : HomologicalComplex C c) {Z : HomologicalComplexUpToQuasiIso D c}
  (h :
    HomologicalComplexUpToQuasiIso.Qh.obj ((F.mapHomotopyCategory c).obj ((HomotopyCategory.quotient C c).obj K)) ⟶ Z),
  CategoryTheory.CategoryStruct.comp
      ((F.mapHomologicalComplexUpToQuasiIsoFactorsh c).hom.app ((HomotopyCategory.quotient C c).obj K)) h =
    CategoryTheory.CategoryStruct.comp
      ((F.mapHomologicalComplexUpToQuasiIso c).map ((HomologicalComplexUpToQuasiIso.quotientCompQhIso C c).hom.app K))
      (CategoryTheory.CategoryStruct.comp ((F.mapHomologicalComplexUpToQuasiIsoFactors c).hom.app K)
        (CategoryTheory.CategoryStruct.comp
          ((HomologicalComplexUpToQuasiIso.quotientCompQhIso D c).inv.app ((F.mapHomologicalComplex c).obj K))
          (CategoryTheory.CategoryStruct.comp
            (HomologicalComplexUpToQuasiIso.Qh.map ((F.mapHomotopyCategoryFactors c).inv.app K)) h)))

Defined in

Mathlib.Algebra.Homology.Localization

Cited by

0 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.CategoryWithHomologyCategoryTheory.CategoryWithHomologyCategoryTheory.MorphismProperty.HasLocalizationCategoryTheory.Functor.AdditiveCategoryTheory.Functor.PreservesHomologyCategoryTheory.MorphismProperty.HasLocalizationComplexShape.QFactorsThroughHomotopyComplexShape.QFactorsThroughHomotopyCategoryTheory.Functor.IsLocalization