Theorems · Theorem · category theory

CategoryTheory.Abelian.Ext.mapExactFunctor_hom

∀ {C : Type u} [inst : CategoryTheory.Category.{v, u} C] [inst_1 : CategoryTheory.Abelian C] {D : Type u'}
  [inst_2 : CategoryTheory.Category.{v', u'} D] [inst_3 : CategoryTheory.Abelian D] (F : CategoryTheory.Functor C D)
  [inst_4 : F.Additive] [inst_5 : CategoryTheory.Limits.PreservesFiniteLimits F]
  [inst_6 : CategoryTheory.Limits.PreservesFiniteColimits F] [inst_7 : HasDerivedCategory C]
  [inst_8 : HasDerivedCategory D] [inst_9 : CategoryTheory.HasExt C] [inst_10 : CategoryTheory.HasExt D] {X Y : C}
  {n : ℕ} (e : CategoryTheory.Abelian.Ext X Y n),
  (CategoryTheory.Abelian.Ext.mapExactFunctor F e).hom =
    CategoryTheory.CategoryStruct.comp ((F.mapDerivedCategorySingleFunctor 0).inv.app X)
      (CategoryTheory.CategoryStruct.comp (e.hom.map F.mapDerivedCategory)
        ((CategoryTheory.shiftFunctor (DerivedCategory D) ↑n).map ((F.mapDerivedCategorySingleFunctor 0).hom.app Y)))

Defined in

Mathlib.Algebra.Homology.DerivedCategory.Ext.Map

Cited by

4 results in Mathlib

Foundations

Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.CategoryCategoryTheory.AbelianCategoryTheory.Functor.AdditiveCategoryTheory.Limits.PreservesFiniteLimitsCategoryTheory.Limits.PreservesFiniteColimitsHasDerivedCategoryHasDerivedCategoryCategoryTheory.HasExtCategoryTheory.HasExt