Theorems · Theorem · category theory

CategoryTheory.Functor.commShiftOfLocalization.iso_inv_app_assoc

∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {E : Type u₃} [inst_2 : CategoryTheory.Category.{v₃, u₃} E] (L : CategoryTheory.Functor C D)
  (W : CategoryTheory.MorphismProperty C) [inst_3 : L.IsLocalization W] {A : Type w} [inst_4 : AddMonoid A]
  [inst_5 : CategoryTheory.HasShift C A] (F : CategoryTheory.Functor C E) (F' : CategoryTheory.Functor D E)
  [inst_6 : CategoryTheory.Localization.Lifting L W F F'] [inst_7 : CategoryTheory.HasShift D A]
  [inst_8 : CategoryTheory.HasShift E A] [inst_9 : L.CommShift A] [inst_10 : F.CommShift A] (a : A) (X : C) {Z : E}
  (h : F'.obj ((CategoryTheory.shiftFunctor D a).obj (L.obj X)) ⟶ Z),
  CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftOfLocalization.iso L W F F' a).inv.app (L.obj X))
      h =
    CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.shiftFunctor E a).map ((CategoryTheory.Localization.Lifting.iso L W F F').hom.app X))
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso F a).inv.app X)
        (CategoryTheory.CategoryStruct.comp
          ((CategoryTheory.Localization.Lifting.iso L W F F').inv.app ((CategoryTheory.shiftFunctor C a).obj X))
          (CategoryTheory.CategoryStruct.comp (F'.map ((CategoryTheory.Functor.commShiftIso L a).hom.app X)) h)))

Defined in

Mathlib.CategoryTheory.Shift.Localization

Cited by

0 results in Mathlib

Foundations

Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Functor.IsLocalizationAddMonoidCategoryTheory.HasShiftCategoryTheory.Localization.LiftingCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShiftCategoryTheory.Functor.CommShift