Theorems · Theorem · category theory

CategoryTheory.HasShift.Induced.add_inv_app_obj

∀ {C : Type u_2} {D : Type u_1} [inst : CategoryTheory.Category.{v_1, u_2} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_1} D] (F : CategoryTheory.Functor C D) {A : Type u_3} [inst_2 : AddMonoid A]
  [inst_3 : CategoryTheory.HasShift C A] (s : A → CategoryTheory.Functor D D)
  (i : (a : A) → F.comp (s a) ≅ (CategoryTheory.shiftFunctor C a).comp F)
  [inst_4 : ((CategoryTheory.Functor.whiskeringLeft C D D).obj F).Full]
  [inst_5 : ((CategoryTheory.Functor.whiskeringLeft C D D).obj F).Faithful] (a b : A) (X : C),
  (CategoryTheory.HasShift.Induced.add F s i a b).inv.app (F.obj X) =
    CategoryTheory.CategoryStruct.comp ((s b).map ((i a).hom.app X))
      (CategoryTheory.CategoryStruct.comp ((i b).hom.app ((CategoryTheory.shiftFunctor C a).obj X))
        (CategoryTheory.CategoryStruct.comp (F.map ((CategoryTheory.shiftFunctorAdd C a b).inv.app X))
          ((i (a + b)).inv.app X)))

Defined in

Mathlib.CategoryTheory.Shift.Induced

Cited by

1 results in Mathlib

Foundations

Depth 34 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryAddMonoidCategoryTheory.HasShiftCategoryTheory.Functor.FullCategoryTheory.Functor.Faithful