Theorems · Theorem · category theory

CategoryTheory.Functor.ShiftSequence.induced_shiftIso_hom_app_obj_assoc

∀ {C : Type u_1} {D : Type u_2} {A : Type u_3} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.Category.{v_3, u_3} A]
  {L : CategoryTheory.Functor C D} {F : CategoryTheory.Functor D A} {G : CategoryTheory.Functor C A} (e : L.comp F ≅ G)
  (M : Type u_4) [inst_3 : AddMonoid M] [inst_4 : CategoryTheory.HasShift C M] [inst_5 : G.ShiftSequence M]
  (F' : M → CategoryTheory.Functor D A) (e' : (m : M) → L.comp (F' m) ≅ G.shift m)
  [inst_6 : ((CategoryTheory.Functor.whiskeringLeft C D A).obj L).Full]
  [inst_7 : ((CategoryTheory.Functor.whiskeringLeft C D A).obj L).Faithful] [inst_8 : CategoryTheory.HasShift D M]
  [inst_9 : L.CommShift M] (n a a' : M) (ha' : n + a = a') (X : C) {Z : A} (h : (F.shift a').obj (L.obj X) ⟶ Z),
  CategoryTheory.CategoryStruct.comp ((F.shiftIso n a a' ha').hom.app (L.obj X)) h =
    CategoryTheory.CategoryStruct.comp
      (CategoryTheory.CategoryStruct.comp ((F.shift a).map ((CategoryTheory.Functor.commShiftIso L n).inv.app X))
        (CategoryTheory.CategoryStruct.comp ((e' a).hom.app ((CategoryTheory.shiftFunctor C n).obj X))
          (CategoryTheory.CategoryStruct.comp ((G.shiftIso n a a' ha').hom.app X) ((e' a').inv.app X))))
      h

Defined in

Mathlib.CategoryTheory.Shift.InducedShiftSequence

Cited by

0 results in Mathlib

Foundations

Depth 40 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.CategoryAddMonoidCategoryTheory.HasShiftCategoryTheory.Functor.ShiftSequenceCategoryTheory.Functor.FullCategoryTheory.Functor.FaithfulCategoryTheory.HasShiftCategoryTheory.Functor.CommShift