Theorems · Theorem · category theory

CategoryTheory.SingleFunctors.map_lift_shiftIso_hom_app_assoc

∀ {C : Type u_1} {D : Type u_2} {E : Type u_3} [inst : CategoryTheory.Category.{u_7, u_1} C]
  [inst_1 : CategoryTheory.Category.{u_6, u_2} D] [inst_2 : CategoryTheory.Category.{u_5, u_3} E] {A : Type u_4}
  [inst_3 : AddMonoid A] [inst_4 : CategoryTheory.HasShift D A] [inst_5 : CategoryTheory.HasShift E A]
  (F : CategoryTheory.SingleFunctors C E A) (G : CategoryTheory.Functor D E) [inst_6 : G.CommShift A] [inst_7 : G.Full]
  [inst_8 : G.Faithful] (Φ : A → CategoryTheory.Functor C D) (hΦ : (a : A) → (Φ a).comp G ≅ F.functor a) (n a a' : A)
  (h : n + a = a') (X : C) {Z : E} (h_1 : G.obj ((Φ a).obj X) ⟶ Z),
  CategoryTheory.CategoryStruct.comp (G.map (((F.lift G Φ hΦ).shiftIso n a a' h).hom.app X)) h_1 =
    CategoryTheory.CategoryStruct.comp ((CategoryTheory.Functor.commShiftIso G n).hom.app ((Φ a').obj X))
      (CategoryTheory.CategoryStruct.comp ((CategoryTheory.shiftFunctor E n).map ((hΦ a').hom.app X))
        (CategoryTheory.CategoryStruct.comp ((F.shiftIso n a a' h).hom.app X)
          (CategoryTheory.CategoryStruct.comp ((hΦ a).inv.app X) h_1)))

Defined in

Mathlib.CategoryTheory.Shift.SingleFunctorsLift

Cited by

0 results in Mathlib

Foundations

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

Assumes

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