Theorems · Theorem · category theory

CategoryTheory.Functor.map_opShiftFunctorEquivalence_unitIso_inv_app_unop_assoc

∀ {C : Type u_1} {D : Type u_2} [inst : CategoryTheory.Category.{v_1, u_1} C]
  [inst_1 : CategoryTheory.Category.{v_2, u_2} D] [inst_2 : CategoryTheory.HasShift C ℤ]
  [inst_3 : CategoryTheory.HasShift D ℤ] (F : CategoryTheory.Functor C D) [inst_4 : F.CommShift ℤ] (X : Cᵒᵖ) (n : ℤ)
  {Z : D}
  (h :
    F.obj
        (Opposite.unop
          ((CategoryTheory.Pretriangulated.opShiftFunctorEquivalence C n).inverse.obj
            ((CategoryTheory.Pretriangulated.opShiftFunctorEquivalence C n).functor.obj X))) ⟶
      Z),
  CategoryTheory.CategoryStruct.comp
      (F.map ((CategoryTheory.Pretriangulated.opShiftFunctorEquivalence C n).unitIso.inv.app X).unop) h =
    CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.Pretriangulated.opShiftFunctorEquivalence D n).unitIso.inv.app
          (Opposite.op (F.obj (Opposite.unop X)))).unop
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.shiftFunctor D n).map ((CategoryTheory.Functor.commShiftIso F.op n).hom.app X).unop)
        (CategoryTheory.CategoryStruct.comp
          ((CategoryTheory.Functor.commShiftIso F n).inv.app
            (Opposite.unop ((CategoryTheory.shiftFunctor Cᵒᵖ n).obj X)))
          h))

Defined in

Mathlib.CategoryTheory.Triangulated.Opposite.Functor

Cited by

0 results in Mathlib

Foundations

Depth 57 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.HasShiftCategoryTheory.HasShiftCategoryTheory.Functor.CommShift