Theorems · Theorem · category theory

CategoryTheory.Adjunction.LeftAdjointCommShift.iso_inv_app

∀ {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] {F : CategoryTheory.Functor C D} {G : CategoryTheory.Functor D C}
  (adj : F ⊣ G) {A : Type u_3} [inst_2 : AddGroup A] [inst_3 : CategoryTheory.HasShift C A]
  [inst_4 : CategoryTheory.HasShift D A] (a b : A) (h : a + b = 0) [inst_5 : G.CommShift A] (Y : C),
  (CategoryTheory.Adjunction.LeftAdjointCommShift.iso adj a).inv.app Y =
    CategoryTheory.CategoryStruct.comp
      ((CategoryTheory.shiftFunctor D a).map (F.map ((CategoryTheory.shiftFunctorCompIsoId C a b h).inv.app Y)))
      (CategoryTheory.CategoryStruct.comp
        ((CategoryTheory.shiftFunctor D a).map
          (F.map ((CategoryTheory.shiftFunctor C b).map (adj.unit.app ((CategoryTheory.shiftFunctor C a).obj Y)))))
        (CategoryTheory.CategoryStruct.comp
          ((CategoryTheory.shiftFunctor D a).map
            (F.map
              ((CategoryTheory.Functor.commShiftIso G b).inv.app (F.obj ((CategoryTheory.shiftFunctor C a).obj Y)))))
          (CategoryTheory.CategoryStruct.comp
            ((CategoryTheory.shiftFunctor D a).map
              (adj.counit.app
                ((CategoryTheory.shiftFunctor D b).obj (F.obj ((CategoryTheory.shiftFunctor C a).obj Y)))))
            ((CategoryTheory.shiftFunctorCompIsoId D b a ⋯).hom.app
              (F.obj ((CategoryTheory.shiftFunctor C a).obj Y))))))

Defined in

Mathlib.CategoryTheory.Shift.Adjunction

Cited by

1 results in Mathlib

Foundations

Depth 39 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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