Theorems · Theorem · category theory

CategoryTheory.Adjunction.LeftAdjointCommShift.iso_hom_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] (X : C),
  (CategoryTheory.Adjunction.LeftAdjointCommShift.iso adj a).hom.app X =
    CategoryTheory.CategoryStruct.comp (F.map ((CategoryTheory.shiftFunctor C a).map (adj.unit.app X)))
      (CategoryTheory.CategoryStruct.comp
        (F.map
          ((CategoryTheory.shiftFunctor C a).map
            (G.map ((CategoryTheory.shiftFunctorCompIsoId D a b h).inv.app (F.obj X)))))
        (CategoryTheory.CategoryStruct.comp
          (F.map
            ((CategoryTheory.shiftFunctor C a).map
              ((CategoryTheory.Functor.commShiftIso G b).hom.app ((CategoryTheory.shiftFunctor D a).obj (F.obj X)))))
          (CategoryTheory.CategoryStruct.comp
            (F.map
              ((CategoryTheory.shiftFunctorCompIsoId C b a ⋯).hom.app
                (G.obj ((CategoryTheory.shiftFunctor D a).obj (F.obj X)))))
            (adj.counit.app ((CategoryTheory.shiftFunctor D a).obj (F.obj X))))))

Defined in

Mathlib.CategoryTheory.Shift.Adjunction

Cited by

2 results in Mathlib

Foundations

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

Assumes

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