Theorems · Definition · category theory

CategoryTheory.Adjunction.compPreadditiveYonedaIso

{C : Type u₁} →
  {D : Type u₂} →
    [inst : CategoryTheory.Category.{v₁, u₁} C] →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        [inst_2 : CategoryTheory.Preadditive C] →
          [inst_3 : CategoryTheory.Preadditive D] →
            {F : CategoryTheory.Functor C D} →
              {G : CategoryTheory.Functor D C} →
                (F ⊣ G) →
                  [F.Additive] →
                    G.comp
                        (CategoryTheory.preadditiveYoneda.comp
                          ((CategoryTheory.Functor.whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat).obj
                            AddCommGrpCat.uliftFunctor)) ≅
                      CategoryTheory.preadditiveYoneda.comp
                        (((CategoryTheory.Functor.whiskeringLeft Cᵒᵖ Dᵒᵖ AddCommGrpCat).obj F.op).comp
                          ((CategoryTheory.Functor.whiskeringRight Cᵒᵖ AddCommGrpCat AddCommGrpCat).obj
                            AddCommGrpCat.uliftFunctor))

If we have an adjunction adj : F ⊣ G of functors between preadditive categories, and if F is additive, then the hom set equivalence upgrades to an isomorphism between G ⋙ preadditiveYoneda and preadditiveYoneda ⋙ F, once we throw in the necessary universe lifting functors. Note that F is additive if and only if G is, by Adjunction.right_adjoint_additive and Adjunction.left_adjoint_additive.

Defined in

Mathlib.CategoryTheory.Adjunction.Additive

Cited by

2 results in Mathlib

Foundations

Depth 36 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.PreadditiveCategoryTheory.PreadditiveCategoryTheory.Functor.Additive