Theorems · Definition · category theory

CategoryTheory.Adjunction.homAddEquiv

{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] → (X : C) → (Y : D) → (F.obj X ⟶ Y) ≃+ (X ⟶ G.obj Y)

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 AddEquiv. 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

12 results in Mathlib

Foundations

Depth 28 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

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