Theorems · Definition · category theory

CategoryTheory.Adjunction.toEquivalence

{C : Type u₁} →
  [inst : CategoryTheory.Category.{v₁, u₁} C] →
    {D : Type u₂} →
      [inst_1 : CategoryTheory.Category.{v₂, u₂} D] →
        {F : CategoryTheory.Functor C D} →
          {G : CategoryTheory.Functor D C} →
            (adj : F ⊣ G) →
              [∀ (X : C), CategoryTheory.IsIso (adj.unit.app X)] →
                [∀ (Y : D), CategoryTheory.IsIso (adj.counit.app Y)] → C ≌ D

If the unit and counit of a given adjunction are (pointwise) isomorphisms, then we can upgrade the adjunction to an equivalence.

Defined in

Mathlib.CategoryTheory.Adjunction.Basic

Cited by

11 results in Mathlib

Foundations

Depth 25 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.IsIsoCategoryTheory.IsIso