Theorems · Theorem · category theory

CategoryTheory.Functor.essImage.unit_isIso

∀ {C : Type u₁} {D : Type u₂} [inst : CategoryTheory.Category.{v₁, u₁} C] [inst_1 : CategoryTheory.Category.{v₂, u₂} D]
  {i : CategoryTheory.Functor D C} [inst_2 : CategoryTheory.Reflective i] {A : C},
  i.essImage A → CategoryTheory.IsIso ((CategoryTheory.reflectorAdjunction i).unit.app A)

If A is essentially in the image of a reflective functor i, then η_A is an isomorphism. This gives that the "witness" for A being in the essential image can instead be given as the reflection of A, with the isomorphism as η_A. (For any B in the reflective subcategory, we automatically have that ε_B is an iso.)

Defined in

Mathlib.CategoryTheory.Adjunction.Reflective

Cited by

2 results in Mathlib

Foundations

Depth 30 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CategoryTheory.CategoryCategoryTheory.CategoryCategoryTheory.Reflective